Mechanical Proofs of the Motion of the Earth1

R. Grammel.

Introduction.—1. Fundamental concepts.—2. Mechanical foundations.—I. Experiments based on the law of motion of the center of gravity.—A. Investigation of azimuthal rotation.—3. Motion of a body thrown horizontally.—4. Plane mathematical pendulum.—5. Conical mathematical pendulum.—B. Investigation of vertical rotation.—6. Scales.—7. Torsion balance.—8. Rotating scales.—II. Experiments based on the law of areas.—A. Investigation of azimuthal rotation.—9. Isotomeograph.—10. Hydraulic experiment.—B. Investigation of vertical rotation.—11. Motion of a thrown body.—12. Free fall.—13. Atwood’s machine.—III. Experiments based on the law of impulse.—A. Investigation of azimuthal rotation.—14. Physical pendulum.—15. Gyroscopic pendulum.—B. Investigation of general rotation.—16. Gyroscope.—17. Gyroscopic inclinometer.—18. Barogyroscope.—19. Determination of the gyroscopic moment by means of scales.—C. Investigation of vertical rotation.—20. Gyroscopic declinator.—Conclusion.—21. Annual revolution of the Earth.—22. Precession and nutation.

Introduction.

1. Fundamental concepts. The heliocentric conception of the astronomical picture of the world, which had appeared by chance already among the ancient Greeks, but was definitively established only by Nicolaus Copernicus and fully explained kinematically and dynamically by Johannes Kepler and Isaac Newton, explains the phenomena of motion on the celestial sphere—as is now undoubtedly known to every person with even some astronomical education—by means of three proper motions of the Earth relative to the Sun: the daily rotation of the Earth about its axis, its annual revolution around the Sun, and the conical motion of the Earth’s axis (precession). The last of these occurs very slowly and, as precise observations show, is accompanied by an extremely complex series of very small oscillations (nutation). Today we can scarcely imagine what enormous difficulties the understanding of the Copernican system of the world presented for

of his contemporaries, but we understand that lively interest, amounting almost to a spiritual exaltation, at the time of the experiments with Foucault’s pendulum, which even at the present time appears in relation to all direct proofs of the motion of the earth.

Lord Kelvin, however, draws a strict distinction between observation and experiment as a basis of proof. The most important foundations for the Copernican system are, of course, astronomical observations (the visible daily rotation of the celestial sphere and the parallaxes of the fixed stars, the aberration of light, the precession of the pole of the world); to these are added, as further foundations, numerous geophysical observations ¹) (the flattening of the earth, the Baer–Ballot law on the deflection of air and sea currents, Baer’s law concerning the stronger erosion of the right bank of rivers in the northern hemisphere and the left in the southern, and, finally, the existence of terrestrial magnetism, the field of which, according to A. Einstein’s hypothesis ²), can be fully explained as the dynamical consequence of a current of electrons arising in the rotation of the earth).

With the large number of these observations, experiment plays only a modest role, but it is nevertheless significant insofar as, without the aid of astronomy, it gives us information concerning the motion of the earth by means of an exceedingly ingenious application of the laws of nature. The number of such experiments is much greater than is commonly thought, and it is all the more worth arranging them into a coherent system ³), for the best known among them are by no means the best, while the most ingenious are least known. Besides the mechanical experiments, to which we shall confine ourselves, there also exist certain electromagnetic ones. Of these the most important—the optical interference experiment of A. Michelson for proving the motion of the earth—acquired its exceptional significance because it led to a negative result. The negative result of all non-mechanical experiments gave grounds for the establishment of the principle of relativity. In view of the important consequences of this principle, it is also necessary to clarify its relation to mechanical experiments.

Here it should be noted that Einstein’s principle of relativity (in its generalized formulation of 1915) deprived all proofs of the earth’s motion of absolute force, for these proofs must rest on Newton’s conception of absolute space. We have long known that the Copernican system is in general not demonstrable, that it finds its justification only in its exceptional—

¹) This is popularly set forth in W. Brunner’s booklet, Dreht sich die Erde? Math. physikal. Bibliothek, Leipzig und Berlin, 1915, Bd. 17, S. 46 ff.

²) A. Einstein und J. W. de Haas, Verhandl. d. dtsch. phys. Ges., 17, 156, 1915.

³) An excellent brief exposition belongs to J. G. Hagen: La rotation de la terre, ses preuves mécaniques anciennes et nouvelles. Pubblicazioni della Specola Astronomica Vaticana, 2 serie, I. Roma 1912.

simplicity, clarity, absence of contradictions, and nothing else. But whereas earlier it was only necessary to admit the generally accepted fundamental laws of mechanics and gravitation in order to be able, on the basis of experiments, to draw the necessary conclusion about the motion of the earth, after the establishment of the theory of relativity it becomes in principle permissible to explain each of these experiments otherwise than by the motion of the earth, without falling into contradiction and without introducing any change into the relativistic laws of nature.

Thus, for example, instead of the rotation of the earth one may assume the influence of enormous rotating distant masses, as E. Mach1 had already done, and as Thirring (H. Thirring)2 has recently strictly substantiated. Consequently, the compelling force of all proofs of the earth’s rotation about its axis extends precisely as far as the subjective conviction that these rotating distant masses are absent is probable.

Of course, nothing can be objected against their existence; they may simply be conceived as the totality of fixed stars; the difficulty consists solely in the fact that an extraordinarily large rotation is ascribed to them. Since all the stars, to the most remote depths of space, would take part in this rotation, we would arrive at velocities that must exceed all limits. This improbability could be avoided only by assuming that the totality of fixed stars fills only one finite closed region of the infinitely extending world-space. But such an assumption, as Einstein3 has shown, is physically unfounded.

From the point of view of exact science, the decision takes shape, both on the basis of the data of experience and within the limits of the principle of relativity of 1915, undoubtedly in favor of the rotation of the earth. But in 1917 Einstein acknowledged that both in Newtonian theory and in his own theory of gravitation of 1915 (inseparably connected with the principle of relativity) there is a fundamental, difficult-to-remove inconsistency arising from the spatial infinity of the world. It seems impossible to avoid this inconsistency in any other way than, instead of a spatially infinite world, to assume a world that is at least unbounded, but finite. Such a bold assumption, however—quite irrefutable from the gnoseological point of view and scarcely disputable from the astronomical point of view—leads [this cannot be developed in detail here4] to an extension of Einstein’s theory of gravitation of 1915, and together with

to the generally strictest and most consistent implementation of the principle of relativity itself, in the sense that all motions without exception must now be regarded as relative not only kinematically, as hitherto, but also dynamically.

Applied to the case of the rotation of the earth, this must mean the following: the kinematically firmly established relative motion of the earth and the world of fixed stars could, according to the previous theory of 1913, be dynamically explained only by the rotation of the earth, and by no means by the rotation of the fixed stars, in view of the unlimited increase of their velocities. The new theory of 1917, on the contrary, need no longer fear velocities, since its world is finite. And dynamically it no longer presents any difficulty to prove that all terrestrial phenomena whose cause we had regarded as the rotation of the earth must proceed in exactly the same way also in the case in which the earth stood motionless and the world of fixed stars rotated; or, in other words, that the gravitational forces with which the rotating finite world of fixed stars acts upon the motionless earth fully explain the phenomena of inertia which, since the time of Copernicus, we had been accustomed to regard as a consequence of the earth’s rotation. This circumstance inevitably leads to the conclusion that all experiments which are supposed to prove the rotation of the earth can now, with equal right, be considered as proofs of the rotation of the fixed stars. The two kinematically equivalent possibilities—the rotation of the earth relative to the fixed stars, or the rotation of the fixed stars relative to the earth—have now become equivalent also in the dynamical sense. Together with this they in general no longer differ from one another and are epistemologically identical; now they play only the secondary role of two different names for one and the same phenomenon—the relative rotation of the earth and the world of fixed stars.

They may perhaps be compared with the terms “down” and “up.” Just as, in the purely physiological perception of world space, we always involuntarily imagine down and up according to a free choice, never arguing about what is thereby called “down” and what “up,” so too, in describing this relative rotation, one may use one term or another, one or another definitely chosen representation. This should follow clearly from psychological data, for we are accustomed directly to perceive only absolute motions, but by no means relative rotation, so that the possibilities we imagine, which exclude one another, evidently coincide completely.

From the point of view of the theory of gravitation of 1917, both assertions—“the earth rotates relative to the resting fixed stars” or “the fixed stars rotate relative to the resting earth”—are equally false, and only the proposition is correct: “the earth and the world

fixed stars rotate relative to one another.” To try to prove more than mutual-relative rotation is entirely superfluous, because, epistemologically and factually, it will yield nothing more than relative rotation. On the contrary, from this extreme point of view there will not for a minute be any doubt as to which of these two expressions—the rotation of the earth or the rotation of the fixed stars—should be preferred in describing the phenomenon itself, since one cannot do without some definite term. In any case, to imagine, it is true, the finite but still rather vast world of the fixed stars at rest, and the small earth rotating, is no more than a matter of taste; but, just as in world space “up” and “down” can not only be interchanged but also turned about in any way whatever, so too for designating relative rotation there exist not only two, but an infinite multitude of equally admissible possibilities—and, perhaps, it would be more acceptable to suppose that the two unequal parts into which the world is divided in an exposition of relative rotation, namely the small earth and the remaining immense world, rotate inversely proportionally to their magnitudes, so that the world of the fixed stars rotates extremely slowly from east to west, while the earth, with a velocity approximately equal to that of its daily rotation, rotates from west to east. This assumption would be a golden mean between the rather proud conception of the ancients, according to which everything rotates around the earth, and the infinitely modest view of Copernicus, according to which the earth, in relation to the rest of the universe, is an absolute nothing; and at the same time it would express very well the fact that, according to Einstein’s theory of gravitation, the entire remaining world must also, indisputably, experience the gravitational action of the relative rotation between it and the earth, although to an exceedingly weak degree.

Thus, even if reason now comes in practice to a decision with the same certainty as before the principle of relativity, we must nevertheless at the present time express ourselves at least cautiously. Namely, if we take into account the fact that the mechanical experiments performed hitherto concerned only the rotation of the earth, and if, at the same time, in the future too, we direct our attention exclusively to rotation, then in the present state of our knowledge of nature we can say only one thing: it is quite possible to prove that the phenomena of motion of terrestrial inert masses, considered relative to a reference system rigidly connected with the earth, occur somewhat differently than they would have to occur under the influence of all the terrestrial forces known to us on the basis of the law of inertia; in other words, this terrestrial reference system does not possess the properties of an inertial system (i.e., a system in which the law of inertia holds),

... differs from the latter in a quite definite way. These deviations can be explained, from the standpoint of relativistic mechanics, by the relative rotation between the earth and the rest of the world, and from the standpoint of classical mechanics only by the rotation of the earth.

Despite the fundamental difference between relativistic and classical mechanics, the latter nevertheless gives, for all terrestrial problems, results that numerically approximate the results of the former with very great accuracy; so that, having considered the question epistemologically, we may subsequently, without embarrassment, take the standpoint of classical mechanics. In this sense we shall then have the right to speak directly of the “rotation of the earth.”

Fig. 1. Azimuthal and vertical rotation.

2. Mechanical foundations. In order to bring into systematic order the extremely large number of diverse experiments that can prove the rotation of the earth, it is useful to make use, figuratively speaking, of a “dynamic construction with a kinematic subdivision.” To come closer to this subdivision, let us represent the angular velocity of the earth’s rotation—the cause of the investigated deviation of the terrestrial reference system from an inertial system—by a vector \(\omega\), drawn from the center of the earth in the direction of the earth’s axis to the north (Fig. 1), of length

\[ \omega=\frac{2\pi}{86164}\,\mathrm{sec}^{-1} \]

(in the numerator stands the angle through which the earth turns in the course of a sidereal day, and in the denominator—the number of seconds in a sidereal day).

According to the well-known rule for resolving vectors, the rotation \(\omega\) may be resolved into two component rotations \(\omega_1\) and \(\omega_2\). The first is the rotation of the horizontal plane of an observation site \(A\) of latitude \(\varphi\) about the vertical with angular velocity

\[ \omega_1=\omega \sin \varphi; \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (1) \]

we shall call it the azimuthal rotation. It vanishes only for observation sites situated on the equator, and at the north and south poles it represents the full rotation. The second component rotation turns the horizon about an axis passing through the center of the earth parallel to the meridian of the observation site, with angular velocity

\[ \omega_2=\omega \cos \varphi, \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

and is called vertical rotation. It vanishes only at both poles and coincides at the equator with the total rotation. In most experiments the question is one of investigating either azimuthal or vertical rotation separately, and only afterward does one proceed to the kinematic separation of these experiments.

