M. A. Usov.

Mining Engineer, scholarship student of the Tomsk Technological Institute
of Emperor Nicholas II.

FEDOROV’S

OR

UNIVERSAL OPTICAL METHOD

FOR THE STUDY

OF ROCK-FORMING MINERALS,

ESPECIALLY

FELDSPARS.

With 10 tables.

Tomsk.

Printing-lithography of the Siberian Association of the Printing Trade, corner of Dvoryanskaya St. and Yamskoy Lane, own building.

1910.

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
List of literature . . . . . . . . . . . . . . . . . . . . . . . . . XI

Part One.

Principal procedures of the universal-optical method of investigation.

Chapter I. The Fedorov, or universal, stage.

§ 1. General description of the Fedorov stage . . . . . . . . . . . . . . 1
§ 2. Spherical segments . . . . . . . . . . . . . . . . . . . . . . . . 3
§ 3. Significance of the spherical segments and correction for refraction . . 4
§ 4. Mounting the specimen . . . . . . . . . . . . . . . . . . . . . . 7
§ 5. Installation and verification of the Fedorov stage . . . . . . . . . . 8

Chapter II. Methods for projecting the results of measurements on the Fedorov stage.

§ 6. Foundations of stereographic projection . . . . . . . . . . . . . . 12
§ 7. Problem 1. To find the pole of a given arc of a great circle . . . . . 15
§ 8. Problem 2. To lay off an angle on a given arc of a great circle from a given point . . . . . . . . . . . . . . . . . . . . . 15
§ 9. Problem 3. To construct an arc through three points, of which two lie at the ends of one diameter . . . . . . . . . . . . . . . . . 16
§ 10. Stereographic ruler . . . . . . . . . . . . . . . . . . . . . . . 18
§ 11. Circular ruler . . . . . . . . . . . . . . . . . . . . . . . . . 18
§ 12. Problem 4. To draw an arc of a great circle through two given points. Problem 5. To divide the angle between two points. Problem 6. To measure the angle between points . . . . . . . . . . . . . . . . 19
§ 13. Three-legged compass . . . . . . . . . . . . . . . . . . . . . . 19
§ 14. Problem 7. To find the projection of an arc of a small circle . . . . 20
§ 15. Use of transparent stereographic nets . . . . . . . . . . . . . . . 21
§ 16. V. V. Nikitin’s hemisphere . . . . . . . . . . . . . . . . . . . 22

Chapter III. Principal procedures for working on the Fedorov stage.

§ 17. Fundamentals of the crystal optics of biaxial minerals . . . . . . . 24
§ 18. Setting the planes of elasticity . . . . . . . . . . . . . . . . . 26
§ 19. Methods for determining the completeness of extinction of the specimen . . . 27

§ 20. Plotting the projections of the planes of elasticity . . . . . . 29
§ 21. Adjustment of the results of measurements in determining the planes of elasticity . . . . . . 32
§ 22. Determination of the position of the planes of the optical axes . . . 34
§ 23. Determination of the axes of elasticity . . . . . . . . . . 35
§ 24. Determination of the position of crystallographic planes . . . 38
§ 25. Determination of the position of a direction in a mineral . . . . 40

Second Part.

Determination of the optical constants of rock-forming minerals on the Fedorov stage.

Chapter I. Determination of optical constants expressing the properties of minerals as physical bodies.

§ 26. Determination of the kind of syngony . . . . . . . . . . 43

Determination of the angle between the optical axes and of the optical sign of a biaxial mineral

§ 27. a) With an arbitrary section of the thin section . . . . . . . 45
§ 28. b) With a section of the thin section close to the plane \(n_g n_p\) . . . 48
§ 29. Determination of the optical sign of a uniaxial mineral . . . 50
§ 30. Refractive index . . . . . . . . . . . . . . 52

Chapter II. Determination of the magnitude of birefringence.

§ 31. Foundations of the methods for determining the magnitude of birefringence . 54

Description of instruments for measuring the path difference of rays:

§ 32. Nikitin’s rotating compensator . . . . . . . . . . 56
§ 33. Quartz wedge compensators . . . . . . . . . . 61
§ 34. Fedorov’s mica compensator . . . . . . . . . . 62
§ 35. M. Lévy’s comparator . . . . . . . . . . . 63

Measurement of the thickness of the thin section.

§ 36. Shaw’s method . . . . . . . . . . . . . . 64
§ 37. Nikitin’s method . . . . . . . . . . . . . 66

Determination of the magnitude of birefringence:

§ 38. a) By the direct method . . . . . . . . . . . 68
§ 39. b) By means of an auxiliary mineral . . . . . . . . 69
§ 40. Determination of the magnitude of birefringence by M. Lévy’s method . 71

Chapter III. Determination of the relation between the optical and crystallographic elements of minerals.

Study of twins.

§ 41. Formation and types of twins . . . . . . . . . . 72
§ 42. Determination of the elements of twinning . . . . . . . 78
§ 43. Significance of twinning elements for the determination of minerals . . . 82

Determination of minerals by means of cleavage planes.

§ 44. Significance of the coordinates of the pole of cleavage . . . . . . . . . 85
§ 45. Extinction angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Part Three.

Determination of feldspars.

Lime-sodium feldspars, or plagioclases.

§ 46. Isomorphism of plagioclases . . . . . . . . . . . . . . . . . . . 89

Chapter I. Fedorov’s method for determining plagioclases.

§ 47. General foundations of the method . . . . . . . . . . . . . . . . 92
§ 48. Laws of twinned formations of plagioclases . . . . . . . . . . . 94
§ 49. Analysis of twins . . . . . . . . . . . . . . . . . . . . . . . . . 99
§ 50. Construction of the twin diagram . . . . . . . . . . . . . . . . 100
§ 51. Analysis of the twin diagram . . . . . . . . . . . . . . . . . . 105
§ 52. On the method of determining plagioclases by means of the coordinates of the pole of the cleavage trace . . . . . . . . . . . . 107

Chapter II. Additional methods for determining plagioclases.

§ 53. Angle between the optical axes . . . . . . . . . . . . . . . . . . 109
§ 54. Magnitude of birefringence . . . . . . . . . . . . . . . . . . . 110
§ 55. Determination of plagioclases according to Fouqué’s principle . . 112
§ 56. Determination of plagioclases by means of the pole of the cleavage plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
§ 57. Determination of new twin laws of plagioclases . . . . . . . . . 115

Chapter III. Potassium-sodium feldspars.

§ 58. Isodimorphism of potassium-sodium feldspars . . . . . . . . . . 118

Basic constants and methods for determining potassium-sodium feldspars.

§ 59. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
§ 60. Morphological properties . . . . . . . . . . . . . . . . . . . . 124
§ 61. Extinction angles . . . . . . . . . . . . . . . . . . . . . . . . 126
§ 62. Twinned formations . . . . . . . . . . . . . . . . . . . . . . . 128
§ 63. Angles between the optical axes . . . . . . . . . . . . . . . . . 130
§ 64. Refraction of light . . . . . . . . . . . . . . . . . . . . . . . 131
Potassium-barium feldspars . . . . . . . . . . . . . . . . . . . . 132
Perthitic intergrowths of feldspars . . . . . . . . . . . . . . . . 133

Appendix. Detailed example of determining a plagioclase . . . . . . 137
Tables.

PREFACE.

Petrographic investigations up to the present time have consisted chiefly in determining the structure and general mineralogical composition of rocks.

The optical methods applied in this work, involving the use of a simple polarizing microscope alone, make it possible to find only approximate values of the constants of minerals; therefore, for a more exact determination of a given rock it is necessary also to subject it to chemical analysis. Of course, this circumstance greatly hinders the precise and rapid investigation of a large amount of petrographic material, not to mention the fact that chemical analysis does not always yield reliable results.

It is another matter if we have a method by means of which the various constants of rock-forming minerals can be found accurately in any grain of the latter. In that case the true nature of each constituent part of the rock will be known to us, as a result of which microscopic investigations may sometimes not only replace chemical analysis, but also provide more definite indications concerning the affiliation of this rock with one type or another.

In addition to the significance indicated for the determination of rocks, such a method considerably broadens the sphere of study of minerals as separate physical systems, and thus brings petrography still closer to mineralogy.

At the present time there exist two principal methods that more or less satisfy the conditions set forth above. These are the Fedorov method and the Becke method. The first is based on the use of parallel light, the second—primarily of convergent light. The first consists chiefly in determining the spatial relations among the various elements of minerals, both optical and crystallographic; the second deals exclusively with the derived optical elements, which are not infrequently obtained by artificial means.

From this it is not difficult to see that the first method is better suited to determining the principal constants of minerals, which characterize the simple properties of the latter, and therefore are of greatest importance for the petrographer. For this reason Fedorov’s method is quite rightly called the universal optical method for the investigation of rock-forming minerals.

However, the universal character of this method is limited by the circumstance that, owing to certain features of the instrument on whose use the method itself rests, and which bears the name of Fedorov’s or the universal stage, only not especially small grains of minerals can be subjected to investigation. But this circumstance is almost the only weak side of Fedorov’s method, and in this respect alone does it yield to Becke’s method.

Despite its advantage, the former method is far from having the same distribution as the latter; it must be said that it has not yet crossed the borders of its homeland. Moreover, many Russian workers engaged in petrography, when going on assignments abroad, sometimes have as their chief aim to become acquainted with Becke’s method, without yet having studied that which is the possession of Russian genius.

The cause of such a phenomenon is, in all probability, the absence of a suitable manual for this method, the study of which presents no small difficulties.

Indeed, from the list of literature given below we see that we have, properly speaking, only two works with a more or less systematic exposition of the method: these are “Universalmethode und Feldspathstudien” (1896–98) by E. S. Fedorov—the creator of the method—and “Determination of Plagioclases by Fedorov’s Newest Method” (1898) by his pupil E. Stratanovich.

But the first work still belongs to the period of development of the method, and therefore the latter is presented in it academically, not sufficiently systematically and not fully for its contemporary development. The second work has, in essence, a special character, representing only a more popular exposition of certain propositions of the preceding work.

Thus it may be said that we still have no manual on Fedorov’s method.

In the work offered below I had in view, above all, to fill, insofar as possible, this gap in the petrographic literature.

In the first two parts of this work the general foundations of the method in its contemporary development are systematically set forth, and the essential procedures of optical investigation on the universal stage of the most perfected type are analyzed in detail.

The third part of the work is devoted exclusively to an analysis of the optical properties and methods of determining the feldspars; it not only serves as an illustration of the majority of the procedures of investigation set forth in the first two parts, but also has independent significance, since feldspars are the most essential constituent parts of rocks and since at the present time Fedorov’s method is occupied chiefly with the determination of these minerals.

The third part contains the most important results of the work.

Thus, Fedorov’s twin diagram, which serves as the basis for determining plagioclases, has been verified and considerably expanded, and the possible cases of its use have been analyzed;

the method of determining feldspars by Fouqué, adapted to the universal method, has been modified, and special diagrams have been constructed for the extinction angles employed in this method;

through a review of the data on potash-soda feldspars, the isodimorphous character of these compounds has been established, making it possible to raise and place on practical ground the question of determining these interesting minerals, which in the natural classification of rocks have, in all probability, no less significance than the plagioclases so beloved by petrographers.

At the end of the work a detailed example is appended of the determination of a plagioclase, with indication of all the successive procedures of investigation on the universal stage employed for this purpose.

In preparing the proposed work, besides pursuing the aim indicated above, I also had the following in view.

At present I am completing, in the Geological Cabinet of the Tomsk Technological Institute, the processing of the Dzungarian collections of Prof. V. A. Obruchev. Applying Fedorov’s method in working up such varied material, I encountered many questions, the solution of which led to certain interesting results concerning both the method itself and the ways of determining feldspars.

Bearing in mind that the publication of the materials of the Dzungarian expeditions will be delayed for quite a long time, and that it is nevertheless necessary to preface the petrographic sketch of Northwestern Dzungaria that I am preparing with a description of the method used in processing the collections, I decided to do this separately, only expanding the scope of this task for the reasons given above.

In conclusion I consider it a pleasant duty to express my sincere gratitude to my mentor, the esteemed Professor V. A. Obruchev, who assisted me in every possible way in all my work, and to my friend, the laboratory assistant P. P. Gudkov, to whom I owe my acquaintance with the method I set forth and who did not leave me without his advice.

M. Usov.

Tomsk, 24 September 1910.

LIST OF LITERATURE.

In accordance with the importance of the questions touched upon in the work, the list of literature is divided into three parts.

For the universal optical method of investigation, this list is, in all probability, exhaustive.

As for the two other groups, they include only those works that were taken into consideration in compiling the present work; moreover, for twin formations the literature is indicated more or less completely, while for feldspars the most important works are cited, especially those concerning methods for determining these minerals.

The order in which the individual articles are arranged is everywhere chronological.

To save space, the titles of the more frequently mentioned journals are given in the following abbreviated form.

Journal title	Abbreviation
Annales de chimie et de physique	Ann. ch. et ph.
Bulletin de la Société chimique de France	Bull. chim. fr.
Bulletin de la Société minéralogique de France	Bull. min. fr.
Comptes rendus hebdomadaires des séances de l’Academie des sciences	C. R.
Transactions of the Imperial St. Petersburg Mineralogical Society	Zap. M. O.
Transactions of the Mining Institute of Empress Catherine II	Zap. G. I.
Neues Jahrbuch für Mineralogie, Geologie und Palaeontologie	N. J.
Sitzungsberichte der Königl. Akademie der Wissenschaften in Wien	Wien. Ak.
Tschermak’s mineralogische und petrographische Mittheilungen	T. M. P. M.
Zeitschrift für Krystallographie und Mineralogie	Z. f. Kr.

The Fedorov or universal-optical method.

1. E. v. Fedorow. Universalmethode in der Mineralogie und Petrographie. Z. f. Kr. 21–22. 1893–94.

2. E. v. Fedorow. Universalmethode und Feldspathstudien.
   I Methodische Verfahren. Z. f. Kr. 26, 1896, 225.
   II Feldspathbestimmungen. Z. f. Kr. 27, 1897, 337.

III. Die Feldspäthe des Bogoslowsk’schen Bergreviers. Z. f. Kr. 29, 1898; 604.

1. E. S. Fedorov. Foundations of Petrography. 1897; 59—86.
2. I. Stöber. Ueber die empfindliche Quartzdoppelplatte. Z. f. Kr. 29, 1898; 22.
3. E. Stratanovich. Determination of plagioclases by Fedorov’s newest method. Zap. M. O. 1898.
4. E. v. Fedorow. Biegungsachse der Feldspäthe. T. M. P. M. 18, 1899; 360.
5. W. Nikitin. Beitrag zur Universalmethode. Zur Bestimmung der Doppelbrechung. Z. f. Kr. 33 B, 1900; 145.
6. E. S. Fedorov. Crystallography. 1901; 366—383.
7. E. S. Fedorov and V. V. Nikitin. Bogoslovsky Mining District. 1901; 107—171.
8. E. v. Fedorow. Ueber die Anwendung des Dreispitzzirkels für krystallographische Zwecke. Z. f. Kr. 37, 1902; 138.
9. E. v. Fedorow. Axendispersionsarten und ihre Bestimmung. Ibidem. p. 343.
10. E. Fedorov. The latest steps in the field of universal-optical investigations. Application to feldspars. Zap. M. O. 40, 1903; 221.
11. E. Fedorov. Determination of refractive indices in thin sections. Ibidem, 305.
12. L. Duparc et F. Pearce. Traité de technique minéralogique et pétrographique. Genève. 1907, 281–300.
13. A. Boldyrev. Crystals of dinitrobromobenzene etc. Zap. G. I. 1907, I; 25.
14. V. V. Nikitin. Some new instruments and techniques of the universal-optical method of investigation. Zap. G. I. 1907, I, 43.
15. V. V. Nikitin. Determination of the magnitude of birefringence. Ibidem, 373.

Twinning formations.

1. Haüy. Traité de minéralogie. 1801, 2; 599.
2. G. Tschermack. Zur Theorie der Zwillingskrystalle. T. M. P. M. 2, 1880; 499.
3. W. Brögger. Mineralien der südnorwegischen Augitsyenite. Z. f. Kr. 16, 1890; 38.
4. A. Cesaro. Zur Theorie der Zwillinge. 1893. Ref. Z. f. Kr. 25, 1896; 355.
5. P. Groth. Physikalische Krystallographie. 1895. 278—599.
6. G. Tschermack. Lehrbuch der Mineralogie. 1897; 94.
7. V. Goldschmidt. Ueber nicht parallele Verknüpfung der Krystallpartikel. Z. f. Kr. 29, 1898; 361.
8. W. Barlow. Geometrische Untersuchung über eine mechanische Ursache der Homogenität etc. Z. f. Kr. 29, 1898; 521.

1. H. Baumhauer. Contributions to the knowledge and theory of twin formation in crystals. Z. f. Kr. 31, 1899; 252.

2. P. Gaubert. Contribution to the study of the formation and growth of crystals. Bull. min. fr. 25, 1902; 223.

3. C. Viola. Contribution to twin formation. Z. f. Kr. 36, 1902; 234.

4. J. Beckenkamp. On twin formation of crystal masses and of molecules. Z. f. Kr. 36, 1902; 466.

5. C. Viola. Contribution to twin formation. Z. f. Kr. 38, 1903; 67.

6. G. Friedel. On twins. C. R. 139, 1904; 465, 618.

7. G. Tschermak. Unified derivation of the laws of crystallization and twinning. Z. f. Kr. 39, 1904; 456.

8. P. Paulov. On the conditions for the formation of crystals of different form in a liquid medium. Z. f. Kr. 41, 1906; 144.

9. V. Vernadsky. Toward a physical theory of crystalline twins. Izv. Ak. Nauk. 1907; 335.

10. A. Johnsen. Investigations on crystal twins and their connection with other phenomena. N. J. B.-B. 23, 1907; 237.

11. A. Johnsen. Tschermak’s theory of twinning and the law of mica formation. Centralblatt für Min. etc. 1907; 400.

12. V. Goldschmidt. On heterotwins and uniaxial intergrowths. Z. f. Kr. 42, 1907; 582.

13. F. Levinson-Lessing. On the question of crystalline twins. Izv. SPB. Politekhn. I-ta. 9, 1908; 511.

Feldspars.

1. A. des Cloiseaux. Manual of mineralogy. I, 1862; 294.

2. Klein. On the feldspar from M.-te Gibele on Pantellaria. N. J. 1879; 518.

3. G. Tschermak. The feldspar studies. Wien. Ak. 50, 1864; 566.

4. A. Streng. Feldspar studies. N. J. 1871; 598.

5. G. Tschermak. The form and the transformation of the labradorite from Verespatak. Min. Mitt. 1874; 269.

6. A. Des Cloiseaux. Memoir on the characteristic birefringent qualities of the four principal triclinic feldspars and on a procedure for distinguishing them immediately from one another. Ann. ch. et ph. 4, 1875; 5.

7. A. Des Cloiseaux. Microscopic examination of orthose and of various triclinic feldspars. C. R. 82, 1876; 1017.

8. A. Des Cloiseaux. Memoir on the existence, the optical and crystallographic properties, and the chemical composition of microcline, a new species of triclinic feldspar with a potash base, followed by remarks on the microscopic examination of orthose and of various triclinic feldspars. Ann. ch. et. ph. 9, 1876; 433.

XIV

1. J. Szabo. On a new method for determining feldspars also in rocks. Budapest. 1876.

2. G. vom Rath. Twinning intergrowth of the triclinic feldspars according to the so-called pericline law, and on a distinction of them founded thereon. N. J. 1876; 689.

3. E. Mallard. On the isomorphism of triclinic feldspars. Bull. min. fr. 4, 1881; 96.

4. M. Schuster. Remarks on E. Mallard’s treatise “Sur l’isomorphisme etc.” Supplement on the optical orientation of plagioclase. T. M. P. M. 2, 1882; 18).

5. M. Schuster. On the optical orientation of plagioclase. T. M. P. M. 3, 1883; 155.

6. A. Des Cloiseaux. New investigations on the separation of the optical axes, the orientation of their plane and their bisectors, and their various kinds of dispersion in albite and oligoclase. Bull. min. fr. 6, 1883; 89.

7. A. Des Cloiseaux. Oligoclases and andesines. Bull. min. fr. 7, 1884; 249.

8. H. Förstner. On the feldspars of Pantellaria. Z. f. Kr. 8, 1884; 125.

9. A. Beutell. Contributions to the knowledge of the Silesian potash-soda feldspars. Z. f. Kr. 8, 1884; 351.

10. J. Iddings. Obsidian cliff of Yellowstone National Park. 7-th Annual Report U. S. G. Survey. 1885; 267.

11. A. Offret. On the variation, under the influence of heat, of the refractive indices of several mineral species within the visible spectrum. Bull. min. fr. 13, 1889; 405.