Next let us recall that the fundamental law of dynamics—the law of momentum (which includes the law of inertia in the narrow sense)—when applied to the motion of a rigid or non-rigid body, breaks up into two laws, of which the first governs the translational motion of the body, and the second its rotation. The first, the so-called law of motion of the center of gravity, says that in an inertial system the center of mass of a body (practically the same thing as the center of gravity) moves as it would if the entire mass were concentrated in it and if all the forces applied to the body, which impart acceleration directed along their resultant, acted on it; i.e. the resultant of the forces is vectorially equal to the acceleration of the center of gravity multiplied by the mass of the whole body.

Fig. 2. A body on a gimbal suspension.

If the resultant is equal to zero, then the center of gravity moves uniformly in a straight line (the case of rest is included here). This particular form of the law of motion of the center of gravity is the law of inertia for the translational motion of a body.

For the rotation of a body, the law of momentum has just as simple an expression as for translational motion, but only in the case where the rotation takes place about a principal axis of inertia, and all the forces taken together also tend to rotate the body about this axis. This occurs, for example, when, besides the forces that support the axis, only one more force acts, perpendicular to the axis of rotation and not intersecting the latter (the action of a handle). Then the rotation is accelerated (positively or negatively) in such a way that the moment of this force, i.e. the product of the magnitude of the force by its distance from the axis of rotation (the arm), is equal to the product of the corresponding moment of inertia by the angular acceleration.

In the general case, when rotation, as for example with a gimbal suspension (Fig. 2), can take place about any axis passing

through the center of suspension \(O\) (the so-called point of support), resolve the vector of the angular velocity of the body \(u\) into three components \(u_1, u_2, u_3\) along three mutually perpendicular axes, intersecting at the point \(O\) and fixedly connected with the body, namely along the three principal axes of inertia of the body with respect to the point \(O\).

Fig. 3. Relation between the vectors \(u\), \(\mathfrak{S}\), \(\mathfrak{M}\) in the rotation of a body about the point \(O\).

Let \(A, B, C\) be the moments of inertia of the body with respect to these three axes; let us increase the components \(u_1, u_2, u_3\), respectively, in the ratios \(A:1\), \(B:1\), \(C:1\), then add the vectors \(Au_1, Bu_2, Cu_3\) thus obtained and obtain the vector \(\mathfrak{S}\), which we shall call the vector of impulse; in general it has by no means the same direction as the vector \(u\), since the numbers \(A\), \(B\), and \(C\) are ordinarily not equal. Here it is understood that, both for the vector \(v\) and for \(u\), and in general for all vectors of axial character, we determine the direction represented by the vector of rotation according to the rule of the screw with a right-hand thread. In particular, we shall apply this definition also to the moment of a given force with respect to the point \(O\). Draw (Fig. 4) through the point \(O\) and the direction of the force the plane \(E\); then the moment, like the force, is represented by a vector perpendicular to the plane \(E\) at the point \(O\) and directed so that, with respect to the direction of rotation of the force, it forms a right-handed screw. Its length will, of course, be equal to the product of the force by the arm. The sum of these moment-vectors \(\mathfrak{M}\) (obtained by ordinary addition of vectors) now determines the rotation \(u\) in a very graphic way, namely: the vector \(\mathfrak{M}\) represents the velocity with which the end of the vector \(\mathfrak{S}\) moves in space (Fig. 3). This proposition (whose derivation from the law of impulse cannot be given here) is known as the theorem of the moment of momentum (Schwung-Satz) or the law of rotational impulse (Drehimpulssatz).

Fig. 4. Vector of the moment of a force.

We have an important special case also when the forces give the moment \(\mathfrak{M}=0\); then the impulse-vector \(\mathfrak{S}\) is constant in direction and

magnitude. One might call this more special case the law of inertia for the rotational motion of a body; owing to one intuitive representation which it admits, it is called the law of areas. The motion itself, however, may in this case be very complicated1. But it again becomes especially simple when the body from the very beginning rotates about a principal axis of inertia. Then, for \(\mathfrak M=0\), the product of the moment of inertia and the angular velocity remains constant throughout the entire motion. If the body is rigid, this means that for a long time it preserves its rotational velocity; if the body is not rigid and if its parts can be displaced relative to one another so that the axis of rotation does not cease to be a principal axis of inertia, while the principal moment of inertia may become now smaller, now larger, then the angular velocity changes in the inverse ratio.

The term “law of areas” comes from the simplest case, when a mass point \(m\)—a planet—moves under the action of a force directed toward one immovable center \(O\), the sun. If one disregards rotation, then the motion of the planet consists in revolution around the sun, and the vector \(\mathbf u\) of this rotation, of absolute magnitude \(u\), drawn from the sun, is perpendicular to the straight line joining the planet with the sun. If \(r\) is the distance of the planet from the sun, and \(mr^2\) is its moment of inertia with respect to the axis of the vector \(\mathbf u\), then the impulse-vector \(\mathfrak S\) has magnitude

\[ S=mr^2u,\ . . . . . . . . . . . . . . . . . . . . (3) \]

and is directed in the same way as \(\mathbf u\).

Since the moving force of attraction has no moment with respect to the axis of rotation, the vector \(\mathfrak S\) is immovable in space, and this directly says that \(\mathbf u\) also remains immovable, i.e. that the planet moves all the time in one plane. Let us further write, instead of (3),

\[ \frac{1}{2}r^2u=\frac{1}{2}\frac{S}{m}; \]

then on the left-hand side there will be the area described by the radius-vector \(r\) in a unit of time (taken sufficiently small). Since for \(\mathfrak M=0\) the value \(S\) also does not change, the motion of the planet takes place with constant sectorial velocity

\[ \frac{1}{2}r^2u \]

(the second law of Kepler).

In the more general case of rotation about any axis, for \(\mathfrak M=0\), the law of constancy of sectorial velocities holds for the projection of the motion onto any immovable plane; it also holds for the projection of the motion onto an immovable plane when the vector \(\mathfrak M\), although not equal to zero, nevertheless remains at all times parallel to this immovable plane.

Passing from the particular to the general, we now obtain the following dynamical scheme:

Law of inertia	Law of areas
Law of motion of the center of gravity	Law of impulse.

If we leave aside the law of inertia (in the narrower sense), since it is practically impossible to isolate a body completely from the influ-

of all the forces, then there remain for us, for bringing our experiments into our system, three other laws. And now the point is to show the differences that arise if these laws are applied at one time to an inertial system not taking part in the rotation (then they explain the ordinary motions of bodies), and at another time to the terrestrial system of reference; then they tell us what motions could have been observed if the earth did not rotate. For checking these differences, in principle all mechanical phenomena are suitable; their choice depends only on the required accuracy of observation, which, because of the small effects, must be sufficiently great.

For practical calculation it is much more convenient to use the laws of motion of the center of gravity and the law of areas also in their integral form, in which it states that the increase of kinetic energy is equal to the work performed by the forces (the law of energy). Here the kinetic energy of a rigid body is composed of the kinetic energy of translational motion

\[ \frac{1}{2}mv^2 \]

(\(v\) is the velocity of the center of gravity) and the kinetic energy of rotational motion

\[ \frac{1}{2}(Au_1^2+Bu_2^2+Cu_3^2). \]

If we are speaking only of gravity as the motive force, then the work it performs is also called the loss of potential energy, and then the law of energy is briefly expressed thus: the sum of the kinetic and potential energies is constant.

I. Experiments Based on the Law of Motion of the Center of Gravity.

A. Investigation of Azimuthal Rotation.

3. Motion of a body thrown horizontally. The law of motion of the center of gravity determines by itself the motion of a body quite fully only when the body has no appreciable size and therefore does not possess rotation from the impulse received as a result of the rotation of the earth. In reality it is sufficient to take, as far as possible, a small body with as large a specific weight as possible, and to eliminate any rotation of the body. If such a “material point” is thrown from the observation point \(A\) so that it is to remain almost in the horizontal plane, not being, however, connected with the terrestrial system of reference by horizontal forces (friction, etc.), then its horizontal projection would have to describe a straight line in the event that the terrestrial system were inertial. The azimuthal rotation \(\omega_1\) of the terrestrial system of reference is expressed, accordingly, in this apparent rotation \(-\omega_1\) of the ray going from \(A\) to the instantaneous horizontal projection of the thrown point with respect to the inertial system. This should appear as a deviation of the shot from the terrestrial trajectory in the northern hemisphere to the right, and in the southern to the left, precisely by the amount \(\omega_1=\omega\sin\varphi\) per second, independently of the angle of departure and direction. The deviation in our latitudes

must give an error of more than 2 m at a mean velocity of flight of the projectile of 600 m/sec, if the target is at a distance of 5 km.

In general, the deviation from the target \(z\), for a horizontal component of the mean velocity of flight of the projectile \(v\) and for a distance to the target \(a\), is determined by the formula

\[ z=\frac{a^2}{v}\,\omega\sin\varphi \]

Numerous attempts to investigate this error have not led to a positive result, because the deviation depends to a strong degree on other influences1. These influences arise from the resistance of the air, which rapidly increases with increasing velocity. In order to weaken them, one must, as far as possible, take slow motions.

4. The plane mathematical pendulum. There is a simple means of maintaining, in a perfectly regular way, such slow, almost horizontal motions—namely, a mathematical pendulum of great length \(l\) and, in comparison with it, small amplitude \(a\). The motion of such a pendulum, brought out of its position of rest by a strictly central horizontal push, would differ from the motion of a projectile, apart from its considerably smaller velocity, only in that it would always be attracted to its point of equilibrium by a force which, for small amplitudes, is proportional to the angle of deflection. This force always lies in the plane of oscillation, and therefore it acts only so as to compel the pendulum all the time to move back in the opposite direction without changing the spatial position of the plane of oscillation. Since, by assumption, the height of the pendulum above the horizontal plane must be quite insignificant, this phenomenon, as rigorous investigation shows, is almost entirely independent of the vertical rotation \(\omega_2\), while at the same time there must also appear here an azimuthal rotation \(\omega_1\) through the apparent rotation \(-\omega_1\) of the plane of oscillation of the pendulum. The apparent rotation takes place in the northern hemisphere in the direction \(NOSW\), and in the southern hemisphere in the opposite direction; at the equator it does not occur at all.

The magnitude of the apparent rotation, measured along the circumference of a horizontal circle \(K\), of radius \(a\), at the point \(A\), will be equal to \(2\pi a\sin\varphi\) in 24 hours. That is, as can easily be calculated, the difference in the length of two terrestrial, parallel circles, one of which passes through the center, the other through the northern or southern point of the circle \(K\).

But in practice it is completely impossible to impart to the pendulum a push that is sufficiently precisely central. Therefore, when this experiment is actually performed, the pendulum is always released from its extreme position \(B\), so that it has no initial velocity relative to the earth. Thus, the pendulum is given at once the whole rotational—

zero velocity of the earth, and it is clear that its motion will in fact proceed otherwise than the ideal motion under a central impulse, and that even without an estimate of the deviation from the ideal motion thus altered, these experiments would have no significance at all.

Since, for the small amplitudes to which we wish to restrict ourselves, the question does not at all concern the vertical rotation \(\omega_2\), we may for the time being regard as an inertial system the horizontal plane connected with the earth at the point \(A\), which does not take part in the azimuthal rotation \(\omega_1\) (in Fig. 5 we have already relied on this assumption). In this inertial system the pendulum, when released, has the velocity

\[ v_0 = a\omega_1, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

tangent to the circle \(K\) in the direction \(\omega_1\). Thus, from the point of view of the inertial system, we are not dealing at all with a plane pendulum, but with a so-called spherical one, i.e., one whose mass no longer oscillates in a single plane, but can move over a sphere of radius \(l\) with its center at the point of suspension.

Fig. 5. Horizontal projection of the trajectory of a pendulum deflected by a central impulse at point \(A\) in an inertial system.

Fig. 6. The same horizontal projection in the terrestrial system.

Fig. 7. Horizontal projection of a pendulum released from point \(B\) relative to the resting earth, in an inertial system.

Having a pendulum on a thread that is sufficiently easy to realize, one can be convinced that the horizontal projection of the swinging pendulum will now be represented not in the form of a straight line (as in Fig. 5), but as an ellipse; moreover, the ellipse itself, under more exact observation, reveals—

undergoes a slow rotation in the direction of the push (Fig. 7); theory shows that rotation with velocity \(\omega\) occurs in the direction \(\omega_1\), with angular velocity

\[ \omega'=\frac{1}{3}\omega_1\left(\frac{a}{l}\right)^2 \ldots \tag{5} \]

so that the apparent rotation of the trajectory of oscillation of the pendulum, observed from the earth (Fig. 8), is now not \(-\omega_1\), but will be: \(-\omega_1+\omega'\), or, in consequence of (1) and (5),

\[ -\omega\sin\varphi\left[1-\frac{1}{3}\left(\frac{a}{l}\right)^2\right] \tag{6} \]

The correction term

\[ \frac{1}{3}\left(\frac{a}{l}\right)^2 \]

must be taken into account in all quantitative experiments. In order to reduce it as much as possible, one makes either the length of the pendulum \(l\) very large (as most experimenters do), or, as Kamerlingh-Onnes (Kammerlingh-Onnes) did, the amplitude \(a\) very small (cf. § 14).

Fig. 8. The same horizontal projection from the terrestrial system.

In calculating the correction term we may at once make use of approximations. For this purpose we determine the position of the pendulum in the inertial system by spherical coordinates \(\vartheta\) and \(\omega\), where (Fig. 9) \(\vartheta\) will be the deflection of the pendulum rod and \(\omega\) the azimuth of its horizontal projection relative to the initial position \(AB\), taking as positive the direction of the earth’s rotation. Then the velocity of the pendulum is composed of a radial component \(l\dot{\vartheta}\) (dots above letters denote derivatives with respect to time) and an azimuthal component \(l\sin\vartheta\,\dot{\omega}\); the first relates to the oscillation of the pendulum, the second to its motion along the ellipse and to the rotation of the ellipse. In the position \(\vartheta\) the pendulum has, relative to its position of rest, the potential energy

\[ mgl(1-\cos\vartheta). \]

The energy equation, after division by \(\frac{1}{2}ml^2\), will be:

\[ \dot{\vartheta}^{\,2}+\sin^2\vartheta\,\dot{\omega}^{\,2}+2c^2(1-\cos\vartheta)=h,\ldots \tag{8} \]

where

\[ c^2=\frac{g}{l}, \]

and \(h\) is a constant, which we shall determine later. Further, the horizontal projection \(l\) has areal velocity

\[ \frac{1}{2}l^2\sin^2\vartheta\,\dot{\omega}, \]

whence, by the law of areas, we have

\[ \sin^2\vartheta\,\dot{\omega}=k,\ldots \tag{9} \]

where \(k\) is also constant.

Fig. 9. Spherical pendulum.

Eliminating \(\varphi\) from (8) by means of (9) and replacing, in the first approximation, \(\sin\vartheta\) by \(\vartheta\) and \(1-\cos^2\vartheta\) by \(\frac{1}{2}\vartheta^2\), we obtain the energy equation in the form

\[ \dot{\vartheta}^{\,2}-\frac{k^2}{\vartheta^2}+c^2\vartheta^2=h. \tag{10} \]

Now we regard as known the fact that the pendulum swings back and forth between a small outer angular amplitude \(\vartheta_1=\frac{a}{l}\) and an even considerably smaller inner one \(\vartheta_2=\frac{b}{l}\), where \(b\) is the radius of the small circle whose horizontal projection touches near the point of rest \(A\). Since in both the outer and the inner positions of the pendulum the velocity \(\dot{\vartheta}\) is equal to zero, on the basis of (10) we obtain

\[ \frac{k^2}{\vartheta_1^2}+c^2\vartheta_1^2=h \]

\[ \frac{k^2}{\vartheta_2^2}+c^2\vartheta_2^2=h, \]

whence \(h\) and \(k^2\) are expressed through \(\vartheta_1\) and \(\vartheta_2\) in the form

\[ h=c^2(\vartheta_1^2+\vartheta_2^2) \]

\[ k=\pm c\vartheta_1\vartheta_2. \]

With the aid of these values, the energy equation (10), after extracting the square root, takes the form

\[ \dot{\vartheta}=\pm\frac{c}{\vartheta}\sqrt{(\vartheta_1^2-\vartheta^2)(\vartheta^2-\vartheta_2^2)}, \tag{11} \]

where the positive or negative sign must be taken according as \(\vartheta\) increases from \(\vartheta_2\) to \(\vartheta_1\) or decreases from \(\vartheta_1\) to \(\vartheta_2\).

Let us return once more to the law of areas. If we substitute there the value of \(k\) and choose the sign so that \(\dot{\varphi}\) is positive, and if instead of \(\sin^2\vartheta\) we take the approximate value \(\vartheta^2-\frac{\vartheta^4}{3}\), then instead of (9) we obtain

\[ \vartheta^2\left(1-\frac{\vartheta^2}{3}\right)\dot{\varphi} = c\vartheta_1\vartheta_2, \tag{12} \]

from dividing equations (11) and (12) and putting

\[ \frac{1}{\vartheta\left(1-\frac{\vartheta^2}{3}\right)} \simeq \frac{1}{\vartheta}\left(1+\frac{\vartheta^2}{3}\right) = \frac{1}{\vartheta}+\frac{\vartheta}{3}, \]

we obtain

\[ d\varphi = \pm\vartheta_1\vartheta_2 \left(\frac{1}{\vartheta}+\frac{\vartheta}{3}\right) \frac{d\vartheta}{\sqrt{(\vartheta_1^2-\vartheta^2)(\vartheta^2-\vartheta_2^2)}}. \]

Integrating with respect to \(\vartheta\) within the limits from \(\vartheta_2\) to \(\vartheta_1\), we obtain on the left-hand side the fourth part of the complete azimuth for an entire oscillation, while the right-hand side will be

\[ \frac{\pi}{2}\left(1+\frac{\vartheta_1\vartheta_2}{3}\right). \]

Thus one obtains

\[ \Psi=2\pi\left(1+\frac{\vartheta_1\vartheta_2}{3}\right). \]

The excess of the quantity \(\Psi\) over \(2\pi\) evidently gives the rotation \(\Delta\Psi\) of the ellipse over the course of one whole oscillation. If \(t_0\) is the period, then the angular velocity with which the ellipse slowly rotates will be

\[ \omega=\frac{\Delta\Psi}{t_0} =\frac{2\pi}{3}\frac{\vartheta_1\vartheta_2}{t_0} =\frac{2\pi}{3}\frac{ab}{t_0 l^2}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (13) \]

To verify the complete agreement of this value with (5), p. 347, we must recall that the small amplitude of oscillation of the ellipse \(b\) arises from the velocity \(v_0\) (4), which is imparted to the pendulum by the rotation of the earth \(\omega_1\); but if one observes the swing in the direction of the major semiaxis of the ellipse \(a\), then it seems that the pendulum swings simply, as a plane pendulum, with amplitude \(b\) and maximum velocity \(v_0\), perpendicular to the major axis of the ellipse. In this case the quantities \(b\) and \(v_0\) are connected with the period of oscillation \(t_0\)—as in every harmonic oscillation—by the relation

\[ v_0t_0=2\pi b, \]

which, in consequence of (4), passes into

\[ \frac{2\pi b}{t_0}=\omega_1 a, \]

whence, on the basis of (13), our desired equation (5) is obtained directly.1

As for the history of the experiment, it is generally known that, after long preparations, Foucault2 performed it in January 1851 with complete success in Paris, using a pendulum 67 m long, with an oscillation duration of 16 sec. It is also known that this experiment, in almost a year, made its triumphant progress over the whole earth; in some cases the rotation of the earth was obtained with an error of up to \(1/2^\circ\) (consequently, the length of the day—with an accuracy of almost up to 7 min), and in one experiment, set up in Cologne,3 even up to \(1/6^\circ\). Less well known is the fact that an experiment with certain quantitatively satisfactory results4 had already been carried out in 1661 by Viviani in Florence and in 1833 by Bartolini in Rimini, of which Foucault in any case did not know. As a curiosity, one may also mention that Calin installed ordinary clocks with pendulums in Barmen, which could rotate on vertical steel points, and was able distinctly to observe the visible rotation of the clocks; the quantitative result, however, was obtained with an error of 10%.

5. Mathematical conical pendulum. The evidentiary force of Foucault’s experiment is diminished, despite its fame, by the fact that it lacks the property of reversibility, which alone makes possible the reliable elimination of all systematic errors; i.e., this experiment cannot be reversed in such a way that all systematic—

systematic errors entered with the opposite sign. This shortcoming, it seems, was first noticed by Bravais [A. Bravais1]. Soon after Foucault, already in May 1851, Bravais undertook a new and quite successful experiment, imparting to the pendulum not plane but conical oscillations in such a way that the mass of the pendulum had to describe a horizontal circle, in one experiment in one direction and in the other experiment in the opposite direction. Depending on the direction of rotation, different periods of revolution were obtained. Namely, if \(\varepsilon_0\) is the true angular velocity of the conical pendulum relative to the inertial system, \(\varepsilon_1\) its apparent angular velocity relative to the terrestrial system in the direction \(NOSW\), and \(\varepsilon_2\) the same velocity, but for the direction \(NWSO\), then:

\[ \left. \begin{aligned} \varepsilon_1 &= \varepsilon_0+\omega_1\\ \varepsilon_2 &= \varepsilon_0-\omega_1 \end{aligned} \right\} \qquad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots (14) \]

and, consequently, on the basis of (1), p. 340

\[ \varepsilon_1-\varepsilon_2=2\omega_1=2\omega\sin\psi \qquad \cdots \cdots \cdots \cdots \cdots (15) \]

Bravais found the difference \(\varepsilon_1-\varepsilon_2\) by imparting to a pendulum 10 mt. long the most nearly exact possible motion along a circular cone by means of a horizontally rotating lever, and by determining the relative periods of revolution by sighting through the pendulum thread at a straight line fixed relative to the earth and coinciding with the position of equilibrium.

For still more precise measurements Bravais used two pendulums, suspended one behind the other in the direction of sighting, which differed in length by \(1/100\) and, consequently, in period of revolution by \(1/200\). For brevity let us introduce the readily understood notation

\[ \left. \begin{aligned} \varepsilon'_1 &= \varepsilon'_0+\omega_1\\ \varepsilon'_2 &= \varepsilon'_0-\omega_1 \end{aligned} \right\} \qquad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots (16) \]

Then from (14) and (16) it follows that

\[ (\varepsilon'_1-\varepsilon_2)-(\varepsilon'_2-\varepsilon_1)=4\omega_1=4\omega\sin\psi \qquad \cdots \cdots (17) \]

The first experiment, in which the shorter pendulum rotated in the direction \(NOSW\), and the longer one in the direction \(NWSO\), gave, by observing the coincidence of both pendulums and by calculating the oscillations between two coincidences, the first of the differences of the left-hand side of (17), while the second experiment, with the opposite direction of revolutions, gave the second.

Taking also into account the corrections caused by deviations of the pendulum trajectories from the exact form of a circle (this form is established gradually), Bravais found the value of \(\omega\) with one pendulum to an accuracy of \(3.8\%\), and with two—almost to \(1\%\).

if, despite this, his experiment, in contrast to Foucault’s experiments, is far less well known and was never repeated, this can be explained by the fact that observations of discrepancies in time for Bravais’s pendulum make less of an impression than observations of spatial deviations in Foucault’s experiment—and, in general, Foucault’s experiment acts upon people much more directly and convincingly than Bravais’s experiment, though the latter is just as valuable scientifically. It should be noted, however, that the experiments of Foucault and Bravais are only the last in a whole series of possible experiments on the oscillations of a spherical mathematical pendulum (see § 14).

B. Investigation of vertical rotation.

6. Scales. A complete investigation of the earth’s rotation \(\omega\) without astronomical observations, which would be necessary for determining the geographical latitude \(\varphi\), requires, strictly speaking, also the determination of the vertical rotation of the place of observation \(\omega_2=\omega\cos\varphi\). For this purpose repeated attempts have been made to apply the phenomenon of inertia of a rotating body, which is usually called centrifugal force.

All terrestrial bodies are subject to a centrifugal force arising from the earth’s rotation \(\omega\). Vertical rotation \(\omega_2\), in particular, has as its consequence the circumstance that the weight of a body at rest on the earth’s surface is somewhat less than that which would result from the earth’s attraction alone. One could immediately draw a conclusion as to the magnitude of \(\omega_2\) if we were able to separate the earth’s attraction and the centrifugal force; but this cannot be done directly with scales, for they always indicate only the difference of the two forces. Nevertheless, if one raises a body to a height \(h\) above the earth’s surface, then both the centrifugal force and the earth’s attraction change. Since this change takes place according to different laws, the required separation of the two component parts can be achieved by a new weighing, after which the value of \(\omega_2\) can readily be calculated. It goes without saying that in both weighings the weights must be at one and the same height; therefore the scales must be set at the height \(h\), and the body weighed once on the pan of the scales itself, and a second time by attaching it to the pan of the scales by means of a tared thread of length \(h\).