12. J. Hyland. On the rocks of Kilimanjaro and its surroundings. T. M. P. M. 10, 1889; 249.

13. E. v Fedorow. On a remarkable property of anorthite. T. M. P. M. 12, 1891; 443.

14. Pockels. On the calculation of the optical properties of isomorphous mixtures from those of the mixed pure substances. N. J. B.—B. 8, 1893; 117.

15. F. Becke. On the determinability of rock constituents, especially plagioclase, on the basis of their refractive power. Wien. Akad. 102, 1893; 358.

16. F. Becke. Determination of lime-rich plagioclases by interference figures of twins. T. M. P. M. 14, 1894; 415.

17. F. Fouqué. Contribution to the study of the feldspars of volcanic rocks. Bull. min. fr. 17, 1894; 283.

18. M. Lévy. Study on the determination of feldspars in thin sections. 1894.

19. N. Ussing. Mineralogical-petrographic investigations of Greenland nepheline syenites and related rocks. Meddelelser om Grönland. H. 14; 1894. T. 1.

1. M. Lévy. Recherches des axes optiques dans un mineral pouvant être considéré, comme un mélange deux minéraux déterminés. Application aux plagioclases et à la verification de la loi de Tschermack. Bull. min. fr. 18, 1895; 79.

2. F. Wallerant. Calcul des constantes optiques d’un mélange des substances isomorphes. Application aux feldspaths. Bull. min. fr. 19, 1896; 169.

3. E. v. Fedorow. Die zu den optischen Axen normalen Schnitte der Plagioklase. Z. f. Kr. 25, 1896; 94.

4. G. Becker. Reconnaissance of the Gold fields of Southern Alaska with some notes on general geology. Plagioclase determination. 18-th Annual Report of the U. S. G. Survey. 1896-7; P. III, 30.

5. A. Lacroix. Minéralogie de la France et de ses colonies. II, 1897; 23.

6. C. Hintze. Handbuch der Mineralogie II, 1897; 1332.

7. C. Viola. Ueber Feldspathbestimmung. Z. f. Kr. 30, 1899; 23.

8. C. Viola. Zur Kenntniss des Anorthits vom Vesuv. Z. f. Kr. 31, 1899; 484.

9. F. Becke. Die Orientierung der optischen Axe A in Anorthit. T. M. P. M. 18. 1899; 201.

10. E. v. Fedorow. Constatering der optischen Anomalien in Plagioklasen. Z. f. Kr. 31, 1889.

11. E. v. Fedorow. Mikroskopiche Bestimmung des Periklingesetzes. Z. f. Kr. 32, 1900.

12. V. Tarasenko. Materials for a judgment on the chemical structure of calcareous-soda plagioclases. Zap. Kiev. Society of Naturalists. 16, 1900; 365.

13. F. Becke. Optische Orientierung des Albits von Amelia, Virginia. T. M. P. M. 19, 1900; 321.

14. A. Lavrskii. Plagioclase-augite rocks between the Yenisei and Lena. Transactions of the Society of Naturalists at the Imperial Kazan University. 34, issue 1, 1900; 42–56.

15. F. Becke. Optische Orientierung des Oligoklas-Albits. T. M. P. M. 20, 1901; 55.

16. L. Duparc et F. Pearce. Sur les données optiques relatives à la macle du péricline. C. R. 133, 1901; 60.

17. C. Viola. Ueber die optische Orientierung des Albits und das Tschermack’sche Gesetz. T. M. P. M. 20, 1901; 199.

18. G. Wulff. Untersuchungen im Gebiete der optischen Eigenschaften isomorpher Krystalle. Z. f. Kr. 36, 1902; 1.

19. O. Wenglein. Ueber Perthitfeldspäthe. Kiel, 1903.

20. H. Tertsch. Optische Orientierung von Feldspäthen der Oligoklas-Gruppe. T. M. P. M. 22, 1903; 159.

21. J. Strandmark. Bidrag dill Kännedomen om Celsian och andra Baryt fältspater. Geol. Fören. i Stockholm Förhandl. 26, 1903.

22. G. Tschermack. Ueber die chemische Constitution der Feldspäthe. Wien. Ak. 112, 1903.

XVI

1. F. Becke. Ueber Mineralbestand und Struktur der krystallinischen Schiefer. Wien. Ak. 75, 1903; 55.

2. F. Suess. Ueber Perthitfeldspäthe aus krystallinischen Schiefergesteine. Jahrbuch d. K. k. geol. Reichsanstalt. 54, 1904; 417.

3. L. Duparc. Sur une nouvelle variété d’orthose. C. R. 138, 1904; 714.

4. F. Becke. Die optischen Eigenschaften der Plagioklase. T. M. P. M. 25, 1906, 1.

5. F. Becke. Zur Physiographie der Gemengtheile der krystallinen Schiefer. Wien, 1906.

6. J. Neugebauer. Die Krystalltracht von einfachen Krystallen und Karlsbader Zwillinge des Orthoklases. T. M. P. M. 25, 1906; 413.

7. V. V. Nikitin. A case of secondary overgrowth of primary feldspars of porphyry. One new combination of twin laws of the tetragonal system. Twins along the 1st axis and perpendicular to it. Zap. G. I. I 1907; 236.

8. Ph. Barbier. Recherches sur la composition chimique des feldspaths potassiques. Bull. min. fr. 31, 1908; 152.

9. Barbier et Prost. Sur l’existence d’un feldspath sodique monoclinique, isomorphe de l’orthose. Bull. chim. fr. III, 1908; 894.

10. F. Gonnard. Notes cristallographiques sur les feldspaths potassiques. Bull. min. fr. 31, 1908; 276.

11. F. Gonnard. Sur l’analogie de certaines macles d’orthose nouvelles de Four la Brouque et des macles d’orthose signalées par Tschermack. Bull. min. fr. 31, 1908; 292.

12. K. Brauns. Sanidin von Leilenkopf bei Niederlützingen. N. J. 1909; 1. 43.

13. F. Angel. Ueber einen Natronsanidin von Mittrovitza. N. J. B.—B. 30, 1910; 254.

14. A. Zavaritskii. Petrographic observations in the vicinity of the Miass plant. Zap. G. I. III, 1910; 59.

Also Nos. 2, 5, 9, 12, 20, and 30.

PART ONE.

Basic Methods of Universal-Optical Investigation.

Chapter I.

The Fedorov, or Universal, Stage.

§ 1. General description of the Fedorov stage.

The Fedorov, or universal, three-axis stage in its perfected form is an instrument of small dimensions, fitting onto the object stage of an ordinary polarizing microscope and allowing every possible position to be imparted to a plate of the mineral under investigation, which is achieved by rotating the specimen about several axes of the instrument.

On the lower plate of the latter (Fig. 1 and photograph, Plate I), attached to the microscope stage by special screws, stand two flat columns \(S\), which support the main axis \(J\), carrying the stage of the instrument proper.

This axis is set into rotation by means of the knurled head \(J^{1}\) and is secured by the clamping screw \(i\); the angle of rotation about the axis is measured on the vertical circle \(K\), whose vernier \(n^{1}\) is placed at the top of the adjoining column.

The stage itself consists of several parts rotating relative to one another. The innermost part is a glass disk \(G\), upon which the thin section under investigation is placed. This disk is set into a metal ring, together with which it permits rotation in its own plane, the angles being read on the surrounding graduated ring by means of a special index mark made on the inner metal ring.

The rotation is effected by means of small screws of a special glass spherical segment placed upon the thin section.

This entire system, constituting, as it were, a single whole, is attached to another ring \(R\) on hinges \(H\), as a result of which it can be rotated about an axis lying in the plane of the inner circle and passing through these hinges; the rotation is produced simply by inclining one or the other half of the system, and the inclined position is maintained by fastening the clamping screw \(h\). The method of reading the angles of inclination about this axis is somewhat complicated and will be described below.

The ring \(R\), together with the preceding system, rotates in its own plane; the rotation must be performed by holding only the narrow serrated edge \(m\), not sharply separated from the lower ring \(u\) that accommodates it and is rigidly connected with the axis \(J\); both rings are fastened by the clamping screw \(r\). To determine the angle of rotation of the ring \(R\), degree divisions are marked on it, and the vernier \(n''\) is connected with the axis \(J\) and is located at the second column.

Thus, the Fedorov stage makes it possible to subject the section under investigation to rotation about four axes, of which only one axis, \(J\) (immobile), is fixed, while the other axes change their position relative to the parts of the instrument.

The axis of rotation of the ring \(R\), i.e. perpendicular to the plane of this ring, while remaining at all times normal to the principal axis \(J\), changes its position with the rotation of the latter; this axis is called \(M\) (mobile).

The axis passing through the hinges always remains in the plane of the ring \(R\) and of the glass circle, but can form any angle with the axis \(J\); it is called \(H\) (Hülfsachse).

The axis of rotation of the inner glass circle in its own plane, being always perpendicular to the axis \(H\), may be positioned arbitrarily relative to the first two axes. E. Stratanovich gave it the name \(N\) (normal), as the axis that always remains normal to the plane of the section situated on the glass circle.

Of these four axes, in practice for the most part only three are in use: \(N\), \(H\), and \(J\), which is why the instrument itself is called a “universal stage with three axes.”

Finally, in connection with an ordinary microscope, the entire instrument can rotate about the axis of the latter.

Despite a certain complexity of the instrument, after a little practice work with it proceeds quickly, without delays, since all the parts are close at hand; all the rubbing parts of the instrument, as made by the mechanic Fuess, are well fitted and require only a thin lubricant. Of course, like any more or less complex instrument, the Fedorov stage must be kept properly clean, but its maintenance requires so little time that under no circumstances can it affect the speed of work.

An essential and, moreover, irremediable drawback of the Fedorov stage must be considered the impossibility of working at high magnifications, which is conditioned by the following circumstances.

1) The thickness of the glass segments placed upon the preparation exceeds the focal length of objectives stronger than No. 4¹.

2) Because of the rotations of the instrument about the axes \(H\) and \(J\), the microscope tube must be raised sufficiently high. Completely free handling of the stage is possible only with objective No. 0, which has a small—

Table I.

Fedorov or universal stage (about \(3/4\) natural size).

Nikitin hemisphere (about \(2/5\) natural size).

a greater height and a greater focal distance; with slight rotations about the aforementioned axes one may still use objectives Nos. 1–3.

From what has been said it follows that the Fedorov stage is adapted to work only in parallel light, which, however, does not limit its significance as a universal instrument, because even under this condition it makes it possible to determine all the essential elements of crystals.

§ 2. Spherical segments.

A very important component of the Fedorov stage is formed by the spherical segments, the position of which is shown schematically in Fig. 2.

They are arranged in such a way that their centers are common and coincide with the center of the cross drawn on the glass circle of the stage; moreover, for convenience in working with stronger objectives, the upper segment is usually made smaller. Since the latter is placed upon the thin section, it is removable, whereas the lower segment may be installed once for a long time.

In older constructions of the stage both segments were fastened only by means of glycerin. This fastening is, of course, insecure, since the segments may be displaced by an incautious movement or may slide down under their own weight when the instrument is steeply inclined. To avoid this inconvenience, in the newest instruments the segments have been given the following arrangement¹.

The lower segment $A$ (Fig. 3) is provided with a flat metal ring of such diameter that it freely enters the annular holder of the universal stage, which supports the rotating glass circle.

To fasten this segment, a small drop of glycerin or another suitable liquid is placed on its plane, after which the segment is inserted into the holder and lightly pressed against the glass circle.

Adhesion holds it firmly in place, and a considerable effort is required to detach it from the glass by the handles $pp$.

The mount of the upper segment contains screws $bb$, by means of which it is fastened to the metallic inner ring of the stage, which has corresponding sockets (several pairs of them are shown in the photograph).

The small washers of these screws, located beneath the segment mount, are made so that the screws can rotate freely in the latter without falling out of it.

On the other hand, they protect the thin section, pressed against the inner ring by the segment, from being crushed.

§ 3. The significance of spherical segments and the correction for refraction.

Rotation about the axes of the stage makes it possible to give the mineral under investigation almost any position. But this alone is still insufficient for making the observation itself possible; for this it is necessary to overcome the following difficulties:

1) Owing to the different refraction of the glass, the mineral under investigation, and the air, when the inclination is already at a comparatively small angle, total internal reflection occurs, and only rays reflected from the surface of the thin section enter the microscope tube.

2) Even if there is no total internal reflection, the mineral grain nevertheless shifts from its place in the field of view of the microscope, as a result of which the accuracy of the observations suffers.

3) Finally, owing to the same refraction, when measuring the angles of inclination about the axes \(J\) and \(H\) of the instrument, values are obtained that do not correspond to the true ones.

All these difficulties disappear with the use of spherical segments. Let us consider a system consisting of the latter, a glass disk, object and cover glasses, and the mineral. In order that the whole system be more or less homogeneous, it is necessary to remove the air interlayers, which is achieved by cementing all the enumerated parts with some heavy liquid having a suitable index of refraction; this liquid, as we have seen, is also required for attaching the segments themselves.

If the system is homogeneous, then, as shown in Fig. 4, the central bundle of rays \(A^1A^2\), entering normally into the lower segment and also leaving normally from the upper segment, changes its direction not at all, whatever the angle through which the stage is turned about the axis \(J\). Obviously, the strength of this bundle of rays, which is almost not refracted, will be directly proportional to the radius of the spherical segments.

If, however, one takes the system as a whole, then it will act upon the transmitted light as a biconvex lens, and the latter, from parallel light, becomes convergent; but this convergence, shown exaggeratedly in the figure, is so small even within the limits of the field of view with objective No. 0 that phenomena characteristic of convergent light, i.e. the formation of a black cross, etc., are not observed¹, and only when stronger objectives are used is a weakening of the illumination noticed.

However, the influence of the latter circumstance is to a considerable extent compensated by the use of a special condenser, the system of lenses of which has a long focal distance, corresponding to the height of the universal stage.

As a consequence of the convergence of the light rays, the object appears enlarged, which, of course, is only desirable; this magnification is inversely proportional to the radius of the segments. At the same time the preparation itself appears more transparent, fresher, and cleaner, since the partial scattering of light that occurs in ordinary observations under the microscope owing to the rough surface of the ground section is eliminated.

Thus, provided the entire system is homogeneous, the segments eliminate the harmful effects of refraction of light; it is only necessary to note that, for greater accuracy of observation, the grain under investigation must be located at the main center of the instrument, usually coinciding with the intersection of the cross drawn on the glass circle.

But how is the above-mentioned homogeneity of the system achieved? Since Fedorov’s instrument is still, up to the very latest time, used chiefly for the determination of feldspars, the segments are made of glass having a refractive index \(n = 1.524\), equal to the smallest refractive index of these minerals, which, as will be seen below, is the most convenient. As the binding and cementing liquid, glycerin is used, having approximately the same refraction. In more exact work, the glass plates for the ground section must also be prepared with the same refractive index.

However, in ordinary petrographic investigations one need not require absolute homogeneity of the system, since deviations in the hundredths of the refractive index have almost no effect on the results of measurements. But in the case when a mineral with a refractive index considerably different from that given above is investigated, the homogeneity of the system is greatly disturbed, and a noticeable deviation of the light rays from their original direction occurs; as a result, the measured angles of inclination will be either greater than the true ones, if the mineral has a higher refractive index than the segments, or smaller than the true ones—in the opposite case.

These angles may be reduced to the true ones either by calculation or by means of a diagram constructed by E. S. Fedorov¹.

In fact,

\[ \operatorname{Sin} x = \frac{n}{n^{1}} \operatorname{Sin} \alpha, \]

where

\(n^{1}\)—is the refractive index of the mineral, which may be found approximately in the corresponding tables if the species of the given mineral is known; if the latter is not known, the appropriate result is obtained by successive approximations.

\(n\)—is the refractive index of the segments, usually equal to 1.524.

$a$—the observed angle.

$x$—the true sought angle.

The construction and application of the above-mentioned diagram (Fig. 6) are based on the formula given and may be explained in the following manner¹.

From the formula it is evident that $\sin x$ is the leg of a right triangle with hypotenuse $\dfrac{n}{n^1}$ and with the angle opposite this leg equal to $\alpha$.

Let us draw two concentric circles with radii $OR = 1$ and $OR^1 = \dfrac{n}{n^1}$ (Fig. 5). If $\angle AOB = \alpha$, then from the right triangle $AOB$ it is seen that

\[ AB = \frac{n}{n^1}\sin \alpha. \]

Now draw the straight line $AA^1$, parallel to $OC$; then, obviously,

\[ AB = A^1B^1 = \sin \angle A^1OB^1 = \frac{n}{n^1}\sin \alpha = \sin x. \]

On this basis, the diagram represents a quadrant in which a series of concentric arcs is drawn, whose radii are equal to the corresponding value

\[ \frac{n}{n^1} < 1. \]

In order to give the diagram a more useful form, opposite each arc is placed the value of the reciprocal of the radius, since in the case $n = 1$ no preliminary calculations need be made.

In addition, the diagram contains auxiliary radii and straight lines, parallel to the initial horizontal radius, every $5^\circ$.

To show how the diagram is to be used, let us suppose that it is required to find the true angle from the measured angle of $60^\circ$ for a mineral with refractive index $n^1 = 1.716$. If the refractive index of cement is $n = 1.524$, then $\dfrac{n^1}{n} = 1.126$.

Draw through the division $60^\circ$ a radius up to its intersection with the arc corresponding to this value $\left(\dfrac{n^1}{n}\right)$, and from the point of intersection—a horizontal straight line, which will pass through the division $50^\circ 15'$, which will be equal to the sought angle.

According to the formula

\[ \operatorname{Sin} x = \frac{n}{n^1}\sin \alpha \]

\[ x = 50^\circ 16'. \]

Thus the diagram must be used when \(n\) is less than \(n^{1}\), which occurs most often. If, however, the refractive index of the segments is greater than that in the mineral under investigation, then one must take the ratio

\[ \frac{n}{n^{1}} \]

and find the true value of the angle in the reverse order, i.e., from the horizontal straight line passing through the corresponding degree division to the radius, which will indicate the sought angle.

Thus, in one way or another, homogeneity of the instrument system is, in general, attained quite easily. Moreover, in practice the angles of inclination of the specimen cannot exceed a certain magnitude, because with a very steep inclination the metal holder of the segment enters the field of view, and the plate of the mineral appears very narrow and thick, which greatly hinders observations or makes them impossible.

Usually the upper segment is given a diameter of \(14 \, mm.\), with a limiting angle of observation of \(68^\circ\) on each side.

§ 4. Cementing the specimen.

From the preceding description of the instrument it follows that the specimen to be examined on the Fedorov stage must be prepared in the proper manner.

First of all, in view of the limited area of the inner glass circle and—most importantly—of the segments, the thin section must have definite dimensions and shape, so that every grain of the specimen can be placed at the center of the field of view. It is most convenient to prepare thin sections on round plates \(20 \, mm.\) in diameter, although square glasses of the same size may also be used.

The thickness of the entire thin section is determined so that, when the upper segment is placed on it, the center of the latter coincides with the center of the inner circle.

However, to achieve this, there are special small washers beneath the lugs, the thickness of which must equal the thickness of the specimen, so that the latter is pressed by the segment and cannot move from its assigned position.

Before placing the thin section, all planes of the latter, of the glass circle, and of the spherical segment must be well cleaned with a cloth or—better—chamois leather. Then, on the thin section, on the side of the plate of the investigated substance onto which it is better not to cement the cover glass, a drop of glycerin \(2\text{–}3 \, mm.\) in size is placed at its center with a glass rod. The thin section is quickly turned over and carefully placed on the inner circle; with gradual pressure the glycerin spreads under the thin section, displacing the air. After this the thin section can be moved over the circle until it is properly set.

Then the same drop of glycerin is placed on the plane of the segment; the latter is quickly turned over and screwed to the circle evenly with both screws. If in doing so the thin section shifts from its place, the screws may be loosened slightly and it may be moved with some sharp object, for example the point of a pencil, and then the segment finally secured. When a sufficient quantity of glycerin has been applied, this displacement occurs easily without disturbing the connection between the separate planes; but, on the other hand, there should not be too much glycerin, since then it will flow out from under the objects being shifted by it and will get between the rubbing parts of the instrument, which is harmful to the latter; moreover, the specimen will slide about before the segment is put on. With a little practice the mounting of the specimen is done quickly—more quickly than it can be described; and this circumstance, of course, is of special importance when working through a large amount of material.

After the thin section has been removed, all parts moistened with glycerin must be carefully and cautiously wiped dry.

§ 5. Setting up and checking the Fedorov stage.

Work with the Fedorov stage, in its essential features, amounts to bringing the various planes observed in the mineral under investigation into the plane of symmetry¹ of the microscope, and to bringing the various directions into coincidence with the optical axis of the latter.

This is achieved, in the various manipulations, most easily in the case when the principal axis of the stage \(J\) proves to be perpendicular to the plane of symmetry of the microscope, and the axis \(H\) coincides with this latter plane. Further, for the accuracy of the determinations it is necessary that all the axes of the universal microscope intersect at one point, at which the grain of the specimen under investigation is placed, thus preserving its position under all rotations.

Finally, the axes \(J\) and \(H\) must be perpendicular to the axis \(M\), which in turn, when the plane of the outer ring \(R\) is brought into coincidence with the plane of the inner circle, must coincide with the axis \(N\).

All these conditions require a whole series of preliminary adjustments and checks, which may be arranged in the following order.

1) First, by the usual method, a simple polarizing microscope is checked, in which the optical and geometrical axes of the tube and the axis of rotation of the object stage must coincide.

2) The universal stage is fastened to the stage of the microscope so that, in some position of the former, the center of the cross of the ocular threads coincides with the center of the cross ruled on the surface of the glass circle.

3) Before proceeding to the checks, the stage must be given a certain setting, which we shall regard as preliminary.

a) The inner circle is turned about the axis \(H\) so that its surface is level with the outer ring \(R\), i.e., so that the axes \(M\) and \(N\) coincide.

b) Opposite the \(0^\circ\) of the vernier of the outer ring, the 9th (i.e., \(90^\circ\)) division of the latter is set; in this position the axis \(H\) must be more or less normal to the axis \(J\).

c) The axis \(J\) is set in a vertical plane perpendicular to the plane of symmetry of the microscope; to this end the following procedure is used. The stage of the microscope is turned so that the axis \(J\), to the eye, is perpendicular to the indicated plane; then, by rotating the instrument about this axis away from oneself, the shining edge of the outer ring, representing, as it were, the trace of the upper surface of the latter, is brought into coincidence with the center of the ocular cross, and, in order to see this edge clearly, i.e., to reduce the focal distance of objective No. 0, with which alone work is possible, the Bertrand lens is inserted into the tube; finally, by careful rotation about the axis of the microscope, the edge under observation is brought into coincidence with the horizontal thread of the ocular cross (Fig. 7), and in this position the clamping screw of the microscope is tightened, the reading on the limb of the latter having been noted. The microscope stage usually remains in this position for almost the whole duration of the work.

d) The axis \(J\) is fixed in such a position that the axes \(M\) and \(N\) coincide with the axis of the microscope, while the \(0^\circ\) of the vernier of the vertical circle must more or less coincide with the \(0^\circ\) of the latter.

4) As was said above, all the axes of the universal stage must intersect at one point, represented by the center of the cross ruled on the surface of the glass circle. First and easiest of all is to check the axis \(H\). Since this axis is real and fully accessible, the check can be made by a purely mechanical method: the axis must lie on the continuation of the vertical branch of the indicated cross.

5) When the preceding check has been made, one can determine whether the axis \(J\) is perpendicular to the plane of symmetry of the microscope. For this, the inner circle is inclined about the axis \(H\), and, again using the Bertrand lens, the upper shining edge of this circle is brought into coincidence with the vertical thread of the ocular cross; for this it is sometimes necessary to turn the stage about the axis \(M\). The required condition is fulfilled if, upon rotation about the axis \(J\), the edge does not leave the thread; otherwise it is necessary to—

place something under the lower board of the instrument on the proper side. This is clear from the fact that the axis \(J\) has already been set in the vertical plane perpendicular to the plane of symmetry of the microscope.

6) During the preceding check, the correction is noted for the graduated outer ring \(R\), at which the axis \(H\) will in fact be normal to the axis \(J\).

7) The passage of the axis \(N\) through the center of the cross of the inner circle may be checked by means of compasses. If one leg is placed at the center of the cross, then the circumference described by the other leg with a suitable radius must coincide with the periphery of the circle, since the latter serves as the circle of rotation.

8) It is now possible to find the correction for the vertical circle. For this it is necessary to use the method proposed by Prof. E. S. Fedorov¹. “The upper lens is unscrewed from the eyepiece, a small tube with a Gauss glass is placed on top, and a suitable magnifier is placed on this tube. The Gauss glass is turned until the light reflected from it inside the microscope, directed from a lamp at the side, appears on a piece of paper placed on the glass circle of the stage. Then the paper is removed, and a Bertrand lens is inserted in the path of the ray; this lens is moved along the axis of the microscope until the image of the ocular cross reflected from the stage is visible in the upper magnifier. This will occur, obviously, when the system consisting of the objective and the Bertrand lens has its principal focus at the center of the ocular cross. In this case the rays issuing from this point, after emerging from the objective, will take a parallel position and, having been reflected from the glass, will again give a real image of the cross in its own plane.” In the general case the cross and its image do not coincide; this means that the surface of the inner circle is not perpendicular to the axis of the microscope, or that the axis \(N\) does not coincide with the latter.

The coincidence of these axes is achieved in two steps.

a) First, by rotating the axis \(J\), the horizontal thread of the cross and its image are brought into coincidence. The reading on the vertical circle gives the correction for the latter.

b) By the preceding operation the axis \(N\) has been brought into the vertical plane perpendicular to the plane of symmetry of the microscope. For the final coincidence of the axis \(N\) and the axis of the microscope, the inner circle is inclined about the axis \(H\) until the vertical threads of the cross and of its image also merge. This position of the axis \(N\) may also be fixed on the vertical circle. For this purpose the stage is turned about the axis \(M\) through \(90^\circ\) in the direction such that the dihedral angle formed by the surfaces of the inner circle and of the outer ring is directed

toward the observer. Then, by rotation about the axis $J$, the bright edge of the inner circle is brought into coincidence with the horizontal thread of the ocular cross, using, of course, the Bertrand lens in the ordinary state of the microscope. The reading on the vertical circle, more or less close to $90^\circ$, will fix the relative position of the correctly set axis $N$.

9) It remains to deal with the axis $M$. Since in ordinary work it is rarely necessary to use it, and for the most part only for auxiliary manipulations, there is no special need to pursue absolute accuracy in its setting. This axis is checked by means of E. S. Fedorov’s method.

The stage is set so that the ocular cross coincides with its image. If the outer ring $R$ is now rotated, then, with the correct arrangement of the axis $M$, there should be no divergence of the crosses.

An incorrect arrangement of this axis cannot be changed: here everything depends on the mechanic who made the instrument.

10) Finally, it is also necessary to check the correctness of the mounting on the glass circle of the spherical segments. In stages of the improved type these segments, as we have seen, are fastened by means of special screws and thus remain immobile. If, when the segments are mounted, the ocular and glass-circle crosses that had previously been brought into coincidence do not separate, then the segments have been fitted correctly, i.e., their centers coincide with the principal center of the instrument. Ordinarily the stage always satisfies this condition.

The foregoing exhausts the adjustments and checks of the stage. In work not requiring special accuracy, for example, in the general determination of minerals in thin sections from a large collection of rocks, one may confine oneself to what has been said in points 1, 2, 3, 5, and 6.

The principal condition that the Fedorov stage must satisfy—especially for purposes of petrographic investigations—is that the rotating parts should be well fitted to one another, since nothing so delays the work and tires the worker as much as stiff or irregular motion when rotating about the numerous axes of the instrument.

As regards the accuracy of the setting of the stage and of the measurements made on it, it must be said that although the verniers of the limbs (except for the inner limb, which make it possible to read $5'$), nevertheless, in the treatment of petrographic material this accuracy rarely reaches $1/4^\circ$.

Chapter II.

Methods for projecting the results of measurements on Fedorov's stage

§ 6. Foundations of stereographic projection

Work on Fedorov's stage is conducted in parallel with a graphic representation of the data obtained; this is necessary not only for various constructions, by which many questions are resolved, but also for directing subsequent work.

Since the universal stage is, in essence, a goniometer, the projection that must be used for the graphic representation of the results of measurements must satisfy those conditions under which angles retain their magnitude, and under which various constructions are made easily and quickly, for only in the latter case does the given method acquire practical significance in petrography. The stereographic projection best satisfies the stated conditions, and it was adopted by E. S. Fedorov at the very beginning of the development of his method.

The general foundations of the application of stereographic projection consist in the following.

Let us imagine a sphere of arbitrary radius (Fig. 8), through whose center \(O\) pass all the planes and directions observed in the crystal; in practice it may be assumed that this center coincides with the center of the stage.

All these planes and directions, in intersection with the surface of the sphere, will give, respectively, arcs of great circles and points; moreover, the measurement of angles in the crystal may be replaced by the measurement of the projections of these angles on the sphere. If \(ACBD\) is taken as the equatorial plane, and \(CS^{1}DS\) as the plane of the first meridian, then the position of each direction is determined by two spherical coordinates of its projection, also called the pole of the direction: by longitude, measured along the equator from the point \(C\), clockwise, to the meridian passing through the given pole, and by the height of the latter relative to the point \(S^{1}\). Thus, for the straight line \(OE\), the coordinates are: longitude = arc \(CK\) and height = arc \(ES^{1}\).

Since a plane is determined by two straight lines, its position is indicated in an analogous manner. For this one should take the trace of the equatorial plane on the given plane and the line of intersection of the latter with the plane perpendicular to this trace. Since these straight lines are mutually perpendicular, the plane will be completely determined by the longitude of the first straight line and the height of the second straight line.

Thus, the coordinates of the plane \(HLJF\) will be \(CABH\) and \(FS^{1}\).

The described method of projection onto the sphere, called the linear method, is sometimes replaced by another, in some cases more convenient—

The latter consists in the fact that what is projected is not a plane, but a perpendicular to it; moreover, the projection of this perpendicular is called the pole of the plane, and instead of a line one projects the plane perpendicular to it. Obviously, this method of projection, called gnomonic, is advantageously applied when planes are chiefly being observed—for example, in goniometric investigations of crystals.

Thus, the first approach to solving the question of projecting the data of observations on the Fedorov stage has already been made by us. It must be said that direct projection onto the sphere is accomplished by using Nikitin’s special porcelain hemisphere, which will be described below. Its use, however, is inconvenient in certain respects and is not always permissible; therefore, in order to be able to carry out the construction on paper, it is necessary to pass from projection on the sphere to projection on a plane.

For this purpose, as the projection plane (the picture plane), or the plane of the paper, we shall take the equatorial plane \(AOB\), with which, incidentally, it is customary to make the plane of the section coincide.

As the point of view let us take one of the poles of this plane, for example \(S\). Then, joining the point of view with the projections lying on the upper hemisphere by straight lines or conical surfaces, at the intersections of the latter with the picture plane we obtain the stereographic projections of the given elements.

Among these projections, grammastereographic and gnomostereographic projections are distinguished, corresponding respectively to the linear or gnomonic methods of projection.

Stereographic projection possesses properties very convenient for our purposes. Indeed—as was said above—all the elements of measurements on the Fedorov stage are determined by two coordinates. One of the latter—the longitude—can be laid off directly on some circle representing the equator. Usually such circles, 20 cm in diameter, with certain auxiliary systems of lines in the form of so-called nets, are used already prepared (see the tables).

If on this net there are drawn, at a known number of degrees, diameters which are in fact the projections of meridional planes, and concentric circles corresponding to the polar parallels, then, guided by the latter, it is easy to lay off the second coordinate as well. For the convenience of subsequent constructions, two more equatorial-stereographic systems are superposed on the net, representing the projections of meridians and parallels on two mutually perpendicular meridional planes with points of view lying on the equator.

On the nets it is evident that the linear distance between the parallels gradually increases as one approaches the periphery.

Indeed, if \(AB\) represents the picture plane with the point of sight \(S\) (Fig. 9), and if the radius of the sphere is taken as unity, then

\[ \begin{aligned} a_1 &= \operatorname{tg}\alpha,\quad a_1+a_2=\operatorname{tg}2\alpha;\\ a_2 &= \operatorname{tg}2\alpha-\operatorname{tg}\alpha,\quad a_1+a_2+a_3=\operatorname{tg}3\alpha;\\ a_3 &= \operatorname{tg}3\alpha-\operatorname{tg}2\alpha \quad \ldots\\ &\ldots\\ a_n &= \operatorname{tg}n\alpha-\operatorname{tg}(n-1)\alpha. \end{aligned} \]

Since \(\operatorname{tg}\) increases faster than the angle, the segments \(a\) also increase toward the periphery.

Although linear distances are distorted in the projection, angular quantities, thanks to the degree net, can be measured with a greater degree of accuracy.

Another important property of stereographic projection is the preservation of the form of a circle, although the position of the geometrical center of the latter changes. Let \(BC\) (Fig. 9) represent the diameter of a circle drawn on the sphere. The rays connecting the point of sight \(S\) with all points of this circle form a cone \(SBC\), intersecting the picture plane in some closed curve \(DEB\). The plane \(FEG\), drawn through the center of this figure parallel to the circle \(CB\), will obviously also cut the surface of the cone in a circle. It is not difficult to see that the sections \(FG\) and \(DB\) are equally inclined to the generators of the cone; therefore, if \(FG\) represents a circle, then \(DB\) will also be a circle, which was to be proved.

The center of the latter circle \(E\) cannot be the projection of the center of the circle \(BC\), for \(DE=EB\), while from the preceding we know that the projections of equal arcs drawn on the sphere are not equal to one another.

It remains only to note that if \(BC\) represents the diameter of a circle, and \(AS^1BS\) is one of the meridional planes, then the projection of the center of this circle and the geometrical center of the projection of the latter must lie on one diameter of the stereographic net.

The only serious drawback of the projection described is the circumstance that, within the limits of the circle of the net, only elements situated on one side of the picture plane are depicted. Of course it is easy to obtain projections of points belonging also to the other hemisphere—as shown in the figure—but making constructions and measurements in this case becomes very difficult, or even impossible, since the linear distances between projections increase greatly, and for points lying near the point of sight become infinitely large. However, in studying the optical properties of minerals this inconvenience almost disappears, in view of the fact that such properties do not depend on the direction of the vector, and the picture plane divides the crystal into two completely symmetrical halves.

Having become acquainted with the foundations of stereographic projection, let us proceed

... we shall proceed to the solution of construction problems that may be encountered in the graphical representation of the results of measurements on the Fedorov stage.

All cases of the necessary constructions can be reduced to the solution of the following seven problems.

§ 7. Problem 1. To find the pole of a given arc of a great circle (Table VI).

Let the given arc of a great circle be \(CED\). As was said above, the pole of a plane is the projection of the normal to it. Obviously, this normal must lie in the plane perpendicular to the given circle; but such a plane will be nothing other than a meridional plane, which on the net is represented by one of its diameters. The latter is very easy to find, since it will be perpendicular to the diameter \(CD\), as a straight line lying in the plane of the given circle, and its position will be determined if from any end of this diameter \(CD\) one counts off \(90^\circ\) along the circumference of the net.

On the diameter drawn through the point thus obtained must lie the required pole \(P\). To determine the position of the latter, it is necessary from the point \(E\), as the line of intersection of the planes under consideration, to count off \(90^\circ\), using the polar parallels.

From the construction one may observe that the pole is always situated on the side of the concave part of the arc. Moreover, the counting along the parallels may be made not from the point \(E\), but from the end of the meridian \(G\), laying off the angle which this point \(E\) forms with the center of the net.

In the special case when the given plane is meridional and is represented in projection by a diameter, its pole will lie on the circumference of the net.

§ 8. Problem 2. To lay off a known angle on a given arc of a great circle from a given point (Table VI).

Suppose that on the arc \(CED\), from the point \(L\), it is necessary to lay off from oneself \(32^\circ\). For this, we connect the pole of the arc \(P\) by a straight line with the given point \(L\), lay off from the point of its intersection with the circumference of the net the arc \(L^1T^1 = 32^\circ\), and, connecting the points \(T^1\) and \(P\) by a straight line, at the intersection of the latter with the arc we find the point \(T\), distant from the point \(L\) by the prescribed number of degrees.

For proof, let us turn to schematic drawing 10¹. If the points \(S\) and \(F\) are situated symmetrically with respect to the great circles \(s\) and \(f\), then, obviously, the arcs of the latter, \(L^1T^1\) and \(LT\), will be equal to one another. Now imagine that the point \(S\) is the point of vision of the stereographic projection, and the plane \(s\) the picture plane—

the latter, and the point \(F\) will represent the pole of the given plane \(f\). And in this case the points \(S\) and \(F\) will remain symmetrical with respect to their planes, since each of them is \(90^\circ\) distant from all points of the circle of intersection of the corresponding plane with the sphere, and therefore the equality of the arcs will not change.

From this one may derive the following proposition: planes passing through the point of sight of the stereographic projection and through the pole of a given plane cut off, on the circles of intersection of the latter and the picture plane with the sphere, arcs equal to one another.

But in the stereographic projection, planes passing through the point of sight are represented by straight lines, and this makes the construction shown possible.

Obviously, by the same method one may determine the angle between two points lying on a given arc of a great circle.

§ 9. Problem 3. To construct an arc from three points, of which two represent the ends of the diameter on which the arc will rest, and the third lies on the diameter perpendicular to the first (Table VI).

If we turn to the conditions of the first problem, then the given points will be precisely \(C\), \(D\), and \(E\).

First of all we construct the pole \(P\). Joining the latter with the point \(C\) (or \(D\)) by a straight line, we cut off on the circle of the net the arc \(DP^1\). Then we lay off the arc \(P^1K =\) the arc \(DP^1\), and, joining the points \(K\) and \(C\) by a straight line, at its intersection with the prolongation of the diameter \(EP\) we find the point \(Q\), as the geometric center of the required arc.

This construction may be proved in the following way.

Let us depict in Figure 11 what has been drawn on the net. First of all let us note that the required center of the arc must lie on the straight line \(EE^1\), as the perpendicular to the chord which subtends the given arc and which bisects it.

Let us suppose that the diametrically opposite point of the circle to which the arc belongs is the point \(E^1\). Dividing the straight line \(EE^1\) in half, we find the position of the required center \(Q\) and the radius of the circle

\[ EQ = \frac{EE^1}{2}. \]

Then, joining the points \(P\) and \(Q\) with the point \(C\), we shall show that \(x = 2a\).

From the right triangle \(OCQ\) we find that \(OQ = \operatorname{tg} x\). On the other hand,

\[ O = EQ - OE = \frac{EE^1}{2} - OE. \]

Thus

\[ \operatorname{tg} x = \frac{EE^1}{2} - OE. \]

Find the values of \(\overline{EE^1}\) and \(\overline{OE}\).

From the right triangle \(EOC\) we have that \(OE = \tg\) of the angle \(ECO\). If one imagines that the plane \(MN\) is the picture plane of the stereographic projection, then the arc \(LP^1 = 90^\circ\),

\[ \text{and } \angle ECO=\frac{90^\circ-2\alpha}{2}, \text{ and} \]

\[ OE=\tg\frac{90^\circ-2\alpha}{2}=\frac{1-\sin 2\alpha}{\cos 2\alpha}. \]

Further, from the right triangle \(OCE^1\), in which the angle \(OCE^1\), as resting on the diameter \(EE^1\) of the required circle, will be right, we find that

\[ EE^1=\frac{CE}{\sin OE^1C}. \]

But

\[ CE=\frac{1}{\Cos\frac{90^\circ-2\alpha}{2}}, \quad \text{and } \angle OE^1C=\angle ECO=\frac{90^\circ-2\alpha}{2}, \]

so that

\[ EE^1=\frac{1}{\Sin\frac{90^\circ-2\alpha}{2}\,\Cos\frac{90^\circ-2\alpha}{2}}. \]

Thus:

\[ \operatorname{tg} x = \frac{1}{2\Sin\frac{90^\circ-2\alpha}{2}\cdot\Cos\frac{90^\circ-2\alpha}{2}} - \frac{1-\Sin 2\alpha}{\Cos 2\alpha} = \]

\[ = \frac{1}{\Sin(90^\circ-2\alpha)} - \frac{1-\Sin 2\alpha}{\Cos 2\alpha} = \frac{1}{\Cos 2\alpha} - \frac{1-\sin 2\alpha}{\Cos 2\alpha} = \tg 2\alpha. \]

Thus, \(x=2\alpha\).

Hence \(\angle DCP^1=\angle PCQ\) and—consequently—\(\smile DP^1=\smile P^1K\), which was what had to be proved.

What, then, is the proper value of the segment \(PQ\), representing the distance of the geometrical center of the arc from the pole of the latter?

If we take the diameter \(MN\) as the meridional plane and the point \(C\) as its pole, then, according to the preceding problem,

\[ \smile OP=\smile DP^1 \text{ and} \]

\[ PQ=\smile P^1K, \text{ but since} \]

\[ \smile DP^1=\smile P^1K, \text{ then also} \]

\[ \smile OP=\smile PQ. \]

Thus the point \(Q\) is located from the center of the net at twice the polar distance, which is represented especially clearly in Figure 12. Therefore, if the pole of the arc is less than \(45^\circ\) distant from the center of the net, then the geometrical center of this arc is easily found by means of parallels.

§ 10. Stereographic ruler.