If \(m\) is the mass of the body, \(g\) the acceleration of gravity at the earth’s surface, and \(R\) the radius of the earth, then the centrifugal force (more precisely, its vertical component) at the earth’s surface will have the value \(mR\omega_2^2\). The weight determined will then be

\[ G=mg-mR\omega_2^2. \]

At the height \(h\) above the earth’s surface the determined weight will be expressed—if one takes into account that the decrease of gravity is proportional to the square of the distance

from the center of the earth, while the increase of the centrifugal force is proportional to the first power of this distance,—then:

\[ G' = mg\frac{R^{2}}{(R+h)^{2}} - m(R+h)\omega_2^{2}. \]

Assuming, with sufficient accuracy,

\[ \frac{R^{2}}{(R+h)^{2}} \simeq 1-\frac{2h}{R}, \]

by subtraction we obtain

\[ G-G'=\frac{2mgh}{R}+mh\omega_2^{2}, \]

and therefore

\[ \frac{G-G'}{G}=\frac{h}{R}\cdot \frac{2g+R\omega_2^{2}}{g-R\omega_2^{2}} \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (18) \]

If by weighing the quantity

\[ \frac{R}{h}\cdot\frac{G-G'}{G}=A \]

has been found, then from (18) we obtain

\[ \omega_2^{2}=\frac{g}{R}\frac{A-2}{A+1} \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (19) \]

It is true that, despite certain attempts, up to now it has not been possible to carry out weighing with such accuracy that the investigation of vertical rotation by this method could be regarded as completed.

Fig. 10. Vertical and horizontal components of the centrifugal force of the earth at different heights.

7. Torsion balances. Somewhat greater success may be expected if this experiment is modified in such a way that, instead of ordinary balances, torsion balances are used. The use of torsion balances, taking into account their extraordinary sensitivity, is based on the fact that, owing to the addition of the azimuthal rotation \(\omega_1\), the direction of the centrifugal force is in reality not vertical, but parallel to the plane of the terrestrial equator (Fig. 10). We have already spoken about the vertical components \(Z\) and \(Z'\) of the centrifugal force; but in addition there are also southern components \(S\) and \(S'\), whose difference \(\Delta S\), as a horizontal force, should be well determined by means of torsion balances.

The centrifugal force has, at the surface of the earth, the magnitude

\[ F=mR\cos\varphi\cdot\omega^{2}, \]

and at height \(h\) above the earth

\[ F'=m(R+h)\cos\varphi\cdot\omega^{2}. \]

Its southern components

\[ S=mR\omega^{2}\cos\varphi\sin\varphi=mR\omega_1\omega_2, \]

\[ S'=m(R+h)\omega^{2}\cos\varphi\sin\varphi=m(R+h)\omega_1\omega_2 \]

have the difference

\[ \Delta S=mh\omega^{2}\cos\varphi\sin\varphi=mh\omega_1\omega_2 \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (20) \]

The corresponding experiment, it seems, had already been carried out in 1832 by Hengler¹); it was repeated in 1910 by Hagen²) with a better result, but still without satisfactory accuracy. Yet it was significant at least in that it undoubtedly showed the expected deflection of the torsion balance. If the balance, which in Hagen’s case consisted of a bifilarly suspended block with a horizontal axis, is set from east to west (the axis of the block from south to north), and if at the beginning the eastern mass hangs lower than the western one, then as soon as the western mass is lowered and the eastern one raised, the balance must deflect in the direction \(NOSW\).

1. Rotating balances. Both with simple balances and with rotating balances, the experiments are unusually complicated by the circumstance that in both cases effects of the order of magnitude \(\omega^2\) must be measured. This is due to the fact that in the product \(R\omega^2\), which enters into the magnitude of the centrifugal force, up to now only the first factor has been varied. But instead of this one can make the first factor constant and vary the second, namely in such a way as simply to impart to the body under test a horizontal velocity \(v\) in the direction east or west. In the first case, so to speak, the vertical velocity \(\omega_2\) is increased; in the second it is decreased by the same amount, so that the mass \(m\) of the body under test is in the first case subjected to an increased, and in the second to a decreased, centrifugal force, and thus its weight is correspondingly decreased or increased.

Namely, the vertical component of the centrifugal force during the motion will be

\[ mR\left(\omega_2 \pm \frac{v}{R}\right)^2. \]

Expanding the brackets and at the same time neglecting \(v^2\), as very small in comparison with \(R^2\omega_2^2\) (the square of the circumferential speed of the earth’s rotation) for not very high geographic latitudes, we obtain the magnitude of the loss or increase in weight

\[ \Delta G = \mp 2mv\omega_2 = \mp 2mv\omega \cos \varphi, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (21) \]

which for bodies weighing \(1\) kg, at a speed of \(1\) mt/sec, in our latitudes already gives about \(\mp 10\) mgr.

These differences in weight were pointed out by Eötvös (R. Eötvös) in connection with Hecker’s measurements of gravity at sea (1901–1908). At Eötvös’s³) suggestion, the best way to carry out this experiment in the laboratory is with the aid of a balance beam carrying equal masses at its ends, which can swing about a horizontal axis. If a pulley is mounted on the support of the balance beam (Fig. 11) and it is rotated uniformly about a vertical axis, then

¹) Hengler. Dinglers polyt. Journ. 43, 81. 1832.
²) J. G. Hagen, loc. cit., p. 151.
³) R. Eötvös. Ann. d. Physik (4) 59, 743. 1919. Cf. also D. Pekar. Naturwissenschaften, 7, 389. 1919.

Fig. 11. Arrangement of Eötvös’s experiment.

the masses move alternately eastward and westward; their weights pulsate in the rhythm of the rotation and produce oscillations of the beam about the horizontal axis. These oscillations will be greatest and most readily observable when they come into resonance with the proper oscillations of the beam. Eötvös, who performed such a demonstration experiment in 1917, also showed how, in this case, measurements of high accuracy could be made by comparatively simple means.

In addition, we note that all the experiments of this group (§§ 6, 7, and 8) possess the property of reversibility.

II. Experiments based on the law of areas.

A. Proof of azimuthal rotation.

9. Isotomeograph. The year 1851 is known not only for the experiments on the pendulum by Foucault and Bravais. In the same year Poinsot¹) proposed applying the law of areas to prove azimuthal rotation. Poinsot proceeds from the fact that a body at rest relative to a system connected with the earth, whose principal axis of inertia is vertical and whose moment of inertia relative to this axis is equal to $A$, receives, as a consequence of the azimuthal rotation $\omega$, an impulse $A\omega$ (cf. Introduction, p. 342), whose vertical vector changes neither its magnitude nor its direction relative to the terrestrial system, provided that the rotating moment $\mathfrak{M}$ relative to the vertical axis is equal to zero. To achieve this it is sufficient to suspend the body so that there is neither friction nor torsion. If, under the action of internal forces alone, displacements of masses occur within the body, as a result of which the moment of inertia receives a new value $A'$, then the angular velocity must change so that the impulse does not change its value, i.e.

\[ A'\omega_1 = A\omega_1, \]

for the internal forces, by the law of action and reaction, cancel in pairs and therefore give no moment. After the displace—

¹) L. Poinsot. Comptes rendus 32, 206. 1851; see also Tessan, ibid., p. 504.

of the masses there occurs a decrease or increase of the actual angular velocity by such an amount as corresponds to the increase or decrease of the moment of inertia; thus there takes place an apparent azimuthal rotation of the body relative to the Earth system with angular velocity

\[ \varepsilon = \omega'_1 - \omega_1 = \omega_1\left(\frac{A}{A'} - 1\right), \ldots \ldots \ldots \ldots \ldots \ldots \ldots (22), \]

which we shall regard as positive in the direction \(NWSO\).

The experiment was carried out with great success in 1910 and 1911, and with tenfold accuracy in 1919 by Hagen\(^1\) in the following manner: a horizontal beam (Fig. 12) was suspended by means of a bifilar suspension and was provided with movable auxiliary masses, which could be moved in complete symmetry with respect to the middle of the beam to either side.

Fig. 12. Hagen’s isotomeograph.

Let \(m\) denote the sum of these auxiliary masses, \(a\)—their smallest and \(b\)—their greatest distance from the vertical middle axis of the beam; suppose that at first the beam is at rest relative to the Earth, with the auxiliary masses occupying the nearest position relative to the axis of the beam; then

\[ A = A_0 + a^2m,\qquad A' = A_0 + b^2m, \]

where \(A_0\) is the moment of inertia of the beam itself. After the masses are moved to the ends of the beam, rotation in the direction \(NOSW\) must appear at the first instant; its angular velocity, on the basis of (22), will be

\[ \varepsilon_1 = \frac{(b^2-a^2)m}{A_0+b^2m}\,\omega_1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots (23) \]

Of course, the braking moment of the suspension will immediately begin to affect this rotation, giving rise to horizontal torsional oscillations of the beam; the initial angular velocity \(\varepsilon_1\) will then be obtained in the known manner from the amplitude, the period of oscillation, and the logarithmic decrement. In exactly the same way, if the auxiliary masses, while the beam was at rest, were at the ends of the beam, then moving the masses toward its axis will produce rotation with angular velocity

\[ \varepsilon_2 = \frac{(b^2-a^2)m}{A_0+a^2m}\,\omega_1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots (24) \]

in the direction \(NWSO\). This velocity will be greater than \(\varepsilon_1\), since \(a < b\).

Hagen, at the suggestion of Stephanos (C. Stephanos), called his apparatus an isotomeograph (from the Greek: \(\ἴσος\)—equal, \(\tauομεύς\)—area of a sector (cf. Kepler’s 2nd law, p. 343)) and modified it further in that respect—

\(^1\) J. G. Hagen, loc. cit., p. 135, also supplement 2, p. 9, and Zeitschrift f. Instr.-Kunde 40, 65. 1920.

... (Fig. 13), that instead of the beam he took a vertically suspended frame \((b)\), in which there are three quadrilaterals made of aluminum tubes and capable of rotating about a horizontal axis \((a)\); (in drawing I the three quadrilaterals cover one another, while in drawing II they are side by side). The added masses \((m)\) are placed at the opposite corners of each quadrilateral in such a way that on the front and rear quadrilaterals there is \(\frac{1}{4}m\) each, and on the middle one \(\frac{1}{2}m\).

Fig. 13. Second form of Hagen’s isotomeograph.
Front view and side view.

The front and rear quadrilaterals were turned by \(90^\circ\) to one side, and the middle one by \(90^\circ\) to the other, so that before rotation all the masses lay either on the vertical axis of the apparatus or on the horizontal axis perpendicular to the axis of rotation of the quadrilaterals.

The great accuracy which Hagen ultimately attained (\(\omega_1\) was obtained with an accuracy to the third decimal place) depended very substantially on the circumstance that these experiments possess the valuable property of reversibility, as is evident from formulas (23) and (24).

10. Hydraulic experiment. If the law of areas is applied to flows of liquid, as was first done by Perrot (Perrot)\(^1\) and Combes (Combes)\(^2\) in 1859, then one obtains an interesting modification of the experiment just described. The experiment was carried out, with certain quantitative results, by O. Tumlirz (O. Tumlirz)\(^3\) in 1908, in such a way that water flowed into the space enclosed between two horizontal round glass plates in a strictly radial direction and flowed out at the center.

When the jets, colored in an appropriate manner, became visible, they revealed a noticeable spiral structure

\(^1\) Perrot. Comptes rendus 49, 637. 1859.

\(^2\) Combes. Ibid., p. 775.

\(^3\) O. Tumlirz. Sitzungsber. Wien, 117. 1908; Math. Naturw. kl., 2 Abt., p. 819.

(Fig. 14) instead of the purely radial one which they would have had to discover in an inertial system.

Indeed, each liquid ring, at the moment when all its particles simultaneously enter from outside into the region between the plates, possesses, owing to the azimuthal rotation \(\omega_1\), a certain momentum which it tends invariably to preserve. To the same extent as the ring contracts and thereby diminishes its moment of inertia, its angular velocity of rotation in the direction \(\omega_1\) must increase; hence there follows directly the spiral flow of the colored jets.

It is also easy to obtain the form of the spiral curves. Let \(a\) be the radius of the plate; let us consider only one liquid ring with mass \(dm\), from the moment when it, with moment of inertia \(a^2 dm\), enters the region between the plates with angular velocity \(\omega_1\), until the moment when it has contracted to radius \(r\) and has moment of inertia \(r^2 dm\) and angular velocity \(\omega'_1\). Denote its angular velocity of rotation relative to the earth system by \(\dfrac{d\chi}{dt}\); then, on the basis of (22), p. 355, we shall have

\[ \frac{d\chi}{dt}=\omega_1\left(\frac{a^2}{r^2}-1\right)\ldots\ldots (25). \]

If we take into account that the quantity of liquid flowing per unit time through the surface of a cylinder coaxial (coaxial) with the plates, of radius \(r\), at radial velocity \(\dfrac{dr}{dt}\) and with distance between the plates \(b\), is equal to

\[ 2\pi rb\,\frac{dr}{dt}, \]

Fig. 14. Horizontal radial flow under the action of the earth’s rotation

and that, by continuity, it must be one and the same for all values of \(r\), we may put

\[ \frac{dr}{dt}=\frac{c}{r},\ldots\ldots\ldots\ldots\ldots\ldots (26) \]

where \(c\) is a constant which is easy to determine. Dividing equations (25) and (26) by one another, we obtain the differential equation of the streamlines in the form:

\[ \frac{d\chi}{dr}=\frac{\omega_1}{c}\left(\frac{a^2}{r}-r\right), \]

whose integral is

\[ \chi-\chi_0=\frac{\omega_1}{c}\left(a^2\lg r-\frac{r^2}{2}\right)\ldots\ldots\ldots\ldots\ldots (27) \]

gives, in polar coordinates \((r,\gamma)\), the equation of the family of spiral curves observed by Tumlirz (\(\gamma_0\) is a parameter corresponding to a given curve of the family).