The method described for finding the center of the arc of a great circle is ordinarily employed only rarely, since it requires the expenditure of a comparatively large amount of time, which in petrographic work is very undesirable. Usually, in solving the given problem, one uses the so-called stereographic ruler, which is essentially a scale for the centers of arcs resting on one and the same diameter of a net of a definite size. In the absence of a ready-made one, it is very easy for the worker himself to construct such a scale. For this purpose the beginning of a straight line drawn on the standard sheet of paper is made to coincide with the center of the net, and on this line, by means of the method described, are marked the centers for arcs resting on the diameter perpendicular to this straight line. It is sufficient to find and mark the centers for arcs taken at every \(2\frac{1}{2}^{\circ}\) or \(5^{\circ}\).

In Figure 13 such a scale is shown at a scale of \(1/4\). The numbers placed along the scale indicate by how many degrees the corresponding arc is deflected from the horizontal equator.

On the other hand, according to the preceding, each mark represents half the number of degrees by which the corresponding division is separated from the center of projections. Thus the stereographic ruler may be used, with the indicated correction, also for laying off angular magnitudes from the center of the net.

The ruler presented is incomplete. First, it does not show the divisions between the marks \(0^{\circ}\) and \(45^{\circ}\). This is done because in this interval the centers are easily found by means of polar parallels, as was shown above.

§ 11. Circular ruler.

Second, on the stereographic ruler there are no divisions above \(75^{\circ}\), corresponding to very gently sloping arcs close to the vertical plane. In view of the fact that the radii of such arcs are very large, the latter are much more conveniently drawn with the aid of a special circular ruler designed by E. S. Fedorov.

It consists of a copper base board \(A\) (Fig. 14) and a polished elastic steel plate \(BC\). This plate is bent by means of the projections \(D\) and \(E\) of a transverse rod, upon which the screw \(F\) presses when the latter is screwed into the nut \(g\), attached to the board. The distances of the projections from the support points \(h\) of the plate are chosen so that this bending occurs along an arc of a circle with an accuracy sufficient for ordinary drafting purposes; moreover, the angle of inclination of the arc to the vertical plane must not exceed \(25^{\circ}\); in general, this ruler may be bent by another \(5^{\circ}\).

It should be borne in mind that the indicated ruler, with a constant distance between the projections \(D\) and \(E\), is suitable only for drawing arcs of a great circle on nets 20 cm in diameter.

Therefore, in order to make the ruler applicable in other cases as well, the designated projections are sometimes made movable.

Recently, another design of circular ruler has appeared, in which the bending of the elastic plate is produced directly by turning a special eccentric; the latter is connected with a pointer indicating, on a dial, the angle of bending.

This ruler is a more delicate device than the one described above, and, moreover, the greatest angle of bending of the plate does not exceed \(20^\circ\), so that, as it seems to us, it has no special advantages.

§ 12. Problem 4. To draw an arc of a great circle through two given points (Table VI).

Two points \(L\) and \(T\) are given; it is required to draw through them an arc of a great circle. For this purpose we take the given points as poles and construct the corresponding arcs; the construction is based on the preceding problems and is evident from the table.

Obviously, the point of intersection \(P\) of the arcs drawn will represent the projection of the straight line perpendicular to the plane determined by the points \(L\) and \(T\), as projections of two straight lines; that is, it will be the pole of the arc passing through these points.

Problem 5. To divide the angle between two points (Table VI).

To solve the problem we draw through the given points \(L\) and \(T\) an arc of a great circle, if there was none; from the pole of this arc we project, according to Problem 2, the given points onto the circumference of the net and divide the arc \(L'T'\) cut off on it in the required ratio; the points of intersection with another \(LT\) of the straight lines joining the points of division with the pole divide the angle between the given points in the same ratio.

Problem 6. To measure the angle between two points.

According to the preceding, we draw through the given points an arc of a great circle and, projecting the latter from the pole of the arc onto the circumference of the net, directly read off on it the required angle.

§ 13. The three-legged compass.

In practice, the last three problems are solved much more simply by using the three-legged compass,\(^{1}\) in which, besides the two usual legs rotating on a hinge, there is also a third, [[unclear: continuation cut off on this page]]

\(^{1}\) E. v. Fedorow. Ueber die Anwendung des Dreispitzzirkels für krystallographische Zwecke. Z. f. Kr. 37, 138.

can rotate about two mutually perpendicular axes, so that by this pair of compasses any three points lying on the net can be marked.

For more convenient superposition with these points, the legs of the compasses are made compound and can change their length.

The measurement with such compasses of the angle between two points is based on the following.

Obviously, if the given points lie on one meridian of the equatorial-stereographic system of the net, then the angle between them is read off directly along the parallels normal to the indicated meridian. On the other hand, this angle will remain unchanged if the points are moved in such a way that neither their mutual position nor their distance from the center of the net changes. Therefore, if one leg of the compasses is placed at the center of the net, and the two others are made to coincide with the given points, then, upon rotating the compasses about the central leg, these points will move in accordance with the latter condition. Having placed them on one meridian, we at once determine the sought angle. Thus, in Table VI the points \(L\) and \(T\) have been moved by the compasses into the position \(L_1\) and \(T_1\), and the angle between them is found to be \(32^\circ\).

With the aid of these compasses one may also find the pole of the arc of a great circle passing through the given points. To do this we move these points onto one meridian, find on the diameter perpendicular to this meridian the pole of the arc of the latter, then transfer the third leg of the compasses into this pole and, finally, carefully move the whole compasses into the position in which the first legs coincide with the given points. Then the third leg of the compasses will indicate the pole of the desired arc, after which the latter is easily constructed.

In the table the pole of the arc of the meridian passing through the points \(L_1\) and \(T_1\) is represented by the point \(P_1\), which, after the compasses are transferred, will occupy the position of the required pole \(P\).

§ 14. Problem 7. Find the locus of points lying at an equal number of degrees from a given point (Table VI).

Obviously, the given point represents the projection of the center of a circle drawn on the sphere in the proper manner. It was shown above (§ 6) that this projection does not coincide with the geometric center of the projection of the circle, but, together with the latter, lies in one meridional plane. Since this plane is represented on the net by a diameter, the geometric center of the required circle must lie on the diameter drawn through the given point.

By laying off from the latter, with the aid of polar parallels or a stereographic ruler, the required number of degrees, we obtain on the diameter two opposite points of the circle; and, by bisecting the linear distance between these points, we find the geome-

tric center of the circle sought. Thus, on the table the geometric locus of points lying \(40^\circ\) from the given point \(S\) is represented by a circle with its center at the point \(k\).

In the construction described, the stereographic ruler is sometimes also insufficient for finding one of the diametrically opposite points of the circle. In this case one must resort to a construction based on Problem 2. Taking the diameter passing through the given point as the meridional plane, we project from the pole of the latter this point onto the circumference of the net, where we lay off, to one side or the other, the assigned number of degrees. Then, joining the division of the circumference of the net thus obtained with the pole, we find, at the intersection of the straight line drawn with the diameter, the required point of the circle.

§ 15. Application of transparent stereographic nets.

The arrangement of a net of three stereographic systems, the use of the circular and stereographic rulers, and of the three-legged compass greatly simplify all constructions with stereographic projections. But sometimes even this is insufficient; for example, it is almost impossible, with the aid of a three-legged compass, to measure the angle between two points lying close to the ends of one diameter. Such particular cases of construction can be conveniently carried out with the aid of an additional stereographic net drawn on transparent paper. Such nets, incidentally, are appended to each copy of E. S. Fedorov’s books Foundations of Petrography and Crystallography, and earlier were generally used in work on the universal stage. If a transparent net is superposed on the working net so that their centers coincide, then by rotating the former one can always, for example, place the two given points on one meridian of this net and thus immediately read off the angle between them along the equatorial parallels. This idea—of using two nets, one of which serves as a template—was expressed earlier by the creator of the method, who designed a special drawing board that makes it possible to carry out all constructions without any auxiliary appliances.

In the central part of this board¹, flush with its upper surface, there is a light-colored rotating disk on which a stereographic net of the usual size is drawn in black lines (Fig. 15). As the working net there is used a transparent net, which is superposed on the disk so that their centers coincide and, with the aid of pieces of wax, is fastened to the board itself.

By rotating the lower circle, it can be set in such a position that the points marked on the upper net are placed on one meridian or diameter of the circle’s net, or in general are arranged symmetrically with respect to the zero diameters of the latter.

In such a position it is very easy not only to carry out various constructions over the angles, but also to draw arcs of large circles—freehand.

It must be said that, instead of a transparent net, one may with the same success use ordinary tracing paper or waxed paper.

Of course, this method of graphical solutions of problems in universal-optical investigation is less accurate than the preceding ones, since the centers of the nets do not always coincide sufficiently well, and the upper transparent net will inevitably warp in various places. Nevertheless, it possesses an accuracy quite sufficient for ordinary petrographic work, and one can only be surprised that it is almost never mentioned, while in the catalogues of firms manufacturing optical instruments and accessories for them there is no indication concerning the drawing board described here, the schematic drawing of which was made solely on the basis of a brief description by E. S. Fedorov.

§ 16. V. V. Nikitin’s Hemisphere¹.

The first stage in the graphical representation of the results of measurements on Fedorov’s stage is projection onto the sphere. It has already been stated above that this is achieved by the use of Nikitin’s porcelain hemisphere.

On the surface of this hemisphere (photograph, Plate I) a degree net is incised at intervals of 10°, and the metal ring serving as its base is divided into 360°.

In solving problems involving various constructions in stereographic projection, we have seen that one almost always has to deal with two planes and directions perpendicular to one another. On the hemisphere these planes find their real expression in the form of two metal arcs, which are fastened to the horizontal ring framing the base of the hemisphere, and together with it can rotate about the vertical axis.

One of these arcs can be inclined to the horizontal position, sliding along the fixed arc perpendicular to it. Both arcs carry divisions from 0° to 90° on both sides.

Let us now consider how the above-enumerated problems are solved on this instrument.

To draw an arc of a great circle, one must align the base of the fixed metal arc with the corresponding division of the horizontal limb, which plays the role of the circumference of the stereographic net, and, having inclined this arc at a definite angle, draw along it the required projection of the plane. The pole of the latter will lie at the edge of the fixed arc and can at once be marked in the manner already known to us.

In carrying out this construction, it is necessary to bear in mind that the edges of the two arcs along which the marks are made on the sphere must be perpendicular to each other; this is easily determined from their trace on the horizontal limb. Usually the edges of the arcs along which the degree divisions are inscribed satisfy this condition.

To set off some angle on a given arc of a great circle is exceedingly simple: for this it is only necessary to align the movable metal arc with this arc, and the degree divisions found on the latter will make it possible at once to mark the required angle.

In exactly the same way one may divide a given angle in the required ratio, draw an arc of a great circle through two given points, and measure the angle between the latter.

And the last problem, concerning the drawing of arcs of a small circle about a given point, is solved on the hemisphere very simply, since this point will also serve as the geometrical center of the arc. For the convenience of drawing circles, E. S. Fedorov added to the hemisphere a special platform with a recess for the leg of the compass; this platform can move along the fixed arc and be fastened to it in any position corresponding to the center of the circle.

Thus, all constructions on the hemisphere are performed much more simply and incomparably more quickly than on the stereographic net; but, on the other hand, they are less exact owing to the coarseness of the instrument and, moreover, by the very nature of the method of projection, cannot be fixed. These circumstances also determine the conditions under which the hemisphere may find application.

In conclusion it should be noted that lines on the porcelain hemisphere must be drawn only with a pencil, and that the removal of traces of constructions no longer needed is carried out with a soft cloth moistened with vaseline.

Chapter III

Principal Methods of Working with the Fedorov Stage

We shall now proceed to a description of the actual operations on the Fedorov stage.

As was indicated earlier, the essence of the principal methods of these operations is reduced to bringing into coincidence with the plane of symmetry of the microscope certain planes and directions observed in the mineral under investigation. Of these, the greatest importance belongs to the planes of the axes of elasticity and to the optical axes, for the latter can be observed in any grain and, moreover, occupying a position quite definite for each mineral species relative to its crystallographic axes, they are the most important optical elements.

Therefore the principal methods of work on the Fedorov stage will consist chiefly in determining the position of the planes of elasticity and of the optical axes. In what follows, the exposition of these methods will be connected with the study of biaxial minerals, since the latter represent the most general case.

§ 17. Fundamentals of the Crystal Optics of Biaxial Minerals

The crystal optics of biaxial minerals is in general rather complex. Nevertheless, when using the universal method, despite the importance of the results obtained, all conceptions of optical phenomena can be considerably simplified and, so to speak, schematized.

If all possible directions in a biaxial mineral are represented as issuing from one point, then the geometrical locus of the ends of the radius-vectors denoting a greater part of the quantities characterizing the propagation of light energy will be an ellipsoid with three mutually perpendicular axes.

Fresnel, who created the theory of light as the motion of particles of the ether, placed the elasticity of the ether at the basis of the construction of such an ellipsoid. This quantity, despite its more or less abstract character, is the most suitable for the indicated purpose, because, according to the theory, it is measured in a plane perpendicular to the direction of propagation of light, i.e. in the plane in which the particles of the ether oscillate; and this avoids the possibility of any misunderstandings. Indeed, a given elasticity of the ether in the direction \(AB\) (Fig. 16) fully characterizes the ray \(OC\) perpendicular to it. If, on the contrary, we were to ask for some quantity relating to the ray \(OC\) itself—for example, the velocity of light or the refractive index—then it would remain unclear in which direction, lying in the plane \(ADB\), the oscillation of the ether takes place.

Thus, at the foundation of all crystalloptical conceptions lies the Fresnel ellipsoid, in which mutually perpendicular—

the principal axes of least, mean, and greatest elasticity are denoted respectively by \(\alpha, \beta, \gamma\)¹), and the three planes passing through each two of these axes are called the principal planes of elasticity, or the principal sections of the crystal. In the universal-optical method, however, it is customary to name the principal axes of the ellipsoid according to the corresponding index of refraction of the rays coinciding with these axes.

With this notation of the axes as \(n_p\) (small), \(n_m\) (mean), and \(n_g\) (large), the form of the ellipsoid does not change. Indeed, the index of refraction, as is known, is inversely proportional to the velocity of light, while the latter, according to the formula \(v = \sqrt{\frac{e}{d}}\), is directly proportional to the elasticity of the ether in the direction of vibration; thus the index of refraction of a ray is inversely proportional to the elasticity of the ether measured in the direction perpendicular to that ray. Suppose now that a ray is propagated along \(OB\) (Fig. 17). The vibrations of the ether will occur in the plane \(CDFE\), in which lie the axes of least and mean elasticity; consequently, according to the preceding, the index of refraction of one of the rays coinciding with the given axis will be the greatest. Therefore the axis of greatest elasticity \(\gamma\) and of the greatest index of refraction \(n_g\) coincide. With the same limitation it can be shown that the axes \(\alpha\) and \(\beta\) correspond to the axes \(n_p\) and \(n_m\).

The ellipsoid of elasticity makes it possible simply to explain all phenomena occurring in a biaxial crystal.

Let us take a section of the mineral normal to the entering ray of light. This section, referred to the Fresnel ellipsoid, will in the general case represent an ellipse whose radius-vectors denote the elasticity of the ether in the corresponding directions. Thus the ray entering the crystal must vibrate in a plane characterized by a non-uniform tension of the ether in different directions.

As observations show, vibrations of rays can occur only in the directions of the axes of the ellipse, so that, if the direction of vibration of the entering ray does not coincide with any one of the indicated axes, then this ray begins to vibrate at once along both axes, being thereby split into two rays polarized in mutually perpendicular planes and possessing different velocities and different refractions, owing to the different elasticity of the ether in the direction of vibration of these rays.

Such are the conditions in all sections of the mineral except two corresponding to the circular sections of the Fresnel ellipsoid. These circular sections pass, of course, through the mean principal axis of the ellipsoid \(n_m\), and are inclined at equal angles to the principal section po-

¹) or by the corresponding letters of the German alphabet.

…of the latter, containing the same axis. Since the radius-vectors of these sections are identical, then, disregarding certain secondary circumstances, we may assume that a ray of light normal to the circular section undergoes no change in passing through the mineral, and that the latter, in the two indicated directions, behaves as a perfectly isotropic medium. These special directions, called optical axes, lie, of course, in the plane determined by the axes \(n_g\) and \(n_p\), or, as we shall say below, in the plane \(n_g n_p\), the latter being the bisectors of the angles between the optical axes. When the bisector of the acute angle, denoted by \(2V\), is the axis \(n_g\), the mineral is called positive; when the acute bisector is \(n_p\), it is considered negative.

Thus, having recalled the fundamental principles of crystal optics, let us return to the determination of the position of the planes of elasticity.

§ 18. Setting the planes of elasticity.

In an ordinary polarizing microscope, the transmitted parallel light is always perpendicular to the plane of the thin section, in which the vibrations of the rays will also take place; moreover, these vibrations will coincide with the axes of the ellipse of elasticity corresponding to the section of the thin section. When one of these axes, upon rotation of the microscope stage, coincides with the principal section of the polarizer, then the ray of light polarized in the plane of the latter passes through the mineral without changing the direction of its vibrations and is extinguished in the analyzer placed perpendicular to the polarizer.

In this way the position of the axes of the ellipse of elasticity in the section of the thin section is determined by the angle of extinction, referred to some crystallographic element, for example, to the trace of a cleavage crack. But the axes of the ellipse of an arbitrary section are not parallel to the principal axes of elasticity, whose position alone is characteristic for the given mineral; therefore, in order to obtain the necessary results of the optical investigation, one must either choose definite sections and prepare thin sections from them, or resort to the aid of the universal stage, to which we now turn.

When the stage is rotated about its axes \(J\) and \(H\), the ray of light normally encounters ever new sections of the mineral, in which the axes of the ellipse of elasticity generally occupy different positions, and therefore the extinction angles will be different for different sections, which is observed along one of the horizontal limbs.

But if we set the grain of the mineral under investigation so that one of the principal axes of elasticity coincides with the stage axis \(J\), which, as is known, is placed perpendicular to the plane of symmetry of the microscope, then, upon rotation of the instrument about this axis \(J\), the axes of the ellipse of each section of the mineral will coincide with the principal sections of crossed nicols, and therefore darkness will be observed the whole time. From this one may also derive the converse conclusion: if, during rotation about the axis \(J\), the obst…

... the preparation, set in the proper way, will never give constant extinction with the crosshairs; then one of the principal axes of elasticity of the mineral is combined with the axis of the stage, \(J\), and the plane of elasticity perpendicular to it coincides with the plane of symmetry of the microscope, which in fact is what has to be determined.

Thus we have found a method for distinguishing the principal planes of elasticity of the mineral: for this it is necessary only to give different positions to the inner circle of the stage, on which the preparation under investigation is located, and to test them for constant extinction by rotation about the axis \(J\).

Such a position of the preparation, at least in determining the first plane of the axes of elasticity, is found, so to speak, by touch; one can only make use of certain practical procedures by which these searches require the expenditure of the least possible amount of time.

For the latter purpose one may proceed as follows. With the aid of the axis \(J\) we incline the preparation away from ourselves through the greatest possible angle and, by rotating the inner circle by the segment screws about the axis \(N\), bring the grain under investigation into a position of extinction. Then we turn the stage about the same axis \(J\) toward ourselves into the other possible extreme position and, since in the general case enlightenment of the grain will occur, we attain darkness by inclining the preparation already about the axis \(H\). By this procedure we make the first approximation toward bringing the principal axes of the desired plane of elasticity into a position more or less parallel to the plane of symmetry of the microscope. Sometimes the extinction is obtained more perfectly if, when turning the stage away from oneself, the preparation is inclined about the axis \(H\), and the inner circle is rotated in its plane at the other extreme position of the stage.

Repeating one or the other procedure several times, we finally obtain constant extinction; and this occurs the more quickly, the closer the axes of the ellipse of the section of the thin section are situated to the principal planes of elasticity. Usually 3–4 repetitions are enough to achieve the desired result, and the whole procedure, even with little experience, takes 2–3 minutes, or even less; only in the event that special accuracy of observation is required, or when studying a not very fresh or incorrectly formed mineral, is it necessary to spend a considerably greater amount of time.

§ 19. Methods for determining the completeness of extinction of the preparation.

The accuracy of setting the plane of the axes of elasticity, or in general of setting for darkness, depends on how perfectly the observer’s eye can detect shades of extinction, and in this consists, to a certain degree, a weak aspect of the method. Usually, over the course of...

over time, through practice, the observer attains a certain perfection in this respect, but it is better to resort to some auxiliary measures. Thus, first of all it is better to carry out observations with a strong source of light: for this purpose an ordinary electric lamp may be used. Then various stauroscopes may be employed. Among the latter, the easiest to obtain is a gypsum plate which, between crossed nicols, gives the sensitive violet color of the first order if it is inserted into the aperture of the microscope tube, between the nicols. This sensitive violet color is obtained in the presence of a birefringent mineral only in the case when the latter is completely extinguished. However, not every eye is receptive to this shade, so that the use of a gypsum plate very often does not give reliable results. Stauroscopes based on the principle of addition are of greater help; these include the Brézina plate, the Calderon double plate, the Bertrand plate, the Bravais double plate, Fedorov’s microdichroscope, the Stöber plate, and others. Of these we shall dwell on Stöber’s plate, mentioned in the literature on Fedorov’s method¹. This plate is cut from quartz parallel to its optical axis and consists of two perfectly symmetrical parts, the axes of which form a right angle (Fig. 18). The plate is placed in the focus of an ordinary eyepiece so that the seam is brought into coincidence with the vertical thread of the cross. With such an arrangement the optical axes of both parts of the plate form equal angles with the plane of symmetry of the microscope. Therefore, if the stauroscope is between crossed nicols, both its parts are colored in the same interference color, depending on the thickness of the plate. The same picture is obtained also in the case when the specimen lying on the microscope stage is in the position of complete extinction. But as soon as the specimen leaves this position even slightly, a certain path difference of the rays is produced, which in one part of the stauroscope is increased still more, and in the other is decreased by the same amount, as a result of which a rather considerable difference in the color of these parts is obtained, easily detected by the eye.