Up to now it has not been possible to carry out an experiment that is fully satisfactory in quantitative respect, just as it has not been possible to carry out the converse experiment, i.e. a radial flow from the middle outward with the opposite, but smaller, effect.

B. Proof of the vertical rotation.

11. Motion of a thrown body. In order to prove, by means of the theorem of areas, the vertical rotation \(\omega_2\), it is necessary first of all to make use of motions in the vertical plane \(E\), perpendicular to the meridian plane of the place of observation \(A\), and consequently also to the vector \(\upsilon_2\) (cf. Fig. 1, p. 340). Let us imagine, for example, two bodies thrown in the emptiness with equal initial velocities \(v\) at equal angles of elevation \(a\), one to the east and the other to the west (Fig. 15); then to the vector \(\eta\) there must be added, according to the rule of vector addition, the velocity vector of the place of observation \(\upsilon\), directed eastward. The absolute value of \(\upsilon\) is

Fig. 15. Apparent and true directions of departure in eastern and western shots.

Fig. 16. Trajectories of flights in eastern and western shots.

\[ \upsilon = R\omega_2 = R\omega\cos\varphi, \]

where \(R\) is the radius of the earth. If \(a\) is not equal to zero, then the true angle of elevation \(a_o\) for the body thrown to the east will obviously be smaller than the same angle \(a_w\) for the body thrown to the west; at the same time the true initial velocity \(v_o\) of the eastern one will be greater than the same velocity \(v_w\) for the western one. The bodies describe arcs of Keplerian ellipses, one focus of which is the center of the earth \(O\) (Fig. 16—the continuation of Fig. 15 in the same plane of the drawing). It is clear that these ellipses must be quite different and that their least distances from the center of the earth are very different. Nevertheless, both motions also proceed differently for an observer on the earth. An exact calculation1 would show

that for a greater angle of elevation the range of flight for a body thrown eastward would be somewhat less than for a body thrown westward, while at a small angle of elevation the action is exactly the opposite. The difference is so insignificant that in reality it disappears completely, owing to the nonuniformity of air resistance and the inaccuracy of throwing. At least, repeatedly repeated experiments with shots, even at angles of elevation of \(90^\circ\), have not revealed the slightest effect. It may, however, be assumed with sufficient probability that the effect would be obtained if (cf. §§ 3 and 4) the experiments were carried out with very small velocities, but so far this has not been done.

Although air resistance, as has already been mentioned, strongly affects the trajectory, nevertheless the range of flight, at least for both limiting cases (for the horizontal direction eastward and westward and for the vertical direction upward), can be calculated without taking air resistance into account. The result, of course, has significance only in a qualitative sense.

Let us first suppose that a body is thrown horizontally at a height \(h\) above the surface of the earth with velocity \(v\) eastward or westward. Then its true velocity relative to an inertial system at rest at the center of the earth will be

\[ \begin{aligned} v_0&=v+u=v+(R+h)\omega_z,\\ v_A&=v-u=v-(R+h)\omega_z \end{aligned} \qquad\}\ . . . . . . . . . . . . .\ (28), \]

where the positive direction is taken, respectively, to be eastward and westward.

The double sectorial velocity of the radius vector drawn from the center of the earth to the body thrown eastward has the value \((R+h)v_0\), but later, when the body has fallen to the height \(z\), it will be \((R+z)^2\dot{\psi}\), if \(\psi\) is the angle between the initial radius vector and the radius vector for the height \(z\). By the law of areas, i.e. by Kepler’s second law, we have

\[ (R+h)v_0=(R+z)^2\dot{\psi}. \]

Taking into account that \(h\) and \(z\) must be very small in comparison with \(R\), so that their squares and products may be neglected, we obtain

\[ \dot{\psi}=v_0\frac{R+h}{(R+z)^2}\simeq \frac{v_0}{R}\left(1+\frac{h}{R}\right)\left(1-\frac{2z}{R}\right)\simeq \frac{v_0}{R}\left(1+\frac{h}{R}-\frac{2z}{R}\right); \]

after integration over the entire time of flight \(t_0\), the angle described by the radius vector is expressed as

\[ \psi_0=\frac{v_0t_0}{R}\left(1+\frac{h}{r}\right)-\frac{2v_0}{R^2}\int_0^{t_0} z\,dt. \]

Within the limits of accuracy with which we are calculating here, one may use for the vertical projection of the motion the ordinary formulas of falling, i.e. without air resistance,

\[ z=h-\frac{1}{2}gt^2 \]

\[ h=\frac{1}{2}gt_0^2, \]

so that the integral

\[ \int_0^{t_0} z\,dt = h t_0 - \frac{1}{6} g t_0^3 = \frac{2}{3} h t_0 . \]

Thus we obtain

\[ \zeta_0 = \frac{v t_0}{R}\left(1 - \frac{1}{3}\frac{h}{R}\right), \]

or, in consequence of (28),

\[ \zeta_0 = \frac{v t_0}{R}\left(1 - \frac{1}{3}\frac{h}{R}\right) + \omega_2 t_0\left(1 + \frac{h}{R}\right) \left(1 - \frac{1}{3}\frac{h}{R}\right) \simeq \frac{v t_0}{R}\left(1 - \frac{1}{3}\frac{h}{R}\right) + \omega_2 t_0\left(1 + \frac{2}{3}\frac{h}{R}\right). \]

And the very same result is obtained for a body thrown to the west, assuming in the first approximation that \(t_w = t_0\), namely,

\[ \zeta_w \simeq \frac{v t_0}{R}\left(1 - \frac{1}{3}\frac{h}{R}\right) - \omega_2 t_0\left(1 + \frac{2}{3}\frac{h}{R}\right). \]

Finally, taking into account that the point of departure, owing to the rotation of the earth, will move eastward by the distance \(R\omega_2 t_0\), we obtain, finally, for the ranges of eastward and westward flight the expressions:

\[ \left. \begin{aligned} x_e &= R\zeta_0 - R\omega_2 t_0 = v t_0\left(1 - \frac{1}{3}\frac{h}{R}\right) + \frac{2}{3}\omega_2 h t_0,\\ x_w &= R\zeta_0 + R\omega_2 t_0 = v t_0\left(1 - \frac{1}{3}\frac{h}{R}\right) - \frac{2}{3}\omega_2 h t_0 \end{aligned} \right\} \qquad\qquad (29) \]

Thus, the range of the eastward flight, for an angle of elevation equal to zero and equal initial velocities, will in fact be somewhat greater than the westward one.

Let, next, a body be thrown vertically upward. In this case the law of areas gives

\[ R^2\omega_2 = (R+z)^2 \dot{\zeta}_1, \]

whence

\[ \dot{\zeta} = \omega_2 \frac{R}{(R+z)^2} \simeq \omega_2\left(1 - \frac{2z}{R}\right); \]

the angle described by the radius vector during the whole time of fall until return to the earth, \(t_1\), will be

\[ \zeta_1 = \omega_2 t_1 - \frac{2\omega_2}{R}\int_0^{t_1} z\,dt . \]

Here also, with sufficient accuracy, the formulas for fall with initial velocity \(v_1\) are applicable,—

\[ z = v_1 t - \frac{1}{2}gt^2 \]

\[ t_1 = \frac{2v_1}{g}, \]

\[ h = \frac{1}{8}gt_1^2, \]

so that, after a short calculation, we obtain

\[ \zeta_1 = \omega_2 t_1 - \frac{4}{3}\frac{\omega_2 h t_1}{R}. \]

It follows from this that the point of fall will lag behind the point of departure to the west by the amount

\[ x_1=R\omega_2 l_1-R\dot{\varphi}_1=\frac{4}{3}\omega_2 h t_1 \ldots \ldots \ldots \ldots \ldots \ldots (30) \]

12. Free fall. In contrast to vertical projection upward, free fall downward gives an easterly deflection, which Newton had already indicated, and which, at the same height, if air resistance is not taken into account, is four times smaller than the westerly deflection required by the theory (and just calculated) for vertical projection upward.

The easterly deflection of the point of the plumb line is obtained from the first formula (29), if the initial velocity is put \(v=0\), namely

\[ x_0=\frac{2}{3}\omega_2 h t_0 \ldots \ldots \ldots \ldots \ldots \ldots (31), \]

as was already found by Laplace \(^{1}\). This deflection is indeed equal to one fourth of \(x_1\) (30), since \(t_0=\frac{1}{2}t_1\).

The easterly deflection increases, for one and the same height of fall, together with the time of fall, so that, owing to air resistance, which should increase the time of fall, it increases. For the theory it presents no substantial difficulty to take into account the influence of air resistance. However, the many experiments already carried out are such that it is not worth making a more exact calculation. In all these experiments there were obtained deflections in all directions far greater than those expected; only the mean value usually agreed in direction and approximately in magnitude with the theoretical value. We shall mention the experiments \(^{2}\) of Guglielmini in Bologna (1790/92), Benzenberg in Hamburg (1802), Schlebusch (1804), Reich in Freiberg (1831), Hall in Cambridge (Massachusetts) (1902), and Flammarion in Paris (1903). If, in each series of experiments, we plot the points of fall on cardboard, we shall see a picture resembling, as Hagen aptly observed, a target (cf. Fig. 17, p. 362) at which marksmen of varying skill have fired. The probability that one of the bullets will fall east of the vertical line, even in the very best experiments, is only \(2:3\). Under such circumstances it would hardly have been possible to say that experiments on free fall can serve as incontrovertible proof of the rotation of the earth.

\(^{1}\) P. S. de Laplace, Mécanique céleste, 1805, p. 300.
\(^{2}\) For the literature on this question see Encykl. d. Math. Wiss., 4, Nr. 1 II, Heft 1: Ph. Furtwängler, Die Mechanik der einfachsten physikalischen Apparate, pp. 6 and 50.

Fig. 17. Points of fall in Flammarion’s experiment, with their center of gravity \(S\) in natural size.

The experimental difficulties1 (the exact determination of the point of the plumb line, the isolation of the falling body from extraneous influences) are too great for one to be able to think of an exact quantitative determination of the deviation; at a fall height of 80 m it has a magnitude of about 1 cm. However, a comparison of Hall’s experiments—fall height 23 m, mean error \(3.3\%\)—and Flammarion’s—fall height 68 m, mean errors \(22\%\)—indicates that better results can be obtained only at small fall heights.

13. Atwood’s machine. In order to obtain a well-measured effect at small fall heights, it is necessary to increase the time of fall, since according to theory the easterly deviation is proportional to the product of the fall height and the time of fall (cf. formula (31), p. 361). This gave Hagen2 the very fortunate idea

to use Atwood’s machine, which, as is known, makes it possible to reduce the acceleration of falling \(g=9.81\,\dfrac{\mathrm{mt.}}{\mathrm{sec.}^{2}}\) to an arbitrarily small value \(g'\). A mass suspended on a thread falls downward; the thread passes over a pulley and at the other end is loaded with a weight, which produces the retardation of the fall. Hagen carried out his experiments in the Vatican in 1912; at a height of fall of \(23\ \mathrm{m}\) and an acceleration approximately equal to \(\dfrac{1}{25}g\), they gave an easterly deflection of about \(0.9\ \mathrm{mm}\), in complete agreement with the theory to an accuracy of \(1\%\). Thanks to this, the difficulties in experiments on free fall could be regarded as overcome.

To compute the easterly deflection, let us first imagine that the body falls freely downward. We shall adhere to the former notation (p. 359), with only the difference that now \(z\) is to denote not the distance from the earth, but the height traversed; then the law of areas gives

\[ (R+h)^2\omega_2=(R+h+z)^2\psi_2; \]

whence

\[ \psi=\omega_2\frac{(R+h)^2}{(R+h-z)^2}\simeq \omega_2\left(1+\frac{2z}{R+h}\right)\simeq \omega_2\left(1+\frac{2z}{R}\right), \]

and thus the easterly acceleration is obtained with sufficient accuracy:

\[ R\dot{\psi}=2\omega_2\dot{z}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (32) \]

To apply this formula to non-free fall on Atwood’s machine, we must, of course, first of all introduce for the velocity of fall \(\dot{z}\) not the velocity of free fall \(gt\), but \(g't\), and, secondly, add the quantity which we shall obtain by taking into account that the tension of the thread, owing to the easterly deflection, has a component directed westward. To this component, by the fundamental law of dynamics, there corresponds the same component of acceleration, which we must compute and then subtract from the right-hand side of (32).

Let \(x\) denote the easterly deflection of the mass \(m\), after it has fallen from the height \(z\), \(\vartheta\) the angle of the easterly deflection of the thread at this instant, and \(S\) the tension of the thread; then this additional component of acceleration will be equal to

\[ \frac{S}{m}\sin\vartheta. \]

Taking into account the smallness of the angle \(\vartheta\), we may put

\[ \sin\vartheta\simeq \frac{x}{z}. \]

Further, the tension of the thread will be

\[ S=m(g-g'), \]

since the weight \(mg\) of the falling mass, diminished by the tension of the thread \(S\), must be equal to the effective force of fall \(mg'\).

Thus the required westward component of acceleration will be

\[ (g-g')\frac{x}{z}. \]

We subtract it from the right-hand side of (32) and introduce further

\[ z=\frac{1}{2}g't^2; \]

denoting the found true eastward acceleration instead of \(R\dot\psi\) by \(\ddot{x}\), we obtain, instead of (32),

\[ \ddot{x}=2\omega_2g't-2\frac{g-g'}{g'}\cdot\frac{x}{t^2}. \]

This differential equation of non-free fall is easy to integrate. The integral, taken under the initial conditions—at \(t=0\), \(\dot{x}=0\) and \(x=0\)—after the corresponding calculations, is obtained in the form

\[ x=\frac{g'^2}{2g'+g}\,\omega_2t^3. \]

The total deflection to the east will therefore be

\[ x_0=\frac{g't_0^3}{2g'+g}\,\omega_2. \]

Since

\[ h=\frac{1}{2}g't_0^2, \]

one may also write

\[ x_0=\frac{2g'}{2g'+g}\,\omega_2ht_0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (33) \]

and compare the result with the corresponding deflection (31) in free fall, in which \(g'\) must again pass into \(g\).

It should also have been added to this that the exact theory gives, for the fall of a body, also a southern deflection, but it is a quantity of order \(\omega^2\), and thus too insignificant for one to hope to detect it.

The experiment with Atwood’s machine allows the same modification as leads from the isotomeograph to the experiment of Tumlirz (cf. § 10). A vertical tube is filled with liquid, a float is placed exactly in the middle of the surface of the liquid, and a small tap at the bottom of the tube is immediately opened; then the float slowly descends together with the surface of the liquid, and it is to be expected that the float will be deflected to the east. The experiment was carried out by Maillard (Mail-ard)¹) in 1908 with a very doubtful result.

III. Experiments based on the law of impulse (momentum).

A. Investigation of azimuthal rotation.

14. Physical pendulum. In one of his letters to A. Humboldt in 1853, C. F. Gauss²) points out that the experiment with a pendulum, performed by Foucault, can be applied to the physical

¹) L. Maillard, Comptes rendus 147, 524. 1908.
²) Briefe zwischen A. v. Humboldt und C. F. Gauss, Leipzig 1877, p. 66.

pendulum with correspondingly small dimensions. Gauss imagines a pendulum suspended on a gimbal in such a way that it can swing in all directions. Its moments of inertia about horizontal axes passing through the point of suspension (the center of the gimbal suspension) must have an equal value \(B\); the moment of inertia about the vertical axis is equal to \(A\). In considering Foucault’s pendulum, one may completely neglect the quantity \(A\) in comparison with \(B\); with a physical pendulum, however, as Gauss pointed out, this can no longer be done. In consequence of the azimuthal rotation \(\omega_1\), the pendulum receives a small additional impulse \(A\omega_1\), whose vector is directed upward along the vertical axis of the pendulum. The action of such an additional impulse can be clearly observed on a gyroscopic pendulum, i.e. on a physical pendulum whose bob is replaced by a top, to which an intrinsic rotation about the axis of the pendulum can be imparted (Fig. 18).

Fig. 18. Gyroscopic pendulum.

On such a gyroscopic pendulum it is perfectly evident that the plane of oscillation is not fixed in space, as it is in an ordinary physical pendulum, but immediately begins to rotate rapidly about the vertical line, if only the top has a perceptible intrinsic rotation. This intrinsic rotation corresponds to the additional impulse mentioned above. A theory—by no means quite elementary—based on the law of impulse shows that (though this is also quite directly clear) the angular velocity with which the plane of oscillation rotates is proportional to the additional impulse \(A\omega_1\) and inversely proportional to \(B\), and has the exact value \(A\omega_1/2B\), so that the apparent rotational motion given by formula (6), p. 347, must be supplied with a theoretical correction, measured by the fraction \(A/2B\), which in reality is very small. In other words, the apparent rotation of the plane of the pendulum has not simply the angular velocity \(\omega_1\), but (if we denote the quotient \(a/l\), the so-called angular amplitude, by \(\vartheta_1\))

\[ -\omega_1\left[1-\frac{1}{3}\vartheta_1^{\,2}-\frac{A}{2B}\right]\ldots\ldots\ldots\ldots\ldots (34). \]

The last two terms in the brackets give the final correction, namely, first, the influence that imparts to the pendulum the rotation of the earth at the beginning of the motion (cf. p. 346), and secondly, the influence of the inertia of its own mass.

To compute the last correction term, imagine (Fig. 19) a plane \(E\), invariably connected with the pendulum in such a way that it will be perpendicular to the so-called axis of the pendulum, i.e., to the straight line passing through the point of suspension \(O\) and through the center of gravity of the pendulum \(S\).

We shall call the line of greatest slope on the plane \(E\) the transverse axis, and the horizontal straight line on the plane \(E\) the nodal line (the name is borrowed from astronomy). The deflection of the pendulum from its position of rest is measured by the angle \(\vartheta\); then the velocity \(\dot{\vartheta}\) will be represented by a vector directed along the nodal line. If, as we already know, the pendulum does not undergo plane oscillations, but the plane of its swing slowly rotates about the vertical axis, then together with it the plane \(E\) also rotates; we shall measure the resulting horizontal rotation of the nodal line by the angle \(\psi\), and then the velocity of this rotation \(\dot{\psi}\) is represented by a vector directed along the vertical with the positive direction downward, so that the positive direction of rotation will be \(NOSW\). As we already know from carrying out the experiment, the pendulum, beginning its motion from its extreme position \(\vartheta_1\), has the azimuthal velocity \(\omega_1\), which, with our notation, must be considered negative.

Fig. 19. Toward the theory of Gauss’s pendulum.

This initial velocity \(\omega_1\) gives, on the axis of the pendulum, the component \(\omega_1\cos\vartheta_1\), and on the transverse axis, \(-\omega_1\sin\vartheta_1\); on the nodal line the component is zero. All three axes, by our assumption, are principal axes of inertia; the components of angular momentum along these axes are obtained from the above-indicated components of rotation by multiplying them by the corresponding principal moments of inertia (cf. p. 342). The initial components of the angular momentum imparted to the pendulum will then be (see Fig. 19)

\[ \mathfrak{S}_1^0=A\omega_1\cos\vartheta_1, \]

\[ \mathfrak{S}_2^0=B\omega_1\sin\vartheta_1, \]

\[ \mathfrak{S}_3^0=0, \]

and the initial component of angular momentum along the vertical will be

\[ \mathfrak{S}_4^0=\mathfrak{S}_1^0\cos\vartheta_1+\mathfrak{S}_2^0\sin\vartheta_1 =\omega_1\left(A\cos^2\vartheta_1+B\sin^2\vartheta_1\right). \]

The moment \(\mathfrak{M}\) of the force of gravity \(\mathfrak{G}\) is a vector which lies in the nodal line and always remains horizontal, and therefore perpendicular to the axis of the pendulum. Since, by the law of angular momentum (see p. 342), the vector \(\mathfrak{M}\) gives the direction in which the end of the angular-momentum vector moves, in our case this end moves all the time horizontally and therefore perpendicular to the axis of the pendulum. This means that the components \(\mathfrak{S}_1\) and \(\mathfrak{S}_4\) of the angular momentum along the axis of the pendulum and the vertical axis do not change during the whole motion, so that at all times

\[ \mathfrak{S}_1=A\omega_1\cos\vartheta_1 \ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (35) \]

\[ \mathfrak{S}_4=\omega_1\left(A\cos^2\vartheta_1+B\sin^2\vartheta_1\right) \ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (36) \]

also when the deflection \(\vartheta_1\) decreases to any other value \(\vartheta\). The component of the momentum along the transverse axis \(\mathfrak{S}_2\), of course, will change. In general (Fig. 19)

\[ \mathfrak{S}_1=\mathfrak{S}_1\cos\vartheta+\mathfrak{S}_2\sin\vartheta, \]

whence, if (35) and (36) are taken into account, it follows that

\[ \mathfrak{S}_2=\omega_1\frac{A\cos\vartheta_1(\cos\vartheta_1-\cos\vartheta)+B\sin^2\vartheta_1}{\sin\vartheta}\ . . . . . . . . . . . . \tag{37} \]

The motion of the pendulum is composed, first, of an oscillatory motion with velocity \(\dot{\vartheta}\), which gives no component along the transverse axis; second, of rotation with velocity \(\dot{\psi}\), which has on the transverse axis the component \(\dot{\psi}\sin\vartheta\); and, third, of a possible proper rotation of the body of the pendulum about its own axis, which gives no components along the transverse axis. We can express the component of the momentum along the transverse axis also in the following form:

\[ \mathfrak{S}_2=B\dot{\psi}\sin\vartheta\ . . . . . . . . . . . . . . \tag{38} \]

Formulas (37) and (38) give

\[ B\dot{\psi}=\omega_1\frac{A\cos\vartheta_1(\cos\vartheta_1-\cos\vartheta)+B\sin^2\vartheta_1}{\sin^2\vartheta}\ . . . . . . . . . . \tag{39} \]

From this point on we shall carry out the calculations with an approximation that is possible only for small angles of deflection; we shall replace the series of quantities

\[ \cos\vartheta_1;\ \cos\vartheta_1-\cos\vartheta;\ \sin^2\vartheta_1;\ \sin^2\vartheta \]

by the following:

\[ 1;\ \frac{1}{2}(\vartheta^2-\vartheta_1^2);\ \vartheta_1^2;\ \vartheta^2. \]

For abbreviation let us introduce also the number

\[ k=\frac{2B-A}{A}, . . . . . . . . . . . . . . \tag{40} \]

which in fact is always positive and fairly large; then from (35) we obtain our first fundamental equation in the form

\[ B\dot{\psi}=\frac{A\omega_1}{2}\frac{\vartheta^2+k\vartheta_1^2}{\vartheta^2}\ . . . . . . . . . . . \tag{41} \]

Let us now apply the law of conservation of energy. Let \(s\) be the distance of the center of gravity of the pendulum from the point of suspension; then the potential energy, reckoned from the position of rest \(\vartheta=0\) to any position \(\vartheta\), is equal to \(sG(1-\cos\vartheta)\). On the other hand, the kinetic energy consists of three parts, which we can easily calculate. To this end we shall place in one table the components of the angular velocities along all three axes, bearing in mind that, since the component of the momentum \(\mathfrak{S}_1\) does not change, the corresponding component of rotation (in fact, the sum of the forced proper rotation and the component \(\dot{\psi}\cos\vartheta_1\), which is obtained from projecting the vertical vector \(\dot{\psi}\) onto the axis of the pendulum) always has the initial value \(\omega_1\cos\vartheta_1\).

Components of rotation.	For any \(\vartheta\)	For \(\vartheta=\vartheta_1\)
Axis of the pendulum . . . .	\(\omega_1\cos\vartheta_1\)	\(\omega_1\cos\vartheta_1\)
Transverse axis . . . .	\(\dot{\psi}\sin\vartheta\)	\(\omega_1\sin\vartheta_1\)
Nodal line . . . .	\(\dot{\vartheta}\)	\(0\).