When using this instrument it must be remembered that it acts only with the analyzer inserted.

Thus, Stöber’s plate can be of very good help in setting the grain under investigation to darkness, but, unfortunately, it is extremely difficult to prepare it so that both its parts are completely identical; without this condition it already loses its significance.

The “double quartz plate,” recently constructed by F. Wright², also possesses good sensitivity, not dependent

With time and practice the observer attains a certain perfection in this respect, but it is better to resort to some auxiliary measures. Thus, first of all, it is best to make observations with a strong source of light; for this purpose an ordinary electric lamp may be used. Then various stauroscopes may be employed. Among the latter, the easiest to obtain is a gypsum plate, which, between crossed nicols, gives the sensitive violet color of the first order if it is inserted into the opening of the microscope tube located between the nicols. This sensitive violet color is obtained in the presence of a birefringent mineral only in the case when the latter is completely darkened. However, not every eye is receptive to this tint, so that the use of a gypsum plate very often does not give reliable results. Stauroscopes based on the principle of comparison are of greater assistance; these include Brézina’s plate, Calderon’s double plate, Bertrand’s plate, Bravais’ double plate, Fedorov’s microdichroscope, Stöber’s plate, and others. Of these we shall dwell on Stöber’s plate, mentioned in the literature on the Fedorov method¹. This plate is cut from quartz parallel to its optical axis and consists of two perfectly symmetrical parts, the axes of which form a right angle (Fig. 18). The plate is placed in the focus of an ordinary ocular so that the seam coincides with the vertical thread of the cross. With such an arrangement, the optical axes of both parts of the plate form equal angles with the plane of symmetry of the microscope. Therefore, if the stauroscope is between crossed nicols, then both parts of it are colored in the same interference color, depending on the thickness of the plate. The same picture is obtained also in the case when the preparation lying on the microscope stage is in the position of complete extinction. But as soon as the preparation moves even slightly out of this position, a certain path difference of the rays is formed, which in one part of the stauroscope will be increased still further, while in the other it will be decreased by the same amount; as a result a fairly considerable difference in the coloring of these parts is obtained, easily caught by the eye.

When using this instrument it must be remembered that it acts only with an inserted analyzer.

Thus, Stöber’s plate can be of very great help in setting the grain under examination to darkness, but, unfortunately, it is extremely difficult to prepare it so that both its parts are perfectly identical; without this condition it already loses its significance.

The “double quartz plate” recently constructed by F. Wright² also possesses good sensitivity, not depending

at the same time on the precision with which the separate parts of this stavroscope are made. The latter consists of two plates and two wedges made of right- and left-rotating quartz.

The optical axes of the separate parts of the instrument coincide and are normal to the surfaces of the stavroscope, which has the form of an ordinary plate-type instrument of this kind. However incorrectly the individual plates of the stavroscope may be fitted to one another, there will always be a section of the latter in which the thickness of the individual plates of different signs of rotation will be the same. By this section the stavroscope is divided into two parts, which, under crossed nicols, will be colored in the same interference color only in the case when the field of view of the microscope is completely darkened. If tests show that this double plate has no special shortcomings, then its use will render a great service in work on the Fedorov stage.

Be that as it may, when setting the plane of the optic axes one usually has to follow the tints of greatest darkening, which is checked by small rotations about one or the other axis around this mean position. The most reliable results are obtained if the arithmetic mean of several determinations is taken. For this purpose it is necessary to record the coordinates of the plane being found at each determination of its position. This record is made in the margins of the same stereographic net on which the results of the observations are plotted, to which we shall now turn.

§ 20. Drawing the projections of the planes of elasticity.

It was stated above that the plane of section of the thin section is taken as the picture plane of projection, or as the plane of the drawing, and that the position of every plane observed in the mineral is determined by two coordinates. One of these coordinates represents the angle which the straight line common to the given plane and to the plane of the thin section makes with some initial straight line lying in the latter. If as the initial straight line we take the one that coincides with the axis $H$ when the vernier indicates zero on the inner limb, then the reading made on the latter when the plane is set gives directly the indicated coordinate of the latter.

Let us turn to an example. From the entries in the margins of Table VII we see that, in several determinations, the reading on the inner limb for the 1st plane gave the coordinate: $253^\circ$.

To plot this coordinate on the net, we assume that the initial straight line is brought into coincidence with the vertical diameter of the net, and that the zero of the limb corresponds to the lower zero of the latter. In order to find the plane, we had to turn the inner circle from the initial position

to the angle read on the limb clockwise¹; in order to find on the net the corresponding position of the straight line determining this coordinate, one must count off the same number of degrees along the circumference of the latter, of course in the opposite direction, and the point \(D\) obtained on this circumference will be the end, or—as they say—the pole of the above-mentioned straight line, turned toward the observer. Since this straight line lies in the plane of the thin section, i.e. in the plane of the paper, the arc representing in projection the sought plane must rest on the diameter drawn through the point obtained.

The inclination of the arc is determined by the second coordinate, representing the angle of deviation of the specimen from its initial horizontal position when inclined about the axis \(H\). It is, evidently, equal to the angle through which the sought plane had to be rotated about this axis in order to bring it into coincidence with the plane of symmetry of the microscope. This angle finds its real expression in the instrument as the angle formed by the surface of the inner circle, on which the thin section lies, and by the surface of the outer ring \(R\), which in the initial position coincides with the former. Therefore it can be easily and quickly determined by the method of V. V. Nikitin described earlier. For this purpose, having loosened the screw \(r\), we turn the outer ring through \(90^\circ\) to such a side that the dihedral angle is turned with its opening toward the observer. Then we turn the stage away from ourselves and, pushing Bertrand’s lens into the tube, bring the shining edge of the inner circle into coincidence with the horizontal thread of the ocular cross, noting the reading of the vernier on the vertical limb. To bring the other side of the dihedral angle, i.e. the shining edge of the outer ring, into coincidence with the same thread of the cross, is not necessary, since this side remains constant in all measurements, and the reading on the vertical circle corresponding to this coincidence, with the known correction, is a constant of the given instrument. Subtracting the first reading from the last, we obtain the magnitude of the angle of inclination of the sought plane to the plane of symmetry of the microscope.

In the example given, the inclination of the first plane was found by several determinations to be equal to \(24^\circ\). In order to bring this plane into a vertical position, it was necessary to incline the inner circle to the right; consequently, on the net the arc representing the projection of the plane, referred to the initial position of the thin section, must be inclined by the same number of degrees to the left, which is also marked on the fields of the net in the corresponding place.

Above we saw that the coordinate obtained is equal to the magnitude of the dihedral angle formed by the plane of elasticity being set and the plane of symmetry of the microscope. But the dihedral angle is measured

by a linear angle whose plane is perpendicular to the edge of the former angle. In the present case the edge of the dihedral angle is the diameter \(DE\), and the plane perpendicular to it is represented by the diameter \(GH\). Thus, in order to obtain the projection of the side of the linear angle lying in the plane of elasticity being set, one must lay off along the latter diameter from the center of the net \(24^\circ\) to the left, holding the net toward oneself by the end \(D\) of the diameter \(DE\). The point \(C\) thus obtained will lie on the sought arc.

Drawing the arc through the three points \(D\), \(E\), and \(C\), and finding its pole, are carried out according to Problems 1 and 3.

Thus we have found the position of one of the three principal planes of elasticity. Since the third axis of elasticity is perpendicular to this plane, by determining the latter we have thereby also found this third axis: its projection will be the pole of the constructed arc—the point \(f_1\).

It is necessary to note certain special positions of the planes sought. If the plane is normal to the section, i.e. if, in setting it, it is not necessary to incline the specimen about the axis \(H\), then on the net the projection of such a plane will appear as a diameter with a pole lying on the circumference of the net.

Obviously, the two other poles of the axes of elasticity in this case will lie on the diameter that has been drawn and thus can easily be found by readings along the polar parallels.

We cannot observe the other limiting position of the plane—namely, when it coincides with the plane of the section—at all.

If the plane occupies an arbitrary position relative to the section of the section plate, then it is set the more accurately and quickly the less it has to be inclined about the axis \(H\) in order to bring it into coincidence with the plane of symmetry of the microscope. Therefore, if the first plane that has presented itself to us poorly satisfies the latter condition, it is better to leave it for the time being and proceed to the determination of some other plane.

Having obtained, in one way or another, one plane of the axes of elasticity, we proceed to the search for the second plane. This is done in the same way as in finding the first plane, i.e. more or less by feel. True, we have the condition that both planes must be mutually perpendicular, but it is impossible to predict exactly the position of the second plane from the first one available, since we know only one zone of these planes, represented by the axis of elasticity that has been found.

One can only say that, in view of the perpendicularity of the planes, the arcs representing them on the net will rest on diameters, the angle between which approaches a right angle the less the planes themselves are inclined from the position vertical to the section of the section plate. On this basis, after finding the first plane, we turn the inner circle about the axis \(N\) approximately through \(90^\circ\) and then begin to search for the second plane in the manner described above.

In our example (Table VII), for the second plane the coordinates found are: \(336^\circ\) to the right, \(14^\circ\), according to which the projection of this plane is represented on the network by the arc \(KLM\) with its pole at the point \(f_2\).

§ 21. Adjustment of the results of measurements in establishing the planes of elasticity.

Now, using the condition of perpendicularity of the planes, we can verify the correctness of their determination. Indeed, if the planes are perpendicular, then the normal to one plane must lie in the other, and conversely; consequently, in the projection the pole of the first arc must fall on the second arc, and conversely.

In our example this condition proves to be fulfilled, i.e. the pole \(f_1\) lies on the arc \(KLM\) and the pole \(f_2\) is located on the arc \(CDE\). This indicates that the observations were made correctly.

But for the most part such an exact coincidence does not occur.

There are two principal sources of errors arising here: one consists in the incorrect setting of the crystal under investigation to darkness, and the other—in a discrepancy between the refractive index of the mineral and the refractive index of the segments, or in an incorrectly taken correction for the angles of inclination according to the diagram shown in Figure 6. However, as has already been said above, an error in determining the refractive index by several hundredths has almost no effect on the results of the measurements, and within these limits the given quantity can always be established, even if the nature of the mineral is not fully known to us.

Thus the first source of errors is the principal one. We have already seen that, for the best setting to darkness, it is recommended to use certain stauroscopes, but this is not possible everywhere and does not always lead to the goal.

Therefore, the errors arising in setting the planes have a subjective character and will always be encountered.

It follows from this that, in order to adjust the results of measurements, one must resort to the theory of probabilities. But, as Prof. E. S. Fedorov says¹, “in order to attain objective adjustment and the possibility of applying the rules of the theory of probabilities, it is necessary to make a large number of repeated observations, from the results obtained to determine separately for each of the coordinates the weight of the individual observations, and from the whole in aggregate to derive the mean most probable position and, finally, to determine the magnitude of the probable error, again for each coordinate separately. But it is clear that if such exceptional accuracy is required, then it is not even worth using the present method, but one must turn to more exact methods, although this is applicable only in exceptional cases.”

Thus, the correction of errors can, for the most part, be carried out only by a subjective method. The principal basis for bringing the measurements into agreement is the accuracy with which one or another coordinate is found, as determined by the character of the darkening when rotated about the corresponding axis. Complete darkening is attained the more easily—and consequently the coordinate will be found the more accurately—the smaller the limits within which a more or less identical degree of extinction is observed.

Sometimes it is enough to turn the preparation through a very small angle in order to bring it out of the position of extinction; at another time, rotation through as much as \(10^\circ\) produces no noticeable change in its relation to polarized light. It is clear that the coordinate determined in the first case will be far more accurate than in the latter.

Prof. E. S. Fedorov¹ recommends marking the accuracy of the coordinates on the net in the following manner. The mark must be referred to the pole of that arc whose coordinates are under consideration.

In the case of equal sensitivity of both coordinates, a simple thin cross is placed at the corresponding point on the net, one branch of which coincides with the radius of the net, while the other is perpendicular to the latter. If the degree of accuracy in determining the coordinates is different, this is indicated by unequal thickness of the branches of the cross; moreover, we shall assign the line along the radius, for example, to the coordinate determined by rotation about the axis \(N\), and the other line—to the inclination about the axis \(H\). If both coordinates are not sufficiently distinct, then instead of a cross we put a small circle. With such notation, a mere glance at the net shows what can probably be changed more strongly and what more weakly.

The actual adjustment is performed in very different ways, depending on those combinations in the accuracy of determining the coordinates that may arise. Since in all these operations there is much that is subjective, it is impossible to set forth exactly, or even to enumerate, the procedures which must be used in this case.

Below follows one example of such an adjustment of measurement results. In accordance with what has been said above, let us suppose that in Figure 19 the thick line of the cross refers to an insufficiently well determined coordinate. Obviously, if we slightly increase the inclination of the arc having the point \(f_1\) as its pole, and rotate the arc with pole \(f_2\) in the plane of the drawing clockwise, then, for a certain ratio of the corrections, the poles will lie on the arcs. It remains then to place the preparation in the positions corresponding to the corrected coordinates and to test the grain under investigation for completeness of extinction.

By combining the corrections—which, with some practice, does not require much time—we finally arrive at perpendicularity of the planes set for complete darkness.

Thus, we have found the second plane of the axes of elasticity and the axis \(f_2\) perpendicular to it. There is no need to seek the third plane, since it is very easy to construct it on the net, and, moreover, it is not always determined with sufficient accuracy. Indeed, first of all we seek those planes which deviate least from the direction perpendicular to the cleft of the section; therefore the third plane will be closer to the horizontal, and, if the angle of inclination of the specimen about the axis \(H\) when it is set exceeds \(60^\circ\), one cannot vouch for the accuracy of the determination of this plane.

But what interests us, properly speaking, is not the planes themselves, but their poles, or the axes of elasticity; therefore we shall proceed at once to finding the third axis.

Since each plane is determined by two axes of elasticity, the desired axis must lie both in the first and in the second of the planes found, i.e. it must represent the line of their intersection. Thus, on the net the third axis \(f_3\) is represented by the point of intersection of the arcs drawn. From the axis thus found, as the pole of the third plane of elasticity, one may also construct the latter; but for the subsequent constructions this is unnecessary.

The above-described method of checking the correctness of the finding of the planes and of determining the third axis of elasticity can be shortened still further in the following way. Having plotted on the net the first plane and measured the coordinates of the second plane, we then immediately find the diameter on which the pole of the latter must lie, and plot it. With correct setting it must fall on the arc representing the first plane. Such a check can also be made by eye, without encumbering the diagram with superfluous lines.

Having set precisely the first plane and the pole of the second (Fig. 20), we find the third axis of elasticity, which together with the second axis must lie in the first plane of elasticity. Since these latter axes are mutually perpendicular, the determination of the required axis reduces to solving Problem 2 (§ 8): on the arc \(AB\), lay off \(90^\circ\) from the point \(f_2\).

For this it is necessary, from the pole of the arc \(f_1\), to project the point \(f_2\) onto the circumference of the net, count off \(90^\circ\) on the latter, and connect the resulting point \(F_3\) with the pole by a straight line, which at its intersection with the arc will give the projection of the desired axis.

§ 22. Determination of the position of the plane of the optical axes.

Thus, we have found the position of all three axes of elasticity. In order to determine their relative magnitudes, it is first of all necessary to proceed to an examination of the planes of elasticity found, to see whether any of them contains the optical axes.

The latter, as is known, are characterized by the fact that in their direction the rays of light do not undergo double refraction.

Thus, if some optical axis is brought into coincidence with the axis of the microscope, then, with crossed nicols, extinction occurs, which is preserved when the stage of the microscope is rotated.

On this basis the test is carried out in the following manner. Suppose that some plane of elasticity has been brought into coincidence with the plane of symmetry of the microscope. This plane, in the given position, of course contains the axis of the latter, while the axis of elasticity perpendicular to it coincides with the axis of the stage \(J\). Since, when the stage is rotated about this axis, extinction is observed all the time, and there is no possibility of marking the position of the optical axes, if such exist, we bring our plane out of the position of extinction, but in such a way that the axis of the microscope remains in it; for this purpose, having loosened the clamping screw of the microscope, we turn the stage of the latter approximately by \(45^\circ\) counterclockwise. If now the instrument is rotated about the axis \(J\), which in this position remains perpendicular to the plane being tested, then all directions lying in the latter will successively be brought into coincidence with the axis of the microscope—and among them, of course, the optical axes, whose position will be marked by the ensuing extinction.

Everything that has been said is illustrated schematically in Figure 21. Since, when rotating about the axis \(J\), the angle of view is \(140^\circ\), sometimes only one optical axis can be brought into coincidence with the axis of the microscope; the other axis, lying outside the field of view, will be indicated by the position of the interference colors when approaching it.

The position of the optical axis found may be marked on the vertical limb, but this will be discussed below.

§ 23. Determination of the axes of elasticity.

The test just described of the plane of the axes of elasticity for the presence in it of optical axes must in fact be carried out at once, every time each plane is set.

Returning to our example, we must note that in the first plane of elasticity found by us there proved to be no optical axes.

In the second plane, however, extinction is observed at two positions corresponding to the coincidence of one and the other optical axis with the axis of the microscope. Consequently, the second plane of elasticity is at the same time also the plane of the optical axes, and the axis perpendicular to it is the mean axis of elasticity; this we mark on the net, denoting the point \(f_2\) by \(n_m\).

In order to learn the relative magnitude of the two other axes and thereby determine their names, we resort to the aid of so-called comparators, i.e. plates of a definite mineral with a known arrangement of the axes of elasticity.

If such a plate is placed upon the grain under investigation so that the axes of elasticity of both mineral species are parallel, then two cases may arise.

If the relation of the parallel axes is the same, i.e., the longer axis of the plate is parallel to the longer axis of the grain under investigation, and so on, then the superposition of the plate is equivalent to an increase in the thickness of the preparation, which will be revealed by an intensification of the interference colors.

With the opposite relation of the parallel axes, these colors are weakened, since the rays of light which have acquired, in passing through the mineral under investigation, a certain difference of path begin, in the plate, to change their velocities in the opposite relation.

The comparator plate usually has an elongated rectangular form. It is arranged, for the most part, in such a way that the principal axes of elasticity of the section of the comparator coincide with the sides of the rectangle, the longer axis of elasticity being directed along the short side, and the shorter axis of elasticity along the long side.

In determining the majority of rock-forming minerals with slight birefringence—for example, the feldspars—in thin sections of thickness \(0.02\)—\(0.03\) mm, a mica plate of \(1/4\,\lambda\) is usually used, which, in the case of the opposite relation of the axes, weakens the light effects to the point of more or less complete extinction.

If, however, the preparation has a more considerable thickness, as is obtained artificially at large inclinations about the axes \(H\) and \(J\), or if a mineral with high birefringence is being investigated, then the action of the mica plate proves insufficient, and in that case a mica or quartz wedge is used. By pushing the latter into the slot of the tube, i.e., by superposing upon the mineral thicker and thicker parts of this wedge, we shall observe a gradual rise or fall of the interference colors in accordance with the relation of the axes.

Here it will be appropriate to give a list of the interference colors in the order of their intensification.

Black, gray, white, orange, red, blue, green, yellow, orange, indigo, and so on.

Let us suppose that the original interference coloration of the mineral was blue. If, upon inserting the wedge, the parallel axes are in the opposite relation, then the colors red, orange, white appear in succession, and extinction occurs; when like-named axes coincide, shades of green, yellow, orange, and so on will follow. Of course, only the principal colors are given here; in reality they are connected by gradual transitions.

Comparators should be used for solving the problem posed in the following manner.

Since, when the comparator is pushed into the slot of the microscope tube, its axes are situated in a plane perpendicular to the axis of the latter, the axes of the mineral under investigation must first of all be brought into that same plane.

But, as is known, the plane of the net coincides with it. Turning therefore to our diagram, we see that it is easiest of all to bring into the indicated position the axes whose poles are the points \(f_2\) and \(f_1\); when the latter lie on this plane, the third axis \(f_3\) will coincide with the axis of the microscope.