From these the corresponding parts of the kinetic energy are formed by squaring and multiplying by halves of the corresponding moments of inertia (cf. introd., p. 344). The law of conservation of energy expresses the fact that the (double) sum of the kinetic and potential energies for any position must be equal to the same sum for the extreme position:

\[ B\dot{\vartheta}^{2}\sin^{2}\theta+B\dot{\vartheta}^{2}+2sG(1-\cos\vartheta) = B\omega_{1}^{2}\sin^{2}\vartheta_{1}+2sG(1-\cos\vartheta_{1}). \]

Let us transform this equation. First, let us again introduce approximate values for the trigonometric functions; secondly, let us replace \(B\dot{\psi}\) on the right-hand side of eq. (41):

\[ B^{2}\dot{\vartheta}^{2} = \frac{\omega_{1}^{2}}{4\vartheta^{2}} \left[ (4B^{2}-2A^{2}k)\vartheta_{1}^{2}\vartheta^{2} - A^{2}(\vartheta^{4}+k^{2}\vartheta_{1}^{4}) \right] +sGB(\vartheta_{1}^{2}-\vartheta^{2}) \]

or, after a brief calculation, in which we take into account the abbreviated notation \(k\) (40),

\[ B^{2}\dot{\vartheta}^{2} = \frac{\omega_{1}^{2}A^{2}}{4\vartheta^{2}} (\vartheta_{1}^{2}-\vartheta^{2})(\vartheta^{2}-k^{2}\vartheta_{1}^{2}) +sGB(\vartheta_{1}^{2}-\vartheta), \]

or, finally, putting for brevity

\[ \frac{\omega_{1}A}{2\sqrt{sGB}}=\varepsilon, \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (42) \]

where \(\varepsilon\), owing to the factor \(\omega_{1}\), will be very small, we obtain

\[ B\dot{\vartheta} = \pm \frac{\sqrt{sGB}}{\vartheta} \sqrt{(\vartheta_{1}^{2}-\vartheta^{2})(\vartheta^{2}-\varepsilon^{2}k^{2}\vartheta_{1}^{2})} \ . \ldots \ldots \ldots \ldots \ldots \ldots \ldots (43) \]

(Concerning the sign, the same considerations apply as in § 4, p. 349.) The conditions under which the first part has real values show that the magnitude of the angle of deflection \(\vartheta\) can oscillate only within the limits from the greatest value \(\vartheta_{1}\) to the least \(Ek\vartheta_{1}\), which earlier we denoted by \(\vartheta_{2}\) (§ 4).

Dividing now the two fundamental equations (41) and (43) by one another, we obtain

\[ \frac{d\psi}{d\vartheta} = \pm \frac{\varepsilon}{\vartheta} \frac{\vartheta^{2}+k\vartheta_{1}^{2}} {\sqrt{(\vartheta_{1}^{2}-\vartheta^{2})(\vartheta^{2}-\varepsilon^{2}k^{2}\vartheta_{1}^{2})}}. \]

If we integrate the right-hand side (with the positive sign) with respect to \(\vartheta\) within the limits from the minimum value \(Ek\vartheta_{1}\) to the maximum \(\vartheta_{1}\), then we obtain \(\frac{\pi}{2}(1+\varepsilon)\); in the left-hand side there will then be one fourth of the whole azimuthal turn \(\Psi\), which the angular line performs during a complete oscillation; hence

\[ \Psi=2\pi(1+\varepsilon). \]

If the trajectory of the oscillation did not rotate, this quantity would have to be exactly equal to \(2\pi\). Consequently, the excess

\[ \Delta\Psi=2\pi\varepsilon \]

gives the rotation of the plane of swing; it is obtained negative, because \(\varepsilon\) has the same sign as \(\omega_{1}\) and occurs in the direction of the azimuthal rotation of the earth, i.e. in the direction \(NWSO\). The angular velocity,

\[ \omega''=\frac{\Delta\Psi}{t_{0}}, \]

where \(t_0\) is the period of oscillation, must be subtracted also from the velocity of the azimuthal rotation, if the question concerns the apparent rotation of the pendulum relative to the terrestrial frame of reference. From the known expression for the period of a physical pendulum

\[ t_0=2\pi\sqrt{\frac{B}{sG}}, \]

we obtain, taking into account equation (42), the last correction term of formula (34)

\[ \omega''=\frac{A\omega_1}{2B}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (44) \]

The original experiments with a Gauss pendulum, 1.2 mt. long, were carried out in 1879 by Kamerlingh-Onnes (Kamerlingh-Onnes1) in an airless space, with a most exact assessment of the unavoidable errors; in doing so he succeeded in investigating very precisely not only the Foucault oscillations, but also the Bravais oscillations and a whole series of intermediate spherical motions of the pendulum, both with respect to their apparent rotations relative to the earth and with respect to their kinematic form separately. In particular, Lissajous oscillations were also observed, which the pendulum gives in the case when the moments of inertia \(B\) with respect to a horizontal axis are different from one another.—The mean value \(\omega_1\), found by Kamerlingh-Onnes, agrees with the astronomical one to the third decimal place.

15. Gyroscopic pendulum. The gyroscopic pendulum, which we mentioned earlier, can, for clarity, be successfully applied to prove the azimuthal rotation. Instead of an insignificant additional impulse \(A\omega_1\), one can impart to it, by means of rapid rotation of the top, a very large proper impulse relative to its axis, sufficient to make completely imperceptible all the other impulses accompanying the motion of the pendulum. A top of this kind is called a fast gyroscope; the axis of its proper rotation (in the present case this is the freely suspended axis of a pendulum in a state of rest) may be regarded, without noticeable error, as the line of action of the impulse-vector \(\mathfrak{S}\), whose value, with sufficient accuracy, is equal to the product of the axial moment \(A\) and the angular velocity of the proper rotation \(\nu\), i.e.

\[ S=A\nu. \]

(Thus, in principle, we replace the total impulse by the proper impulse). It is now easy to show that, within the limits of our accuracy, the axis of the pendulum, brought out of the position of rest and left

itself, will begin to oscillate not as an ordinary pendulum does, but will describe around the vertical a circular cone with angular velocity

\[ \mu=\frac{sG}{S}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (45) \]

where \(G\) is the weight of the pendulum; and \(s\) is the distance between the point of suspension and the center of gravity, which lies on the axis of the pendulum. If the proper rotation \(\nu\), when observed from above, takes place clockwise, then the rotation \(\mu\), the so-called precession, takes place in the opposite direction.

Indeed, let \(\vartheta_0\) be the angle of deflection of the pendulum from the vertical (Fig. 20); then the moment of the weight of the pendulum with respect to the point of suspension will be \(sG\sin\vartheta_0\), and its vector \(\mathfrak M\) is directed horizontally along the perpendicular to the vertical plane passing through the instantaneous impulse-vector and, consequently, containing the axis of the pendulum. By the law of impulse (Introduction, p. 342), the end of the vector \(\mathfrak S\) will move all the time horizontally and perpendicularly to the plane just mentioned, which, in turn, rotates together with \(\mathfrak S\), and moreover with the constant velocity \(\mathfrak M\). Consequently, the end of the vector \(\mathfrak S\) necessarily describes a circle, whose radius will be \(r=S\sin\vartheta_0\), and consequently the velocity of rotation \(\mathfrak M\) and the angular velocity will be connected with one another by the relation \(|\mathfrak M|=r\mu\), which is identical with (45).

Fig. 20. Precession of a rapid gyroscope.

With the newest technical means, by means of an electric motor one can impart to a gyroscope of weight \(5\) kg, with a moment of inertia of \(0.076\) kgr. cm./sec.\(^2\), a proper rotation with angular velocity \(V=1000\pi\ \mathrm{sec}^{-1}\) (500 revolutions per second)\(^{1}\). With careful construction one can reduce \(s\) to \(2.5\) mm and, in this way, reduce \(\mu\) almost to \(0.0052\ \mathrm{sec}^{-1}\). This gives a precession with a period of 20 minutes. If the gyroscope is rotated once in one direction, and another time, with exactly the same number of revolutions, in the other direction, then the periods of the precessions observed on the earth should differ from one another in our latitudes by almost 26 seconds.

In general the difference of the precession periods will be

\[ \Delta T=2\cdot\frac{2\pi}{\mu}\cdot\frac{\omega_1}{\mu} =\frac{4\pi\cdot S^2}{s^2G^2}\,\omega\sin\varphi; \]

therefore, it is directly proportional to the square of the impulse.

Experiments not yet carried out up to the present promise in the future to introduce substantial improvements into Bravais’ result.

\(^{1}\) See R. Grammel, Der Kreisel. Braunschweig, 1920, pp. 257, 271, and 282.

B. Investigation of the general rotation.

16. Gyroscope. The idea of applying a gyroscope to the proof of the rotational motion of the earth (but, of course, in a completely different way from the one we have just examined) belongs to Person (Person)1; it was carried out by Foucault2 a year after his famous experiment with the pendulum. If we have a body suspended in a Cardan suspension in such a way that it can rotate astatically, entirely without friction, about any axis (precisely so that its center of gravity geometrically coincides exactly with the center of suspension), then, having initially been brought into a state of rest relative to the earth, it can execute, in an inertial system, a rather complicated motion, studied very precisely by A. Puanso3, which, however, will always be of the order of magnitude \(\omega\). If the body has an axis of symmetry which is at the same time an axis of inertia, and the moment of inertia with respect to this axis is \(A\), then it must be imparted, about this axis, its own rotation \(\nu\), very large in comparison with \(\omega\). In this way a rapid gyroscope is obtained, whose own impulse \(A\nu\) suppresses the initial impulse from the earth’s rotation \(\omega\), so that from this moment it may be considered that the impulse vector \(\mathfrak{S}\) coincides with the axis of the gyroscope. Since, according to the assumption, neither friction nor the force of gravity imparts to the gyroscope any moment \(\mathfrak{M}\), the vector \(\mathfrak{S}\), and together with it the axis of the gyroscope, preserve their direction at all times in the inertial system. If this axis is not accidentally parallel to the earth’s axis, then the rotational motion of the earth \(\omega\) must reveal itself in the form of an apparent rotation \(-\omega\) of the axis of the gyroscope relative to the earth.

Fig. 21. Foucault gyroscope.

Foucault performed this experiment with the aid of a top, which he called a gyroscope (from the Greek \(\gammaῦρος =\) circle), its Cardan suspension being hung on a thread as stiff as possible with respect to twisting (Fig. 21). But this did not give satisfactory results, for it proved impossible finally to eliminate friction, and it was impossible to place the center of gravity in the center of suspension so accurately that there would be no noticeable precession, necessarily connected with the eccentricity of the center of gravity and interfering with the experiment (as in the gyroscopic pendulum).

1 C.-C. Person, Comptes rendus 35, 417 and 549. 1852.
2 L. Foucault, loc. cit., pp. 401—420 and 576.
3 See the footnote on p. 343.

17. Gyroscopic inclinometer. When Foucault understood the causes of the imperfection of his gyroscope, he began to seek new methods. Instead of showing the rotation of the earth directly, kinematically, in that same year he began to devise a way to detect this phenomenon dynamically, using the influence of the earth’s rotation on a gyroscope whose axis was compelled to take part in the rotational motion of the earth. When Foucault fastened the outer Cardan ring \(r_1\) of his gyroscope to the stand, and allowed the inner ring \(r_2\) freedom to rotate about its horizontal axis directed from east to west, the axis of the gyroscope \(\mathfrak S\) (Fig. 22) could move only in the plane of the meridian. The plane of the meridian, in turn, must take part in the rotation of the earth \(\omega\); the rotation of this plane may be represented by the vector \(\upsilon\), drawn from the center of the Cardan suspension toward the pole of the world, with length equal to \(\omega\).

Fig. 22. Toward the theory of the gyroscopic inclinometer.

Foucault found that, according to the law of impulse, in such a gyroscope, which is compelled to take part in the rotation \(\upsilon\), there arises, owing to inertia, a force (which is called the force of deviation or, better, the gyroscopic moment) that tends to bring into coincidence the direction of the vector-impulse \(\mathfrak S\) (and together with it the axis of the gyroscope) and the vector of the imposed rotation \(\upsilon\). He called this tendency the rule of unilateral parallelism of the axes of rotation (namely, the proper axis of rotation \(\mathfrak S\) and the imposed axis of rotation \(\upsilon\)).

Suppose that the vector-impulse \(\mathfrak S\) (the axis of the gyroscope) forms with the vector \(a\) the angle \(\delta\). Then, as a result of the rotational motion of the meridian plane, the end of the vector \(\mathfrak S\) will be given a velocity \(\nu\) about the axis \(\upsilon\), whose magnitude will be \(S\sin\delta\cdot\omega\), if \(S\sin\delta\) is the distance of the end of the vector \(\mathfrak S\) from the axis \(\upsilon\). This velocity will be directed perpendicular to the plane of the meridian (in Fig. 22, toward the observer) in such a way that its direction, together with the direction of rotation \(\upsilon\), forms a right-hand screw. By the law of impulse, the rotation \(\upsilon\), which is imparted to the axis of the gyroscope, can be maintained only by the moment \(\mathfrak M=\nu\). The magnitude of this moment, since \(S=A\nu\), will be

\[ M=A\omega\sin\delta,\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (46) \]

its direction of rotation striving to change the angle \(\delta\). This moment is imparted to the gyroscope from outside, if only through the fastening, if its axis rotates together with the meridional plane, while the angle \(\delta\) must not change. By the law of reaction, the top itself gives an exactly equal in magnitude but opposite moment, the so-called gyroscopic moment \(\mathfrak K=-\mathfrak M\).