To achieve this we shall take any one of the planes found. It is most convenient to turn to the plane closest to the vertical position; in our case this condition is satisfied by the second plane. Let us bring the chosen plane into coincidence with the plane of symmetry of the microscope, which we have already done. If this position is regarded as the initial one, then on the net the plane thus set will be expressed by the diameter \(KL\), and the point \(f_3\) will occupy the position of some point \(f_3^1\).

Since, with the given setting of the plane, the straight line perpendicular to it coincides with the axis of the stage \(J\), then, as is clear from the diagram, it is necessary to turn the whole instrument about this axis toward oneself through the angle \(f_3^1O\), so that our third axis of elasticity coincides with the axis of the microscope. The angle \(f_3^1O\) is evidently equal to the angle \(f_3M\), and the latter is determined according to Problem 3. For this, from the point \(f_2\), as the pole of our plane, we draw straight lines through the points \(M\) and \(f_3\), and at the intersection of these straight lines with the circumference of the net we find the magnitude of the required angle \(F_3P = 25^\circ\).

The determination of this angle may, however, also be made by eye, since an error of several degrees will in general not affect the result of the further investigation.

Thus, we turn the stage about the axis \(J\) toward ourselves by \(25^\circ\), fixing it in this position with the screw \(i\).

Now the axis \(f_3\) is aligned with the axis of the microscope, the axes \(f_1\) and \(f_2\) lie in the horizontal plane, with the axis \(f_1\) located in the plane of symmetry of the microscope, and the axis \(f_2\) coinciding with the axis of the stage \(J\).

Since the comparator is inserted into the tube of the microscope at an angle of \(45^\circ\) to the plane of symmetry of the latter, in order that the axes of elasticity of the comparator and of our mineral become parallel, the stage of the microscope must also be turned by \(45^\circ\). If the turn is made counterclockwise, then the axis \(f_1\) will coincide with the smaller axis of the comparator, etc. Observing in the tube of the microscope, we notice that, as the comparator is inserted, a weakening of the interference colors occurs. Therefore the axes of elasticity of the comparator and of the mineral are in the opposite relation, i.e. the axis \(f_1\) is greater than the axis \(f_2\).

But since the latter, as we have already determined, is \(n_m\), it follows that the axis \(f_1\) must be called \(n_g\), and the axis \(f_3\), of course, \(n_p\), which we also mark on the net.

If the microscope stage is turned by \(45^\circ\) clockwise, then there will be the opposite relation in the arrangement of the axes and in the color phenomena.

Everything stated concerning the precise determination of the axes of elasticity may be schematically formulated as follows:

The microscope stage is rotated	Interference colors	The horizontal axis lying in the established plane
counter- clockwise	weaken	the larger
counter- clockwise	intensify	the smaller
clockwise	weaken	the smaller
clockwise	intensify	the larger

§ 24. Determination of the position of crystallographic planes.

Rock-forming minerals rarely have properly and clearly developed boundary elements, and when examined in thin sections these elements are almost completely lost. That is why the determination of minerals in sections of rocks is carried out on the Fedorov stage almost exclusively on the basis of observations of the optical elements of the mineral. But in some cases it is nevertheless possible to make a more or less precise setting of known crystallographic planes, chiefly cleavage cracks, which occur in almost every grain of a mineral possessing some perfect cleavage. The latter, like the plane of the axes of elasticity, is brought into coincidence with the plane of symmetry of the microscope.

For this purpose the inner circle is rotated about the axis $N$, until the trace of the crack coincides with the vertical thread of the ocular cross, lying in the plane of symmetry of the microscope. If the preparation is then inclined about the axis $H$, passing through this trace, then the plane being set will rotate about a straight line lying in the plane of symmetry of the microscope, and therefore, at a certain angle of inclination, can be brought into coincidence with the latter.

During this rotation the walls of the crack, being projected at various angles, will appear as indistinct bands of variable width $a$, as is seen in Figure 22. The required coincidence will evidently occur when the walls of the crack are projected as straight lines, and the crack itself becomes sharpest and thinnest. For the exact finding of this position it is not infrequently necessary to use stronger objectives, but if the section

sufficiently thick, and the cleavage is expressed distinctly, it is better to leave the objective No. 0 ordinarily used, since with a shorter focus of the objective the different parts of the walls of the crack will not be equally well visible.

Having set the plane, one must, as usual, take the index reading on the inner circle of the stage and find the angle of inclination about the axis \(H\), having, of course, first replaced the stronger objective with No. 0. The recording on the margins of the net and the drawing of the projections of the plane are done in the ordinary order.

For accuracy it is necessary to make several determinations, each time bringing the cleavage crack into the position of greatest sharpness now on one side, now on the other side of the latter, and then to take the arithmetical mean of the readings on the vertical circle.

The method adopted by us for determining the angle of inclination about the axis \(H\) may in the present case be simplified in the following way. The cleavage trace is aligned not with the vertical, but with the horizontal thread of the ocular cross, and the cleavage crack is brought into the plumb position by rotation about the axis \(J\). The required angle is read directly on the vertical circle, since the inclination about the axis \(H\) is here replaced by an inclination about the axis \(J\).

It remains now to find the index reading and the direction of inclination of the specimen corresponding to the coincidence of the given cleavage plane with the plane of symmetry of the microscope. This can be done directly, but it is easy also to obtain the necessary data mentally, if one bears in mind that the index reading will differ by \(90^\circ\) from the reading which corresponds to the proper position of the specimen. One may give the following practical rule for finding these marks: if for the indicated coincidence it is desirable to turn the inner circle clockwise, then \(90^\circ\) must be added to the original index reading, and the supposed inclination about the axis \(H\) will be counted to the right if the stage was turned about the axis \(J\) toward oneself,—and conversely.

However, in each particular case it is easy to consider how to pass to these marks.

It is much more rarely possible to find the position of some face of a crystal in thin sections of rocks.

Bringing a face into the plumb position is generally achieved with much greater difficulty and imperfectly, since when the stage is rotated about one or the other axis, the face being set, having the same significance as the wall of a cleavage crack, is projected sometimes through the layer of Canada balsam or an adjacent mineral, sometimes through the thickness of the crystal being examined, whereby finding the mean position of this face is greatly hindered, and in densely colored minerals becomes impossible.

To facilitate the setting of faces it is useful to apply the following: if the illuminating apparatus is diaphragmed or lowered,

then the outlines of the grains of the thin section become much sharper. If, moreover, the light is at least somewhat convergent—which, as we have seen, is generally characteristic of the Fedorov stage—then, when the face is in a vertical position, the latter is separated by a bright band on one side or the other, depending on the difference in the refractive indices of the given mineral and of the surrounding medium¹.

In measuring the angles of inclination corresponding to the setting of some crystallographic plane, it is necessary to pay attention to how the refractive index affects the magnitude of these angles. In resolving this question several particular cases may present themselves.

1) If the walls of the cleavage crack are not distinguishable, and the latter appears as if it were a simple plane, then, evidently, the refractive index of the mineral whose grain contains this cleavage will exert its influence on the magnitude of the measured angles, and the latter must be corrected according to the diagram of figure 6.

2) If, however, the cleavage appears in the form of a gaping crack, then, when the latter is brought into coincidence with the plane of symmetry of the microscope, the rays of light will pass between the walls of the crack without refraction, the walls themselves being marked by bright bands. In this case, of course, the measured value of the angle is the true one, and no corrections need be made.

3) The determination of a crystal face represents a special case of the setting just mentioned.

It may seem indefinite when a cleavage crack is to be regarded as gaping, since, properly speaking, every crack is such.

Here everything depends on the magnification at which this crack is examined. One and the same crack at low magnification may appear as a simple plane, or as a lamellar inclusion, whereas with strong objectives there are already discerned in it walls along which, with the proper setting, sharp, bright bands appear. Obviously, the angle of inclination will be different in the one case and in the other.

The accuracy of setting crystallographic planes is in general less than that of planes of elasticity, and depends on the perfection of the cleavage or on the development of the face. It is comparatively rare that such a setting can be made with an accuracy up to \(1/2^\circ\); usually the latter fluctuates from \(1^\circ\) to \(2^\circ\).

§ 25. Determination of the position of a direction in a mineral.

Up to now we have considered chiefly the determination of the position of some plane observed in the mineral under investigation, and from it have then found one or another direction.

Nevertheless, some directions cannot be determined otherwise: thus, the axes of elasticity themselves are not determinable, since Fedorov’s stage does not in general allow convergent light to be used. But sometimes it is necessary to find certain directions directly—for example, the optical axes.

Let us consider, as an example, precisely the case of the direct finding of the optical axes. For this purpose the microscope stage is turned to \(45^\circ\), and in this position, by rotating the specimen about the axes \(N\) and \(J\), the grain of the mineral under investigation is brought into a state of complete extinction. If the extinction is then preserved upon rotation about the axis of the microscope, this means that one optical axis coincides with the latter.

The coordinates necessary for plotting the axis thus found on the net are determined in the following way.

One may assume that, with the given setting, the optical axis is first brought into the vertical plane perpendicular to the axis \(J\), and then, by rotation about the latter, is made to coincide with the axis of the microscope. But this vertical plane is the meridional plane; and since it is set perpendicular to the axis \(J\), and in the initial position of the stage would coincide with the plane of symmetry of the microscope, the reading of the index on the inner limb of the stage determines the first coordinate.

The angle of inclination of the specimen about the axis \(J\) is read at once on the vertical circle and represents the second coordinate. Both coordinates are marked on the margins of the net, the inclination about the axis \(J\) being counted in the direction opposite to that observed in reality. Thus, in our example (Table VII), for the optical axis \(A_2\) found in this way, the coordinates will be as follows: \(40^\circ\), toward oneself \(15^\circ\).

From these coordinates we find the projection of the optical axis. The vertical auxiliary plane in which this axis lies will be represented by a diameter passing through the corresponding division of the circumference of the net. Turning the latter toward oneself by that end of the diameter which coincides with this division, we lay off from the center of the net along the polar parallels \(15^\circ\), and at the point \(A_2\) find the projection of the desired optical axis.

When any other directions are found, only the method of establishing them is changed, but the coordinates are determined in an analogous way.

Much more often one has to deal with the inverse problem to the one just described: by means of various constructions on the net, to find a known direction, and then, from the read-off coordinates of this direction, to determine its position in the mineral under investigation.

Most often, as we shall see below, this procedure is applied to checking the twin axis found by construction; but for illustration we shall use our example and suppose that an optical axis \(A_1\) has in some way been found and must be checked.

The diameter drawn through the point \(A_1\) represents the projection of the vertical auxiliary plane, which we then align on the stage with the plane of symmetry of the microscope, placing the index opposite the division of the inner circle corresponding to the division of the circumference of the net at some end of this diameter, for example, \(146^\circ_5\). Having counted off along the parallels the number of degrees by which the point \(A_1\) lies away from the center of the net, we incline the specimen about the axis \(J\) by \(57^\circ_5\) away from ourselves, i.e. in the direction opposite to that observed on the net, and in this way bring our optical axis into coincidence with the axis of the microscope; if the latter has been correctly determined, this will be expressed in the preservation of extinction of the mineral when the microscope stage is rotated.

It must be said that, predominantly, the optical and twin axes are those directions which are subjected to independent determination on the Fedorov stage, but even this is comparatively rare; for the most part one has to deal with various planes.

PART TWO.

Determination of the optical constants of rock-forming minerals on the Fedorov stage.

In the first part the general methods of universal-optical investigation in parallel light were set forth. These methods make it possible to establish the position of any plane and direction observed in the crystal under investigation, and to represent them in stereographic projection.

By combining these elements of the mineral in a known manner and determining their mutual relation, we find the so-called constants—quantities characterizing one or another mineral species. Among the whole series of such constants, the greatest importance for determining rock-forming minerals in thin sections is possessed by the type of syngony, the optical sign, the refractive index, the magnitude of birefringence, the angle between the optical axes, and the character of twinning.

The universal stage makes it possible, with a sufficient degree of accuracy, to find almost all these constants, with the exception of the refractive index and, in part, the magnitude of birefringence. In some rather rare cases it appears possible also to determine the crystallographic elements of the mineral and thus to find the extinction angle.

Below only the optical constants enumerated here will be considered.

Chapter I.

Determination of the optical constants expressing the properties of minerals as physical bodies.

§ 26. Determination of the type of syngony¹.

Let us consider first of all how isotropic, uniaxial, and biaxial minerals are distinguished.

It must be said that, when working with an ordinary microscope, sometimes

it is almost impossible to determine to which of these three principal, optically considered, groups of minerals the grain under examination belongs.

On the Fedorov stage this question is resolved at once, and only the mounting of the preparation requires the expenditure of a certain amount of time.

In isotropic substances and in crystals of the cubic system, there is no double refraction in any section; therefore, when the stage is rotated about any axis, the extinction of the preparation between crossed Nicols must be preserved.

In uniaxial minerals all principal sections passing through the optical axis are planes of symmetry of the ellipsoid of elasticity. Therefore, if such a section is brought into coincidence with the plane of symmetry of the microscope, the mineral remains extinct when the stage is rotated about the axis \(J\). But however the plate of a uniaxial mineral may have been cut, there will always be one principal section perpendicular to the plane of the thin section, and the just-mentioned setting of such a plane to darkness will be obtained without tilting the preparation about the axis \(H\). Obviously, the single optical axis must lie in this plane; it is easy to ascertain this if it is at all accessible to observation, for which, as usual, it is necessary to turn the microscope stage through \(45^\circ\) and, by rotating about the axis \(J\), to seek the extinction of the mineral corresponding to the coincidence of the optical axis with the axis of the microscope. If the optical axis lies outside the limits of the field of view, its presence will be indicated by a lowering of the interference colors at the extreme positions of the stage.

Thus, what is characteristic of a uniaxial mineral is the circumstance that, of the two (properly speaking, four) positions of extinction observed when the axes \(N\) and \(M\) are brought into coincidence, one is preserved when the stage is rotated about the axis \(J\), without inclination about the axis \(H\).

But a similar case may also occur in biaxial minerals, when the plate has been cut perpendicular to some plane of the axes of elasticity. To resolve such a case one may proceed in the following manner. We incline the preparation more or less steeply about the axis \(H\), whereby in general the grains become bright; then we place the stage in one of the extreme positions by rotation about the axis \(J\), and bring the grain into the nearest position of extinction by turning the preparation in its plane about the axis \(N\). Obviously, in the case of a uniaxial mineral, a principal section of the crystal again coincides with the plane of symmetry of the microscope, and the extinction will be preserved when rotating about the axis \(J\). This operation may be repeated for various inclinations about the axis \(H\) with one and the same result. In a biaxial mineral, however, cut perpendicular to a plane of the axes of elasticity, a secondary setting to darkness is possible, upon inclination about the axis \(H\), only when we turn the inner circle through \(90^\circ\) from the indicated position, i.e. when

combine with the plane of symmetry of the microscope the second plane of the axes of elasticity. If, however, the plate of a biaxial mineral has an arbitrary section, then setting the plane to darkness without inclination about the axis \(H\) is in general impossible.

In accordance with what has been said, the distinction between uniaxial and biaxial crystals is made as follows.

We set the inner circle in its initial position, bring the plate into the position of darkness, and rotate it about the axis \(J\). If illumination appears, then we turn the inner circle by \(90^\circ\) about the axis \(N\) to the second extinction and again rotate about the axis \(J\). If illumination again appears, then the mineral is undoubtedly biaxial; but if darkness is preserved, then it may generally be assumed that the crystal is uniaxial, and it is only necessary to carry out the above-described verification by inclining about the axis \(H\).

As regards determining the kind of syngony of a mineral, this is generally possible comparatively rarely, since rock-forming minerals for the most part do not have regular elements of delimitation. The belonging of a mineral to the cubic syngony is most easily determined in view of its isotropy. Among uniaxial minerals, a tetragonal crystal can be distinguished from a hexagonal one only in the case where outlines of faces or traces of cleavage planes along the principal axis are present, and if it is possible to bring the optical axis of the mineral into coincidence with the axis of the microscope. With this coincidence, corresponding to a section of the mineral perpendicular to the principal axis, the external faces of a tetragonal crystal form angles of \(90^\circ\) and \(45^\circ\) with one another, while in hexagonal crystals the angles are \(60^\circ\) and \(120^\circ\).

To distinguish the kinds of syngony in biaxial crystals, it is necessary to bring some axis of elasticity into coincidence with the axis of the microscope.

In the case of the rhombic syngony, all three axes coincide with the crystallographic axes; therefore, whichever axis of elasticity is brought into coincidence with the axis of the microscope, upon rotation about the latter there will be observed straight or symmetrical extinction with respect to the outlines of the faces of the crystal.

In the monoclinic syngony, only one of the axes of elasticity coincides with a crystallographic axis, and symmetrical or straight extinction is observed only when this axis is brought into coincidence with the axis of the stage \(J\). Finally, in the case of the triclinic syngony, none of the axes of elasticity coincides with the crystallographic axes, and straight extinction is not observed.

Determination of the angle between the optical axes and of the optical sign of a biaxial mineral.

§ 27. a) With an arbitrary section of the thin section.

To carry out this task, it is necessary first to find the position of all three axes of elasticity of the mineral. Already in the determination

of the relative magnitude of the latter we had to resort to testing each plane of elasticity for the presence in it of optical axes. For this, as we have seen, after setting the plane, it is necessary to turn the microscope stage through \(45^\circ\) and observe whether extinction of the grain under study occurs in some position when the instrument is rotated about the axis \(J\): the mineral is extinguished when its optical axis coincides with the axis of the microscope.

This coincidence we can fix by readings on the vertical circle, and from the latter derive the value of the angle between the optical axes.

Let us suppose that the plane containing these axes has been found by us. Depending on the position of the section of the thin section relative to the ellipsoid of elasticity, the optical axes are observed either one by one or on both sides of the initial position of the zero of the vernier of the vertical circle; or only one axis may be observed. When the optical axes lie on both sides of the perpendicular to the section of the thin section, coinciding at the initial position of the stage with the axis of the microscope, then the angle between them is evidently equal to the sum of the readings (Fig. 21), i.e.

\[ 2V = \alpha_1 + \alpha_2 \]

If \(\alpha_2 > 68^\circ\), then the optical axis \(A_2\) cannot be determined, and one has to confine oneself to observing a single axis \(A_1\). In this case the angle \(2V\) is found indirectly: since the axes of elasticity are the bisectors of the angles between the optical axes, each of the latter is equal to twice the angle measured between the corresponding axis of elasticity and the given optical axis. As for the name of the optical axis (\(A\) or \(B\)), in the general case, when the crystallographic elements of the mineral are unknown, it cannot be determined at all, which, however, is of almost no significance for us; we shall call the axes indifferently \(A_1\) and \(A_2\).

In determining the angle \(2V\), errors can very easily be made; therefore the corresponding readings must be taken especially carefully. The chief cause of errors is that near these axes the optical properties of the mineral change rather slowly, so that the greatest extinction is in general caught with difficulty. To obtain exact results, the readings must be made several times while fixing one and the same axis, bringing the grain under study into the position of apparent greatest extinction first on one side and then on the other side of the latter, and then taking the arithmetic mean of the readings.

In the case of strong dispersion of the axes, the observation must be made in monochromatic light, since in ordinary light the optical axes for each wavelength occupy different positions, and complete extinction cannot be achieved.

It will be appropriate to note here that monochromatic light can easily be obtained by means of a light filter inserted into the recess of the microscope stage, where the illuminating apparatus is placed.

But even in monochromatic light the darkening becomes ever less perfect as the birefringence increases, owing to the strongly manifested conical refraction, in which the ray of light, on leaving the crystal, acquires the properties of an ordinary ray with vibrations in every possible direction, and therefore is not extinguished in the analyzer. For the vast majority of natural minerals the influence of conical refraction is quite negligible.

It is of interest to note here a method for finding the position of the optical axes in minerals with very high birefringence and, consequently, sharply expressed conical refraction. This method was proposed by A. Boldyrev¹ and consists in the following: in the usual manner one finds the plane of elasticity that contains the optical axes; since there should be illumination near the latter, the stage is left in the ordinary position, without being turned through 45°, and the preparation is inclined about the axis \(J\) to the position of greatest illumination, corresponding to the coincidence of one of the optical axes with the axis of the microscope. These observations are distinguished by the same sensitivity as in ordinary cases.

Having found in this or another way the position of the optical axes and having corrected the angles of inclination according to the known diagram, we pass to a graphical representation of the data obtained.

If in the last figure we imagine the observer to be on the left, then, in order to bring the optical axis \(A_1\) into coincidence with the axis of the microscope, he must turn the stage about the axis \(J\) through the angle \(\alpha_1\) toward himself. But since on the net the elements of the mineral are plotted in the position in which they appear in the initial position of the preparation, whose section is taken as the plane of projection of the picture, in the latter the optical axis \(A_1\) must be deflected from the corresponding direction by the same angle away from himself, i.e. on the net the angles are laid off, as in other cases, in the direction opposite to that in which they are read on the stage itself.

In this corrected form the results are also marked on the margins of the net. Thus, for example, from the data of Table VII for the second plane containing the optical axes, the marks for the latter are as follows:

\[ \begin{aligned} A_1) \quad & \ldots\ldots\ldots\ldots && \text{away from himself: } 56^\circ \\ A_2) \quad & \ldots\ldots\ldots\ldots && \text{toward himself: } 6^\circ . \end{aligned} \]

It remains now to find the direction from which these angles must be laid off. This direction, obviously, coincides with the axis of the microscope if, when our plane of elasticity is set, the vertical circle is in the original position. Therefore the required direction also lies in the meridional plane, perpendicular to the diameter \(KL\), and is projected at the point \(M\). In order to lay off from this

the required angles from point to point; we project it from the pole of the arc onto the circumference of the net at the point \(P\). Before setting off the angles on this circumference, we turn the net toward ourselves so that the division corresponding to the index for the given plane is turned toward us, and then we count off \(56^\circ\) away from ourselves and \(6^\circ\) toward ourselves. The points \(A^1_1\) and \(A^1_2\) thus obtained are again joined with the pole by straight lines, and at their intersection with the arc we find the projections of the optic axes \(A_1\) and \(A_2\). Often there is no need to transfer the projections of these axes to the arc, since we are for the most part interested not in the axes themselves, but in the angles which they form with each other and with other elements of the crystal; for this it is sufficient to mark the projections of the optic axes on the circumference of the net, along which the various readings are made.

In our example both optic axes are known, and therefore the angle \(2V\) is determined as the sum of the readings on the vertical circle; but on the net one may verify the correctness of the measurements made. Indeed, the axes of elasticity lying between the optic axes must bisect the angles between the latter.

Projecting the axis \(n_p\) onto the circumference of the net, we find that the point \(P_3\) precisely bisects the arc \(A^1_1 A^1_2\).

In the case when only one optic axis is known, the angle \(2V\), as was said above, is equal to twice the angle between the given optic axis and the nearest axis of elasticity. Since there is no direct check here, it is necessary to determine the position of the optic axis with particular care—all the more so because, in calculating the angle \(2V\), the error made in the measurement is doubled.

As for the sign of a biaxial mineral, it is at once determined on the net. For this it is only necessary to see which axis of elasticity lies in the acute angle between the optic axes: if it is the axis of least elasticity \(n_p\), then the mineral will be negative; the mineral is considered positive if the acute bisectrix is the axis \(n_g\). The optical sign of the mineral is placed before the numerical value of the acute angle \(2V\). In our example the angle between the optic axes should be denoted as follows:

\[ 2V = -62^\circ. \]

§ 28. c) In a section of a thin section close to the plane—\(n_g n_p\).

In determining the angle between the optic axes we have so far assumed that at least one of them can be brought into coincidence with the axis of the microscope. But sometimes the section of the thin section is more or less close to the plane of the optic axes, and then the latter cannot be observed directly. For this case also E. S. Fedorov gave a method¹ for finding the position of the optic axes and the angle between them.

For this purpose, first of all we bring into coincidence with the axis of the microscope the axis of elasticity \(n_m\), which is very convenient to do, since the latter, owing to the special position of the plane \(n_g n_p\), is close to vertical; in making this coincidence it is not necessary to touch the axis \(J\).

Then, rotating the stage about the axis \(M\), we place the preparation in such a position that one of the optical axes coincides with the principal section of the polarizer. Of course, we do not know exactly where such an axis is located, since it is precisely this that we have to find; but by certain indications, for example by the extinction angle, we can form an idea of its approximate position in the crystal of the mineral, the nature of which should be somewhat known to us.

For convenience in the further exposition, let us represent what has been said schematically in stereographic projection (Fig. 23). Let \(OC\) represent the position we have adopted for the optical axis, lying in the plane of symmetry of the microscope. We shall rotate the preparation about the axis of the microscope until complete extinction, which will occur when one of the principal axes of elasticity lying in the horizontal plane \(n_g n_p\) enters the plane of symmetry of the microscope. Having obtained the extinction angle, we lay it off on the circumference of the grid in the direction opposite to the direction of rotation of the stage, and thus find one of these axes, for example \(n_g\). Since the latter is the bisector of the angle \(2V\), the position of the second optical axis \(OA\) can at once be marked, found by laying off the arc \(A n_g = n_g C\). We shall then deflect the preparation from the given position by a greater or lesser considerable angle \(\gamma\) about the axis \(J\) toward ourselves. Suppose that, in doing so, the plane of the optical axes takes the position of the arc \(JcaJ\) with the pole \(p\). Then the new section of the mineral will coincide with the horizontal plane; it will obviously still be represented by the fundamental circle of projection.

Now we can verify the correctness of our assumption concerning the position of the optical axes in the crystal. The verification is based on the following.

According to the well-known theorem of Fresnel, the axes of the ellipse of elasticity of any section of a mineral bisect the angles formed by the traces, on the plane of this section, of the planes perpendicular to the latter and passing through the optical axes of the mineral.

In our example, after deflection of the preparation, the projections of the optical axes will occupy the positions of the points \(a\) and \(c\), as is easy to find according to problem 2 (§ 8), while the above-mentioned planes, perpendicular to the given section of the mineral, will be represented by diameters passing through these points.

Thus the angle that has just been discussed will be equal to the angle \(COA^1\), and the axis of the ellipse of elasticity of the section, which is the bisector of this angle, will be represented by the straight line \(On^1\).

If the optical axes were originally placed by us correctly—

apparently, the extinction of the mineral in the new section will occur when this straight line coincides with the plane of symmetry of the microscope, which we can also test by rotating the entire instrument in the appropriate manner about the axis of the latter. Ordinarily, of course, this almost never happens. If the observed extinction angle proves greater than the expected one, this means that the optical axis does not coincide with \(OC\), but occupies some position \(OC'\); in the opposite case it will lie to the right of the vertical diameter. Knowing the direction of the correction, one need only subject the preparation to a series of small rotations about the axis of the microscope in order, after the constructions described, to arrive at such a position where the observed and expected extinction angles coincide, and one optical axis will actually lie along \(OC\), and the angle \(2V\) will be equal to the angle \(COA\).

As an example, let us take the following observation on oligoclase, given by E. S. Fedorov:

Reading of the horizontal limb.	Extinction angle in the horizontal position.	Extinction angle at an inclination of \(60^\circ\), obtained by observation.	Extinction angle at an inclination of \(60^\circ\), obtained by construction.
\(5^\circ\)	\(46^\circ\)	\(44^\circ\)	\(45\frac{1}{4}^\circ\)
\(6^\circ\)	\(47^\circ\)	\(46\frac{1}{2}^\circ\)	\(46^\circ\)
\(7^\circ\)	\(48^\circ\)	\(47\frac{1}{2}^\circ\)	\(46\frac{1}{2}^\circ\)

From these numbers we see that agreement between the angles observed and those obtained by construction occurred between the readings of the horizontal limb \(5^\circ\) and \(6^\circ\); to this intermediate position there corresponds an extinction angle of about \(46\frac{1}{2}^\circ\); consequently, the angle between the optical axes is

\[ 2V = 46\frac{1}{2}^\circ \times 2 = 93^\circ. \]

Of course, this method of determining the angle between the optical axes is very painstaking, so that in petrographic work it should be used only in the case where, in the given thin section of the rock, there is no other grain of the mineral under investigation with a section more favorable for examination.

§ 29. Determination of the optical sign of a uniaxial mineral.

In uniaxial minerals the axis of elasticity, coinciding with the principal crystallographic axis and also representing the optical axis of the crystal, is unique of its kind and in different minerals is either greater or smaller than the other vectors of the ellipsoid of elasticity. Depending on the relative magnitude of this axis, minerals are divided into positive and negative. Consequently, in order to determine the optical sign of a uniaxial mineral, it is first of all necessary to find the position of its optical axis.

From the preceding we already know how simply to find the principal section of a mineral, perpendicular to the plane of the thin section, and how to fix the position of the optical axis by readings on the vertical circle, if only the latter can be brought into coincidence with the axis of the microscope. But sometimes the optical axis may be situated close to the section of the thin section. In that case it cannot be observed directly, and one has to resort to the following procedure¹. Having found the principal section of the mineral perpendicular to the plane of the thin section, we rotate the inner circle of the stage by \(90^\circ\), whereby this section is made to coincide with the axis \(J\). If the stage is then put back into its original position, the axis \(J\) will lie in one vertical plane with the optical axis. This vertical plane is represented in schematic Figure 24 by the diameter \(JJ^1\).

We shall now bring the section under investigation correctly out of the state of extinction, by inclining the preparation about the axis \(J\) away from ourselves through some angle \(\gamma\), so that the above-mentioned vertical plane assumes the position \(J_o^1J^1\). In order again to obtain extinction of the mineral, it is necessary to rotate the instrument about the axis of the microscope through some angle \(\alpha\), until the vertical plane passing through the new position of the optical axis becomes perpendicular to the plane of symmetry of the microscope. But such a vertical plane will be represented in projection by the diameter \(KK^1\), forming with \(JJ^1\) the angle \(\alpha\), laid off in the direction opposite to the rotation of the microscope stage. It is evident that the point of intersection of this diameter with the arc \(J_o^1J\) will represent the projection of the sought optical axis in its new position. Since, in all the manipulations described, we have not inclined the preparation about the axis \(H\), the angle \(JA\) will determine the inclination of the optical axis to the plane of the thin section, which was what had to be found.

Thus, we can always find the position of the optical axis of a uniaxial mineral. To determine the sign of the latter, it is necessary to compare this axis with the other vectors of elasticity; the result of the comparison will be most distinct if we take the direction lying in the circular section of the ellipsoid of rotation, perpendicular to the optical axis. The investigation itself is carried out with the aid of mica or quartz compensators, described in connection with the determination of the axes of elasticity.

For comparison we bring the principal section of the mineral into the plane of symmetry of the microscope, rotate the microscope stage by \(45^\circ\), and, rotating the instrument about the axis \(J\), place the optical axis in the horizontal plane in the manner known to us. Then we introduce into the slot of the microscope tube a compensator, the long side of which will coincide with the optical axis of the mineral if the instrument is rotated counterclockwise. In the event that a lowering of the interference colors occurs, the optical axis will be the axis of greatest elasticity, and the mineral must be called positive, and

conversely. But if the optical axis is more or less normal to the plane of the thin section, then it cannot be brought into coincidence with the horizontal plane. Nevertheless, in this case as well one must proceed in the manner just indicated, only inclining the preparation about the axis $J$ through as large an angle as possible.

Indeed, as was said above, the character of the relation between the magnitudes of the axes of the circular section of the ellipsoid of rotation and the other elasticity vectors of one and the same uniaxial mineral remains constant. Thus, for us it is to some extent immaterial which two elasticity vectors have to be compared with one another, provided only that one of these directions is the axis of the circular section.

§ 30. The refractive index is the most characteristic optical constant of minerals, since the refraction of light belongs to the most sharply expressed additive properties of physical bodies. But the methods for determining this constant are based either on the principle of total internal reflection, or on Brewster’s law, or else on the use of prisms made from the substance under investigation; for the most part they require complex apparatus and special preparations and therefore have no practical significance in petrography.

Incidentally, it must be noted that the Fedorov stage, with its segments forming a homogeneous sphere, at the center of which a plate of the mineral under investigation is placed, might, it would seem, successfully serve for determining refractive indices in thin sections according to the just-mentioned principle of total internal reflection; but special investigations by E. S. Fedorov himself have shown¹) that the use of the stage for this purpose encounters many difficulties and as yet has no practical significance.

In general, it may be considered that the problem of determining refractive indices in thin sections of rocks is still unresolved.

It is necessary to mention a method for an approximate solution of the problem posed, by observing the position of the planes of elasticity. We know that if the mineral under investigation has the same refractive index as the segments, then, with careful adjustment of these planes, they must prove to be perpendicular to one another, and in projection their poles will lie on the corresponding arcs.

But if the indicated refractive indices have different magnitudes, then such coincidence cannot occur, and in this case it is necessary to correct the observed angles of inclination about the axis $H$ according to the known diagram (Fig. 6). Obviously, by the same route we can solve the ob—

¹) E. S. Fedorov. Determination of refractive indices in thin sections. Transactions of the Imperial St. Petersburg Mineralogical Society, 40, 1903; p. 348.

a separate question: by means of approximations we correct these angles to the proper position of the planes of elasticity in such a way that, on the indicated diagram, different angles correspond to one and the same auxiliary arc, which will also indicate the relative magnitude of the refractive index of the given mineral, whence it is easy to proceed to the true value of this constant.

Of course, this method is in general a very crude one, but E. S. Fedorov considers it sufficient for purely petrographic purposes¹.

It seems to us that this can be accepted with the following essential correction. The mineral under investigation must be perfectly fresh, correctly constructed, and colorless, for only under this condition will the subjective errors accumulating in the setting of the planes of elasticity have no effect on the results obtained. But it is clear that in thin sections of rocks such grains of minerals are encountered very rarely, and therefore the very method of determining refractive indices can hardly have any practical significance.

Chapter II.

Determination of the Magnitude of Birefringence

§ 31. Principal methods for determining the magnitude of birefringence.

The difference between the greatest and the smallest refractive indices of the rays formed in birefringent crystals bears the name of the magnitude of birefringence and is, like the refractive index, a sufficiently characteristic optical constant of minerals. In the case of biaxial minerals, to this magnitude \((n_g - n_p)\) there are added two other, no less characteristic magnitudes, \((n_g - n_m)\) and \((n_m - n_p)\), which together determine the given mineral species with sufficient accuracy.

We have seen above that we have no petrographically convenient method for determining the refractive indices of rock-forming minerals; for this reason the measurement of the magnitude of birefringence can be carried out only by some indirect method. Such a method exists and is based on the following.

As is known, a ray of light, entering an anisotropic crystal, is resolved in it into two rays propagating with different velocities. In consequence of this, on emerging from the crystal, one ray lags behind the other by an amount proportional to the difference of the velocities and to the thickness of the plate, so that in different sections and with different thicknesses of the thin section different values of this quantity are obtained. Since, on the other hand, there is a regular relation between the velocity of light and the refractive index of the medium in the given direction, it is not difficult to find the dependence between the indicated retardation of the rays, or, as it is called, the path difference, and the magnitude of birefringence. This dependence is expressed by the formula¹:

\[ (n_g - n_p) = \frac{Sn_g - Sn_p}{e} = \frac{p}{e}, \]

where \(Sn_g\) and \(Sn_p\) represent the paths traversed by the rays vibrating along the axes of greatest and least elasticity, and \(e\)—the thickness of the mineral plate.

Thus, the measurement of the magnitude of birefringence is divided into two parts.

First of all, the path difference of the rays corresponding to the selected directions is determined, and then the thickness of the thin section is measured. The first of these quantities cannot be determined directly, since this task consists in measuring a part of the wavelength

of the light wave, which requires the use of very sensitive instruments and delicate methods. Therefore, in petrographic work only an indirect method of determining this quantity is possible, consisting in comparing the mineral under investigation with one whose birefringence is precisely known. Quartz is usually used as such a mineral, since it withstands external influences well and has an average magnitude of birefringence.

There are two types of instruments serving this purpose. The first type, to which belong Nikitin’s rotating compensator, the quartz wedge, Fedorov’s mica wedge, the Siedentopf compensator, and others, is based on compensation of the path difference; for this reason the instruments themselves are called compensators.

If a compensator is placed upon a plate of the mineral under investigation so that the unlike axes of elasticity of the two mineral species coincide, then the path difference acquired by the rays in passing through the lower mineral will be diminished by the compensator and, at a certain thickness of the latter, will be reduced to zero, which will be expressed in the extinction of the mineral. Thus the plate of the compensator must have variable thickness and, if it is provided with a scale and it is known what path difference corresponds to each division, then the measurement of the path difference in the given section of the mineral is reduced to determining that division of the compensator at which extinction of the mineral occurs.

Recently Prof. E. S. Fedorov proposed a new method of using compensators for measuring path difference.

The essence of this method consists in the interference color of the plate of the mineral under investigation being brought by the compensator to the sensitive violet of the 1st order.

If this is done in parallel and mutually perpendicular positions of the like axes of elasticity of the mineral and the compensator, then the difference of the two corresponding readings of the latter will determine twice the value of the path difference sought. According to the investigations of V. V. Nikitin¹, this method is 1.7 times more accurate than the ordinary method of measuring this quantity, but its application is limited.

First of all, not every eye is sensitive to the violet tint; then, if the path difference developed by the mineral plate is greater than 0.000575 mm, or if the limits of the compensator are small, this method cannot be used at all.

In instruments of the second type, to which belongs, for example, M. Lévy’s quartz-wedge comparator, the path difference is determined by comparing the interference colors of the mineral under investigation and of the plate of the comparator, placed in the proper position. Obviously, when identical tints are attained, the path differences will be identical; the magnitude of this difference is determined from the reading of the comparator scale.

As regards measuring the thickness of the thin section, this point constitutes the weakest place in determining the magnitude of the birefringence.

In general it must be said that the given method has many shortcomings: it requires a large number of measurements, which often cannot be made with sufficient accuracy. Therefore it must be regarded as a palliative, unavoidable in the present state of microscopic technique; and, as soon as simple methods for determining refractive indices in thin sections are improved, it will in all probability be abandoned.

Description of instruments for measuring the difference in path of rays.

§ 32. V. V. Nikitin’s rotating compensator¹.

This instrument may be regarded as one of the derivatives of the universal-optical method of investigation, reflecting in itself—to a certain extent—some of the ideas laid at the foundation of the universal stage.

An essential component part of the compensator described is a round quartz plate \(K\) (Fig. 25), set into a metal ring, together with which it can rotate about the axis \(AA\), lying in its plane. Rotation is effected by means of the head \(s\), and the angle of rotation is marked by the pointer \(i\) on the vertical limb \(L\), rigidly connected with the box \(D\), which supports the whole instrument.

The compensator is inserted by its box into a slot present in the microscope tube; moreover, in the majority of microscopes rotation of the plate is possible only when the box has been pushed in as far as the limb itself. The firm position of the box in the slot is maintained by the spring \(R\).

The change in the difference of path of two interfering rays is attained in this compensator by rotating the quartz plate. Indeed, if the optical axis of the latter is set perpendicular to the axis of rotation \(AA\), then, when the plate is inclined, the difference between the axes of elasticity of the section ellipse, normal to the axis of the microscope, will increase; consequently, the difference of path of the rays vibrating in the direction of these axes will likewise increase.

This change in the difference of path is also to a considerable degree conditioned by an increase in the thickness of the plate in the direction of the passing light rays. Thus the limits of the magnitude of the birefringence that can be determined by this compensator vary from zero, which corresponds to the coincidence of the optical axis of the quartz plate with the axis of the microscope, to that difference in path of rays which is obtained at the greatest inclination of the compensator. It is not difficult to see,

FEDOROV’S OR UNIVERSAL-OPTICAL METHOD.

that this limit will be the more considerable, the greater the angle formed by the optical axis of the plate with the perpendicular to the latter, and the greater the thickness of the plate itself.

The angle just mentioned, however, must be arranged in order to increase the sensitivity of the compensator for small values of the path difference. In fact, if, when the optical axis leaves coincidence with the axis of the microscope, the plate is brought into a horizontal position, then the path traversed by the ray in the plate will decrease, and this will be expressed by a delay in the increase of the path difference, so that, in order to change the latter by some amount, the compensator must be turned through a larger angle than in the case where the plate is cut normally to the optical axis.

Of course, this will hold only until the plate becomes perpendicular to the axis of the microscope; with further inclination, the change connected with it in the length of the path traversed by the ray in the mineral already acts in the direction of increasing the path difference. Thus the divisions of the compensator limb will have the following meanings.

Let, in Fig. 26, the exit of the optical axis of the compensator plate coincide with the axis of the microscope \(AA\), forming with the normal to the latter an angle \(\varphi\). Then the path difference, whose zero value is answered by the division \(c\), will increase from \(c\) to \(b\) and further to the division \(d\), corresponding to the greatest possible inclination of the plate; moreover, along the arc \(bc\) small values of birefringence will be measured, so that mainly the left part of the limb is in operation.

In order to make use, as far as possible, of the greater part of the latter, it is necessary to increase the angle \(\varphi\). But the magnitude of this angle must be such that, for checking the instrument and measuring very small path differences, it should be possible to bring the optical axis into coincidence with the axis of the microscope, which occurs when the field of view is completely darkened; and this minimum of illumination can be caught satisfactorily only if the plate is brought to this position from both sides. Usually the plate is cut from quartz so that the optical axis makes an angle of \(25^\circ\) with the normal to the plate, which corresponds to \(\varphi = 40^\circ\).

As for the thickness of the plate, increasing it, as noted above, expands the limits of the determinable path differences, but at the same time diminishes the sensitivity of the compensator.

Up to now the plate has been given a thickness of \(0.07\ \mathrm{mm}\), proceeding from the requirement of obtaining the whole gamut of colors of the 1st order, and so that the sensitive violet color, corresponding to a path difference of \(0.000575\ \mathrm{mm}\), should correspond to an inclination of the plate of not more than \(60^\circ\).