It would be useful to note that the moments \(\mathfrak M\) and \(\mathfrak K\) stand in the same relation to one another as the centripetal force \(\mathfrak Z\) and the centrifugal force \(\mathfrak F\) in the case, for example, of a stone rotated in a circle on a string. If the tension of the string \(\mathfrak Z\) is slightly weakened, the stone moves away from the center just as if it were not moving in a circle at all, but were moving only under the action of the force \(\mathfrak F\). In the same way, when the moment \(\mathfrak M\) disappears, the axis of the gyroscope will rotate toward the vector \(\upsilon\) as though the moment \(\mathfrak K\) were acting on it, and it will stop only when it has come into coincidence with the vector \(\upsilon\) (after some oscillations).

Foucault compared his instrument with a magnetic inclinometer, because its axis should become parallel to the earth’s axis and thus indicate the geographical “inclination” $\varphi$ (see Fig. 22), if there did not arise opposing moments due to friction and to the inevitable deficiencies of astatization, which cannot be neglected in comparison with the force producing the parallelism of the axes (and precisely in comparison with the small gyroscopic moment (46)). Therefore the indications of the instrument are very inaccurate.

Fig. 23. Gilbert’s barigyroscope.

18. Barigyroscope. Gilbert (Ph. Gilbert)¹ was the first, in 1882, by means of a simple adaptation, to overcome the principal difficulties, consisting in the insufficiency of astatization. He loaded (Fig. 23) the axis of the gyroscope with a small counterweight $(g)$, whose moment is sufficiently large in comparison with the possible inaccuracy in the position of the center of gravity, but which can have absolutely no effect on the gyroscopic moment. If an impulse is imparted to the rotor whose vector is directed upward, then at first the vertical axis of the gyroscope (see Fig. 24, p. 374), after several oscillations, will take a new inclined position of equilibrium, and then the axis of the gyroscope, by its upper end, will be slightly inclined toward the north. If an impulse is imparted in the opposite direction, and consequently the gyroscope is given rapid rotation in the other direction, a considerably greater inclination toward the south will be obtained (as in Fig. 23).

Fig. 24. On the theory of the barigyroscope.

Gilbert, who also constructed an instrument similar to this one with two gyroscopes, could indisputably establish the existence of these inclinations. The apparatus received the name “barigyroscope.”

In the position of equilibrium the moment of the force of gravity, $-sG\cos(\delta+\varphi)$ (cf. Fig. 24), must be equal to the gyroscopic moment (46)

\[ sG\cos(\delta+\varphi)=A\nu\omega\sin\delta. \]

Let us introduce the angle of inclination of the gyroscope axis to the vertical

\[ \vartheta=90^\circ-(\delta+\varphi); \]

then we obtain

\[ sG\sin\vartheta=A\nu\omega\cos(\vartheta+\varphi), \]

¹ Ph. Gilbert, Journ. de Phys. Paris 2, 106. 1883.

whence it follows that

\[ \tang \delta = \frac{A\nu\omega \cos \varphi}{A\nu\omega \sin \varphi + sG} \tag{47} \]

Likewise, for the angle \(\delta'\), through which the lower end of the axis rises toward the north when the gyroscope is set in rotation in the opposite direction,—

\[ \tang \delta' = \frac{A\nu\omega \cos \varphi}{A\nu\omega \sin \varphi - sG} \tag{48} \]

The deviation \(\delta'\) is evidently, under otherwise identical circumstances, considerably greater than the deviation \(\delta\); therefore the instrument in the second case is more sensitive than in the first.

19. Determination of the gyroscopic moment by means of a balance. From the preceding it is easy to see that the gyroscopic moment can very readily be measured by means of a balance. If the axis of the gyroscope or, more precisely, the vector \(\mathfrak{S}\), is directed horizontally from south to north, then the gyroscopic moment (according to (46) it has the magnitude \(S\omega \sin \varphi\)) tends to raise the northern end of the axis and lower the southern one. If one takes a top of the type mentioned in § 15, places both ends of its horizontal axis on two prisms attached to the pans of a balance, and balances the pressure of the stationary gyroscope, then, as soon as the gyroscope attains its full number of revolutions, the pressures of the ends of the axis on the two pans will give a difference which, with a distance between the prisms of 10 cm, will be equal to about 2.7 g and can be well detected on a balance with a sensitivity of \(1:10000\).

C. Investigation of vertical rotation.

20. Gyroscopic declinator. When Foucault again released the outer ring of the Cardan suspension of his gyroscope (Fig. 21, p. 372), and fastened the inner \(r_2\) to the outer, the axis of the gyroscope could move only in the horizontal plane, like the needle of a magnetic compass. It was now no longer connected with the azimuthal rotation \(\omega_1\), but was compelled to take part in the vertical rotation of the horizontal plane \(\omega_2\). This rotation may be represented by a vector \(a_2\), directed northward1, and the rule concerning the tendency of axes toward one-sided parallelism (p. 372) directly shows that the positive end of the impulse-vector, and with it the axis of the gyroscope, tend to set themselves to the north, and indeed geographically more accurately than the needle of a magnetic declinator. Foucault was fully aware that he had in this way found the principle of the gyroscopic compass, despite the fact that the success of the experiment was greatly hindered by the stiffness of the suspending thread.

A quantitative investigation was first carried out 52 years later by A. Föpple (A. Föppl), who suspended the gyroscope on a trifilar suspension (triple suspension) and, by means of electricity, brought the speed—

the speed of rotation to 2400 revolutions per minute, successfully damped the azimuthal oscillations. The value \(\omega_2\) found by Föppl agrees with the astronomical value to within \(2\%\).

Finally, as a substantial step forward, one may also mention Lord Kelvin’s proposal1 to let the gyroscope float on mercury. From this Kelvin’s idea develops all the way to the modern gyroscopic compass2, which received its best form in the design of Anschütz & Co. and may be called, owing to its high technical execution, the most perfect gyroscopic instrument at the present time, one that can be used for exact quantitative investigations of the rotation of the earth. In experiments now being carried out by M. Schuler in the laboratory of Anschütz & Co., the gyroscopic compass, isolated from shocks, gives the northerly direction, i.e. the horizontal projection of the earth’s axis, with an accuracy of up to \(20''\)3; the accuracy with which the value \(\omega_2\) can be determined by means of the gyroscopic compass has not yet been assessed, but it should far surpass the accuracy of all other non-astronomical determinations.

Concluding Remarks.

21. Annual revolution of the earth. The theoretical-cognitive significance of non-astronomical proofs of the earth’s rotation has changed greatly with the passage of time. From the historical point of view, it undoubtedly consisted originally in the fact that they confirmed, to a considerable degree, the heliocentric interpretation of astronomical observations. But it has long since no longer needed such support. Nevertheless, mechanical experiments have by no means lost their significance, at least insofar as they can lay claim to exact quantitative measurements. On the one hand, in general, they may be counted among the best confirmations of the fundamental kinetic laws of mechanics; on the other hand, they show (and in respect of knowledge this in no way confirms the motion of the earth) that the frame of reference relative to which, on average, the world of fixed stars known to us is at rest possesses, in practice, entirely the properties of an inertial frame.

This conclusion, however, is still not sufficiently precise. This is indicated by the annual parallaxes of the fixed stars. In fact, in the experiments mentioned up to now the question has always concerned only the rotation of the earth, but not its revolution around the sun, to say nothing of the precession and nutation of the earth’s axis. Thus, an additional question is raised:

whether it is possible, or at least conceivable, to prove by experiment that the terrestrial frame of reference differs from an inertial frame, both as a consequence of its annual revolution around the sun and as a consequence of precession and nutation. There is no doubt that this question is to be answered in the affirmative, but, with respect to the annual revolution of the earth, the answer leads to a conclusion which, from the point of view of classical mechanics, is quite unexpected.

It would be quite hopeless to detect the annual revolution by investigating the centrifugal forces arising from it, since these forces could be measured only relative to some terrestrial place of observation. This place itself is subject to the action of exactly the same centrifugal force. According to the principle of relativity, all terrestrial phenomena must proceed in the same way, irrespective of whether the earth revolves around the sun or the sun around the earth1. There is no absolute decision between the two possibilities. However, every terrestrial frame of reference shows—if all terrestrial forces and the rotation of the earth are taken into account—a clear difference from an inertial frame, which is most simply explained by the annual revolution of the earth. This difference is the gravitational field of the sun in the region of the earth.

Any experiment that reveals this gravitational field may with sufficient justification be regarded as a proof of the annual revolution of the earth. From the relativistic point of view, in general nothing more can be proved than the existence of this gravitational field. The negative optical experiments, which had vainly attempted to find the motion of the earth relative to the “stationary ether,” have been replaced, in our view, by the one observation of enormous positive significance: that light rays are deflected by the gravitational field of the sun. No terrestrial mechanical experiment for investigating the gravitational field has yet been devised. But it is possible in principle, as follows from the phenomenon of ebb and flow. The participation of the sun in these phenomena is based simply on the inhomogeneity of the sun’s gravitational field in the region of the earth; consequently, it is enough to create artificial conditions for phenomena similar to ebb and flow. There are many possibilities here (besides hydraulic ones, elastic ones can also be conceived: a rod of finite length has, depending on its position relative to the vector of the gravitational field, different stresses), but the possibility of obtaining indisputable results is very small, owing to the limited measuring means at our disposal in laboratories.

22. Precession and nutation. The situation is still worse with the possibility of detecting by mechanical means the precession of the earth’s axis,

more precisely speaking, caused by its violation of inertiality, quite apart from nutations as small deviations of the second order. Precession consists in the fact that the terrestrial axis, considered in a system fixed to the center of the earth and not taking part in the rotation, describes in 26,000 years a circular cone with an aperture angle of 47°. The vector of the angular velocity of this motion, drawn from the center of the earth (Fig. 25), is perpendicular to the plane of the ecliptic, is directed southward, and has length \(\varepsilon\), equal to \(1/26000.366\) part of the rotation vector \(\omega\). If the two vectors are added by the parallelogram rule, one obtains the resultant rotation vector \(u\), which with approximately exactness has the length \(\omega\), but is deflected from \(\omega\) by a small angle \(\delta\) toward the perpendicular to the plane of the ecliptic.

The vector \(u\), if the earth’s revolution is not taken into account, will be fixed in space and, consequently, during the course of a day describes about its geographic axis a narrow circular cone with an aperture angle \(2\delta\). Moreover, it takes the place of the vector \(\omega\) in all experiments that investigate the rotation of the earth, and the question is only one of increasing the quantitative accuracy of these experiments sufficiently for them to detect the daily relative oscillation of the “true” rotation vector \(u\). A consequence of this oscillation is the fact that the “true” northern direction (i.e., the horizontal projection of the vector \(u\)) during the course of a day undergoes oscillations from east to west within an angle \(\psi\), which for our latitudes has a magnitude of about \(1/75\) of an arc second, and for higher latitudes is larger. (However, these oscillations must not be confused with the almost 12 times larger oscillations of the Euler–Chandler pole, which occur in a 12- or 14-month period.)

Fig. 25. Daily oscillations of the precession of the earth’s axis.

The angle \(\delta\) (Fig. 25) will be

\[ \delta \lessgtr \sin \delta = \frac{\varepsilon}{\omega}\sin 23^\circ,5. \]

Further, from the right spherical triangle, one leg of which is the distance from the place of observation to the nearest pole, and the other, adjacent to the pole, has arc length \(\delta\), we have

\[ \tan \psi = \frac{\tan \delta}{\cos \varphi}, \]

ing, since for not too high geographical latitudes the angle \(\psi\) is sufficiently small,

\[ \psi \leq \frac{\delta}{\cos \varphi} \leq \frac{\varepsilon}{\omega}\,\frac{\sin 23^\circ\!,5}{\cos \varphi}, \]

or, numerically,

\[ \psi=\frac{0.0086''}{\cos \varphi}\, . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (49) \]

A gyroscopic compass with its sensitivity increased by more than a thousand times would probably be capable of directly indicating the diurnal oscillations of the “true” northern direction due to precession. But it is hardly possible to increase by a factor of a thousand the sensitivity of the instrument, which has already now been brought to an extreme degree. And even if this were to happen, it is unknown whether the diurnal oscillations of the earth’s surface would not again damp out the effect to be measured.

It would also be possible to carry out observations at higher geographical latitudes, for the amplitude of the oscillations \(\psi\) increases strongly when approaching one of the poles. But, unfortunately, the directive force of the gyroscopic compass, and with it its sensitivity, decreases at high geographical latitudes in the very same proportion in which the angle \(\psi\) increases, and thus all hope of detecting mechanically the precession of the earth’s axis disappears.

Translated by Nik. Buchholz.

1. This is set forth very clearly in the recently published work of G. Mie, Die Einsteinsche Gravitationstheorie. Leipzig, 1921, p. 47 ff. ↩↩↩↩↩↩↩↩↩↩↩↩

2. See R. Grammel. Der Kreisel. Braunschweig 1920, § 19. ↩↩↩↩

3. On the basis of a written communication from Dr. Schuler. ↩↩↩

4. See J. G. Hagen, loc. cit., p. 8. ↩↩