With such a construction of the compensator, adopted by Fliess, path differences from 0 to \(0.000575\ \mathrm{mm}\) are measured by \(100^\circ\) of the limb, divided into...

a cut of \(5^\circ\); in this case \(40^\circ\) falls in the interval from 0 to 0.000115 mm, and about \(60^\circ\) in the remaining 0.000460 mm.

Thus the error in determining the path difference is expressed in 15–35 millionths of a millimeter and, by repeated observations, can be brought to a smaller value, even to 1 micromicron¹.

Before use the compensator must be checked.

First of all, the perpendicularity of the optical axis of the plate to the axis of rotation of the instrument is verified. For this purpose the compensator, inserted into the slot of the microscope tube, is tilted until darkness is obtained. If the darkness obtained is complete, this may be considered satisfactory; otherwise it is necessary to make certain changes in the instrument itself. To achieve this aim V. V. Nikitin recommends applying the following procedure. The compensator is placed on the microscope stage so that its long side coincides with some definite diameter of the stage; for a more exact coincidence with the given diameter, the latter may first be drawn on the stage with a pencil. If the above-mentioned axes were mutually perpendicular, then, upon rotation of the microscope stage, extinction of the mineral would correspond to the coincidence of the selected diameter with the principal section of one of the nicols. To achieve this, with the given incorrect position of the optical axis of the plate, one must bring the selected diameter into the plane of symmetry of the microscope; for this it is necessary to know the position of the zero of the stage vernier relative to the latter, and then somehow rotate the plate in its own plane until darkness is obtained.

For this purpose one may use either a special key or simply a bent steel wire; by inserting the ends of the latter into two opposite holes present in the annular metal setting of the plate, the required displacement of the latter may be produced by friction.

The next check concerns the correctness of the setting of the compensator pointer, which must correspond to the zero of the limb when the plate is perpendicular to the axis of the microscope. It must be said that the fastening of the head and of the pointer is one of the weak points of the compensator; therefore, at certain intervals of time, it is necessary to carry out this check. In view of the fact that the limb of the compensator is divided only every \(5^\circ\), the following simple method of checking may be applied.

If, after introducing the compensator into the slot of the microscope, all optical systems situated between the compen-

... and the eye of the observer, and if the transmitted light is eliminated, then, with the plate of the compensator in a horizontal position, the observer will see the reflection of his eye from this plate.

The division at which the indicator of the compensator stands for the given position of the plate will be the correction to $0^\circ$ of the limb; in this it is necessary to take into account the signs of the angles.

If in Figure 26 the angle $\varphi$ is taken as positive, then, obviously, all angles measured along the arc $bc$ will likewise be positive, while the angles indicated by the index along the arc $bd$ must be considered negative.

Thus, in the compensator described, it is easy to determine and mark the section that gives the same path difference of the light rays as the given section of the mineral being studied. In order to obtain the numerical value of this path difference, it is necessary to know the value of each division of the compensator scale. To speed the work, it is convenient to construct a diagram showing the relation between the divisions of the compensator and the corresponding values of the path difference.

The graduation of the rotating compensator is carried out in the following manner.

If the thickness of the plate of this compensator can be determined with a sufficient degree of accuracy, then the diagram can be constructed directly on the basis of calculations by the formula derived by V. V. Nikitin¹ for the analogous case for uniaxial minerals.

\[ p = e\,\frac{\operatorname{Sin}^2(\varphi_0 - J)}{\operatorname{Cos} J}\,(n_g - n_p). \]

In this formula:

\[ \begin{aligned} P &\text{— denotes the path difference;}\\ J &\text{— denotes the angle of inclination of the mineral plate in a medium homogeneous with the latter;}\\ \varphi_0 &\text{— the angle of the optical axis with the normal to the plate;}\\ n_g - n_p &\text{— the magnitude of the birefringence, and}\\ e &\text{— the thickness of the mineral plate, in m/m.} \end{aligned} \]

The angle $\varphi_0$ is determined for the given compensator by direct observation, for which purpose the compensator plate is inclined, with crossed nicols, until the greatest darkness is obtained, corresponding to the coincidence of the exit of the optical axis of the plate with the axis of the microscope. Obviously, the required angle $\varphi_0$ is equal to the angle of inclination of the compensator $\varphi$, divided by the refractive index of quartz $n = 1.54$. Since the moment of passage of the optical axis through the axis of the microscope is not determined with sufficient sensitivity, it is necessary to make a large number of observations and take the arithmetical mean of the readings.

In exactly the same way, any generally observed angle of inclination of the compensator plate is

\(J' = nJ = 1.54J\), whence \(J = \dfrac{J'}{1.54}\).

Finally substituting, in the formula given, the constants for the given quartz compensator: \(n_s - n_p = 0.009\) and \(e\), we find the following dependence between the path difference and the corresponding division of the instrument:

\[ p = e \cdot \frac{ \operatorname{Sin}^{2} \dfrac{e - J'}{1.54} }{ \operatorname{Cos} \dfrac{J'}{1.54} }. \]

Plotting on the abscissa axis a series of values of the angle \(J^1\), varying from \(+40^\circ\) to \(-60^\circ\), and on the ordinate axis the values \(p\) calculated by the formula, we obtain the diagram shown in Fig. 27. The curve of the latter from \(+40^\circ\) to \(0^\circ\) runs much more gently than over the remaining extent, since in this part the compensator is made more sensitive, as described above. For convenience in using the diagram, it is best to make it on graph paper.

If the thickness of the compensator plate is not known exactly, then the graduation is carried out with the aid of some other compensator, or of a standard; for this it is necessary to bring the eponymous axes of elasticity of both minerals into coincidence and to incline the plate of the compensator under investigation until complete extinction is obtained.

Knowing the value of \(p\) at the given angle of inclination \(J^1\), we find from the formula the thickness of the plate \(e\), after which the graduation is carried out in the manner already described.

However, in view of the sensitivity of the setting to the violet color of the 1st order, one can dispense with any standard. For this the compensator plate is inclined under crossed Nicols until the field is colored violet; the angle of inclination \(J^1\) will correspond to \(p = 0.000575\) mm, since this value corresponds to the phase difference of the rays whose interference gives a violet color. Substituting the latter quantities in the formula, we find the thickness of the plate \(e\), so that the graduation is again reduced to the method described.

Finally, one may mention one more method of graduating compensators. This method consists in the use of some homogeneous light with a fully determined wavelength.

The easiest to obtain is sodium yellow light, whose wavelength is equal to \(0.000589\) mm.

If such illumination is used, the compensator will extinguish under crossed Nicols each time the path difference of the rays reaches a value containing an odd number of times

\[ \frac{\lambda}{2}, \]

i.e. \(0.000294\) mm.

Thus, beginning to rotate the compensator plate from the na-

of the initial position, we shall obtain a series of extinctions corresponding to

\[ \frac{\lambda}{2},\ \frac{3}{2}\lambda,\ \text{etc.} \]

For the given compensator it is sufficient to find one value of the path difference in order to be able to construct the required diagram.

§ 33. Quartz wedge compensators.

A rotating compensator may serve for measuring comparatively small values of path difference, usually not exceeding 0.000575 mm. Thus this compensator is successfully applied chiefly in the investigation of minerals with a small amount of birefringence—such as zeolites, zoisites, or acid feldspars in thin sections, and the like.

In all other cases one must resort to instruments of another kind, among which quartz wedge compensators enjoy the widest use, owing to the simplicity of their construction.

In its simplest form such a compensator is a quartz wedge with a very small angle, cemented between two glass plates having the form of an elongated rectangle. With a given length of the wedge, any desired path difference can be obtained; but usually the plate is given a length of 70 mm, at which path differences of 0.002 mm can be determined, which is quite sufficient for petrographic purposes.

As a compensator, the wedge is placed upon the plate of the mineral under investigation in such a way that the oppositely named axes of elasticity of the two minerals coincide. Then, at a certain position of the compensator, complete extinction occurs, indicating that the part of the wedge lying over the grain being studied has the same path difference as the latter. To determine this position, a millimetre scale is marked on the upper surface of the compensator, and it is indicated what path difference corresponds to each division of the latter. The slit in the tube of the microscope into which the compensator is introduced must be in the focus of the eyepiece, so that the divisions of the scale falling opposite the grain under investigation can be read. But since microscopes ordinarily do not have such a slit, a special eyepiece is used, in the focus of which the slit is located. The wedge is moved in the latter either by hand or with the aid of a rack-and-pinion mechanism; for the convenience of counting the divisions, one or two hairs are stretched across the centre of the eyepiece (Compensatorocular).

Obviously, when this eyepiece is used the analyser must be placed above it.

The relation between the divisions of the compensator and the corresponding path differences is most conveniently represented in the form of a diagram. The latter must have the form of a straight line. Indeed, the wedge is cut off

from quartz so that the optical axis lies in the lower plane of the wedge, parallel to its edge. Therefore the rays of light entering normally to the lower plane of the wedge will vibrate in different parts of the latter in the same directions, so that the retardation will depend only on the thickness of the wedge. If in Fig. 28 \(ABCD\) represents the transverse section of a wedge with angle \(\alpha\), then, on the basis of the foregoing,

\[ p = e(n_g - n_p) = (l + k)(n_g - n_p)\tg \alpha \quad \text{or} \quad p = A + Bl. \]

That is, the above-mentioned dependence is expressed by the equation of a straight line. Therefore, in order to draw the required diagram, it is necessary to determine only two of its points.

For calibration one may use either monochromatic light or compare the compensator with some standards, among which the commonly used gypsum and mica plates may serve. The first of these gives a retardation of \(0.000575\) mm, corresponding to the sensitive violet color of the first order, while the second has a retardation of the rays of \(1/4\) of the mean wavelength, which is equal to \(0.000127\) mm. In making the comparison it is necessary to place the compensator under investigation upon each of these plates at a right angle and to determine that division of the scale which corresponds to complete extinction.

If the readings obtained are then laid off on the axis of abscissas, and on the axis of ordinates the corresponding retardations, two points will be obtained which determine the straight line of the diagram.

For all the simplicity of their construction and convenience of use, quartz wedge compensators have a very substantial shortcoming, consisting in the fact that the wedge between crossed nicols actually gives a continuous spectrum, as a result of which the sought tint often merges with adjacent ones, and the measurement itself becomes uncertain, especially when there is noticeable dispersion of the axes of the mineral under investigation.

This drawback is not eliminated in the more complex quartz-wedge compensators of Babinet, Bravais, and Khrushchev.

According to the investigation of V. V. Nikitin (Proceedings of the Mining Institute, I, 1907; 385), the mean error in determining retardation with a quartz wedge is equal to \(10\ \mu\mu\).

§ 34. Fedorov’s mica compensator.

This compensator, while preserving the idea of the wedge, is at the same time devoid of the drawback connected with the continuity of the latter. The compensator consists of a packet of mica sheets glued together and arranged one above another in steps, each upper step being 2 mm shorter than the one below. The plates are oriented in such a way that the greater axis of elasticity, lying in their plane, is arranged parallel to the short side of the compensator, as

and in other instruments of this kind. Therefore, using it when measuring the retardation in the grain under examination is in no way different from the usual procedure. The required retardation is determined by the number of steps of the compensator that preceded compensation, and since each step represents a plate of \( \frac14 \lambda = 0.000127 \, mm \), then \( p = n \frac14 \lambda \).

When counting the number of steps one must use a special eyepiece, as for the quartz wedge.

In general, this compensator is very practical; moreover, its accuracy is not especially low, and in the case when adjacent steps darken equally, it rises to \( \frac18 \lambda = 0.000063 \, mm \), which is quite sufficient for practical purposes.

§ 35. M. Lévy’s comparator.

This instrument, constructed on the basis of the principle of comparing the interference colors of the plate of the mineral under examination and of the quartz wedge of the instrument, is the most perfect, although also the most complex, comparator.

It permits, with the same degree of accuracy, the measurement of both large and small retardations; the undesirable property of discontinuity of the wedge is almost eliminated in it, and the very principle of comparing color shades gives more accurate results than the direct determination of the moment of greatest darkening in compensators.

Figure 29 schematically shows the arrangement of this comparator. The eyepiece \(T\) has a branch in which there is a quartz wedge \(q\), moved by means of a rack-and-pinion and screw \(s\). To determine the position of the wedge, the latter is connected with a scale \(t\), on which, by means of a vernier \(v\), tenths of a millimeter can be read.

In order to obtain interference colors, the wedge is enclosed between two nicols \(n_1\) and \(n_2\).

The light propagates in the following way. Reflected from the mirror \(m\), the rays of light are directed by the prism \(p\) along the axis of the tube, where they first meet the polarizer \(n_1\), then the lens \(o_1\), which slightly concentrates them in the quartz wedge, thanks to which they acquire a more homogeneous interference coloration. The diaphragm \(\delta\) with a very narrow aperture also assists this. Having then passed through the analyzer \(n_2\) and the lens \(o_2\), the rays again become parallel and, being reflected from the silvered hypotenuse plane of the prism \(P_1\), enter the field of view of the microscope. For the direct passage of light rays from the mineral being examined, this prism is cemented with another prism \(P_2\), and in the center of the silvered plane a small circular opening is made.

Thus, during observations we see in the field of view of the microscope a small circle colored by the interference color of the mineral and surrounded by a ring of the interference color of the quartz wedge.

By moving the latter, the two tints are brought into coincidence, which corresponds to the same path difference in both minerals.

For a more convenient comparison of the tints, before beginning the investigation it is necessary to equalize the intensity of illumination of the microscope and of the comparator. For this purpose the quartz wedge is removed from the tube, both pairs of nicols are set in a parallel position, and the intensity of illumination is equalized by means of the corresponding iris diaphragm. In order to preserve the absolute equality of illumination thus attained, each time the path difference is measured the plate of the mineral under investigation must be placed so that its axes of elasticity make \(45^\circ\) with the sections of the nicols, because the quartz wedge is also situated at the same angle to its nicols; when this angle is changed, the brightness of the interference color also changes.

On the basis of the latter, equal intensity of the colors can also be achieved during the measurement itself by rotating the stage of the microscope.

The calibration of the comparator is carried out in the same way as for ordinary wedge compensators; but since the interference coloration of the field of view in this case is homogeneous, one may use the setting of the wedge in two such positions in which sensitive violet tints of the 1st and 2nd orders are obtained, corresponding to path differences of 0.000575 and 0.001128 mm.

Measurement of the thickness of a thin section.

As was said above, in order to determine the magnitude of the birefringence it is necessary to know the thickness of the plate of the mineral under investigation.

If the point is to measure the thickness of a separate plate, this can be done fairly accurately and quickly with the aid of various instruments, for example a spherometer, Klein’s universal stage, the so-called mikrometertaster, etc.

But when measuring the thickness of a plate of mineral in a thin section, very many difficulties are encountered.

Up to the present time there exists, properly speaking, only one more or less general practical method of such measurement.

§ 36. Shon’s method.

If, with the thin section in a horizontal position, one successively focuses the objective on any points lying on the upper and lower surfaces of the mineral plate, then the microscope tube must be raised or lowered by an amount proportional to the thickness of this plate. For moving the microscope tube there serves a special micrometer screw, provided with a limb, by which also the determ—

Fig. 1.
Visible labels: \(i\), \(M\), \(M_2\), \(H\), \(C\), \(K\), \(R\), \(m\), \(u\), \(S\), \(g\), \(g'\), \(h\), \(n\), \(N\), \(t\).

Fig. 2.
Visible annotations: “microscope”; “glass circle.”

Fig. 3.
Visible labels: \(A\), \(B\), \(p\), \(v\).

Fig. 4.
Visible labels: \(A_1\), \(A_2\).

Fig. 5.
Visible labels: \(O\), \(a\), \(a'\), \(A\), \(A'\), \(B\), \(B'\), \(e\), \(\alpha\), \(\gamma\).

Fig. 7.

Fig. 8.
Visible labels: \(S'\), \(S\), \(A\), \(B\), \(O\), \(g'\), \(\xi\), \(x\), \(\eta\), \(\chi\), \(\varphi\).

Plate II

Fig. 6.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

Fig. 16.

Fig. 18.

Fig. 19.

Visible labels: \(f_2\), \(x\), \(f_1\), \(x\).

Fig. 21.

Visible labels: \(a\), \(A_1\), \(a_1\), \(m_p\), \(d_2\), \(A_2\), \(m_g\), “section of the thin section,” \(20^\circ\), \(45^\circ\), \(n_m\), “plane of symmetry of the microscope.”

Fig. 20.

Visible labels: \(B\), \(f_3\), \(g_3\), \(f_1\), \(g_2\), \(A\), \(90^\circ\).

Fig. 22.

Visible label: \(a\).

Fig. 23.

Visible labels: \(e\), \(e'\), \(e''\), \(m_g\), \(m'\), \(A\), \(A'\), \(a\), \(O\), \(P\), \(J\), \(J\), \(m_p\), \(\gamma\).

Fig. 24.

Visible labels: \(O'\), \(O\), \(J'\), \(J\), \(A\), \(X\), \(f\), \(f'\).

Table III

Fig. 25.

Fig. 26.

Fig. 27.

Fig. 28.

Fig. 29.

Fig. 30.

Fig. 31.

Fig. 32.

Fig. 33.

Fig. 34.

Fig. 35.

Fig. 36.

Fig. 39.

Table VI.

Fig. 37.

Fig. 38.

No. 901 b
1909

of the Dzhuntar collection.

1) \(15^\circ \ldots\ldots 0^\circ\)

\(A_1\)) away from oneself:

\[ \frac{29^\circ}{29^\circ} \]

\[ \frac{29^\circ_{25}}{25} \]

\[ 2V = +66^\circ \]

2) \(195^\circ,\) to the left:

\[ \frac{35^\circ}{32^\circ} \]

Cleavage by prism:

\(153^\circ,\) to the right:

\[ \frac{13^\circ}{14^\circ} \]

\[ \frac{13^\circ}{12^\circ_{5}} \]

\[ \angle Cn_g = 38^\circ \]

Augite.

Fig. 40.

Visible labels in the diagram: \(N_m\), \(N_g\), \(N_p\); faces \((100)\), \((010)\), \((001)\).

Diagram of \(2v\) of plagioclases.

Fig. 41.

Visible labels: “angle”; \(2v\); \(\%\,An\).

Fig. 42.

Visible curve labels: \(N_g - N_m\), \(N_g - N_p\), \(N_m - N_p\).
Visible horizontal-axis label: \(\%\,An\).

Fig. 44.

Visible labels: I, II; “I microcl.”, “II microcl.”; \(N_g(001)\), \(N_p(010)\), \(N_m(010)\); \(N_g\), \(N_p\).
Visible horizontal-axis label: \(\%\,Na_2O\).

Diagram of extinctions of plagioclases on planes

\[ \perp N_p . \]

Visible side label: “angles.”

Visible curve labels: \(N_m(010)\), \(N_m(001)\), \(N_p(010)\), \(N_p(001)\), \(N_g(001)\), \(N_g(010)\).

Fig. 43.

Table V cont.

Diagram of angles $2V$.

Calibration: feldspar–quartz.

Direction of cleavage.

Fig. 45.

Prismatic row

Monoclinic row

Fig. 46.

$N_{a_2O}$

$[100]$

$[001]$

$o10$

$001$

$100$

Fig. 47.

a

b

c

Fig. 49.

a, b — cleavage along $(010)$,

c — cleavage along $(001)$.

a

Fig. 48.

b

c

Table VI

Plate VIII.

1)

\[ \begin{aligned} &254^\circ\\ &252^\circ \text{ to the left}\ \left\{\begin{array}{l} 25^\circ\\ 23^\circ \end{array}\right.\\ \hline &253^\circ,\ \text{to the right}\ 24^\circ \end{aligned} \]

2)

\[ \begin{aligned} &335^\circ\\ &337^\circ \text{ to the right}\ \left\{\begin{array}{l} 13^\circ\\ 15^\circ \end{array}\right.\\ \hline &336^\circ \text{ to the right}\ 14^\circ \end{aligned} \]

\(a_1)\) [[unclear: handwritten abbreviation]]:

\[ \begin{aligned} &57^\circ\\ &55^\circ\\ \hline &56^\circ \end{aligned} \]

\(a_2)\) [[unclear: handwritten abbreviation]]:

\[ \begin{aligned} &7^\circ\\ &5^\circ\\ \hline &6^\circ \end{aligned} \]

\[ 2V=-62^\circ \]

Visible labels on the diagram include: \(H\), \(K\), \(A_1\), \(A_2\), \(a\), \(b\), \(c\), \(p\), \(k\), and angle marks such as \(35^\circ\), \(104^\circ\), \(126^\circ\).

1. W. Nikitin, Beitrag zur Universalmethode. Zur Bestimmung der Doppelbrechung. Z. Kr. 33 B; 2 H, 145. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

2. F. Wright. On the measurement of extinction angles in the thin sections. Am. Journ. of Sc. 26, 1908; 349. ↩↩