PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II.

Vol. 12. 1908. No. 4.

II.

V. N. Pinegin.

Experiments on Investigating the Dependence of Deformations Produced by Steel Balls in Metals under the Action of Kinetic Energy

1–42.

Investigation of the Deformations Produced by Steel Balls in Metals under the Action of Their Living Force

V. N. Pinegin.

The question of determining the hardness of solid bodies by Brinell’s method has so interested engineers and physicists that a large literature has recently been devoted to it alone.^1

^1 Referring readers interested in the literature on Brinell’s method to the article by P. Ludwik: “Über Härtebestimmung mittels der Brinellschen Kugeldruckprobe und verwandten Eindruckverfahren,” Zeitschr. d. Öster—Ingen—und Archit.—Vereins, 1907, pp. 191, 205, which gives a detailed list and critical discussion of this literature, I mention only, as a supplement to that article, works not indicated by Ludwik and, for the most part, already published after his work had appeared:

1) T. Friesendorff. Ueber die Brinellsche Kugelprobe zur Bestimmung der Härte der Metalle. Baumaterialienkunde, 1906.

2) Malmström. Versuche mit Gusseisen über den Einfluss des Kugeldurchmessers und des Druckes bei der Brinellschen Methode der Härtebestimmung. Dinglers Polyt. Journal, 1907, Vol. 32, pp. 33—36.

3) L. Guillet. Compte rendu du Congrès des méthodes d’essais des matériaux de construction Bruxelles 1906. Mémoires de la Société des Ingenieus Civils de France, 1907, Février, pp. 298—340.

4) Dr. Oswald Meyer. Ueber die Ludwiksche Kugeldruckprobe. Österreichische Zeitschr. für Berg—u. Hüttenwesen, 1907, No. 21, pp. 257—258.

5) Rejtö. Die auf dem Brüsseler Kongresse mitgeteilten Versuchsergebnisse bezüglich der Gesetze der bleibenden Formänderungen. Baumaterialienkunde, 1907.

Finally, one may also point to articles on this question by J. E. Stead and T. Greville (The Iron and Coal Trades Review, 1907, 12 April, pp. 1216—1217) and W. T. Balentine (American Machinist, 1907, 1 June, p. 698).

B. Pinetin.

Meanwhile, this question still remains little clarified; Brinell’s method, at least in the form in which it was proposed, proved not entirely suitable for determining “hardness.” The point is that the so-called “hardness” of a body (the force acting on the ball, referred to the unit area of the resulting indentation) turns out to depend not only on the dimensions of the ball itself, but also on the magnitude of the load on the ball; moreover, this dependence is not at all simple. Of particular interest from this point of view are the results of experiments carried out very recently by Prof. Meyer in Charlottenburg and not yet published.

According to his numerous experiments, it turns out that the ratio $\dfrac{P}{\omega}$ (where $P$ is the force, and $\omega$ is the area of the indentation produced by the ball) is not a constant quantity, and the relation between the force $P$ and the diameter $d$ of the indentation can be expressed by the formula $P = ad^m$, where $a$ and $m$ are constants, different for different metals, and moreover $m$ is generally close to 2 [varying for the metals tested from 2.05 (copper) to 2.32 (steel)], but is not equal to 2, as would follow from Brinell.

Having become interested in these experiments, of which I was an eyewitness, and bearing in mind that Brinell’s method under the dynamic action of a force has been investigated even less than under the static action of a force, I resolved to take up investigations of the deformations produced in metals by steel balls when acted upon by vis viva.

For this purpose, in the spring of 1907 I undertook the experiments described below in the Charlottenburg laboratory. These experiments were carried out in the following manner: placing the metal plate to be tested on the anvil of a drop hammer, and placing a steel ball on the plate (I used balls of various sizes: diameters of 10, 12½, 15, 20, and 25 mm), and causing a definite load to fall upon it, an indentation was obtained on the plate, which could serve as a measure of the deformation for the given vis viva. The measurement of the diameter of the circumference of this indentation was made in two directions deviating from one another by $90^\circ$. The mean of the diameters thus obtained, $a_1$ and $a_2$—

\[ \frac{a_1 + a_2}{2} \]

—was taken as the true value of the diameter $d_m$ of the indentation. From this diameter $d_m$ the area

\[ \left(\frac{\pi d_m^2}{4}\right) \]

of the circle bounding the given indentation was determined; by dividing the vis viva by this area, similarly to the way this is done in determining the hardness of bodies under static action—

the force, we obtain, as it were, the “stiffness” of the body under the dynamic action of the force. We shall evidently obtain the vis viva from the product of \(P\), the falling load, and \(H\), the height of fall of this load.

The falling load for the first series of experiments was chosen by us as \(37.49\ \mathrm{klgr}\) (the weight of the “ram” enclosing sleeve \(P_1 = 17.29\ \mathrm{klgr}\) and the weight of the “ram” \(P_2 = 20.20\ \mathrm{klgr}\)). The heights were usually taken in multiples of \(10\ \mathrm{cm}\), increasing them by \(20\ \mathrm{cm}\) or by \(40\ \mathrm{cm}\), so that in the first case they were \(10\ \mathrm{cm}\), \(30\ \mathrm{cm}\), \(50\ \mathrm{cm}\), \(70\ \mathrm{cm}\), \(90\ \mathrm{cm}\), \(110\ \mathrm{cm}\), etc., and in the second \(10\ \mathrm{cm}\), \(50\ \mathrm{cm}\), \(90\ \mathrm{cm}\), \(130\ \mathrm{cm}\), etc.

In view of the fact that the fall of the load directly onto the steel ball would be accompanied by damage to the end surface of the “ram,” it was necessary to resort to some intermediate body, placed directly on the ball, and to make the load fall already onto this body (we shall hereafter call it the “interlayer”). In order to keep in equilibrium the ball on the body being tested and the interlayer body—the “interlayer”—the following device was arranged: on the ball placed on the tested body there was placed (see Fig. 1) a thin bent strip of sheet metal, having in the middle an opening of considerably smaller diameter than the diameter of the ball, and resting with its ends, depending on the dimensions of the specimen being tested, either on the anvil, or on the body being tested itself. The ball had to protrude only slightly beyond this plate; on top of this plate was placed the plate—the “interlayer”—which had a recess exactly corresponding to the protruding part of the ball; it was always possible to ensure that the “interlayer” lay as nearly as possible horizontally on the sheet-metal plate and at the same time rested on the ball.

But whereas, under the static action of a force, the material of the interlayer does not play a special role—at least if metals are taken for the interlayer (as follows partly from my preliminary experiments, partly from the experiments of Prof. Meyer)—under the dynamic action of a force the material of the “interlayer” and its condition already have enormous significance. The following table gives the results obtained with “interlayers” of different materials, and also of one and the same material in different states, which can be attained, for example, by placing the interlayer on the ball each time with the same place, and consequently this place will become more and more compacted.

Table 1.

Type of backing plate	Diameters: \(a_1\), in mm	Diameters: \(a_2\), in mm	Mean diam. \(\dfrac{a_1+a_2}{2}\), in mm	Area of indentation \(\omega\), in sq. cm	\(\dfrac{L}{\omega}\)	Note
Steel, for the 1st time	7,576	7,551	7,563	0,448	4162	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.
Copper . . . . . .	6,748	6,793	6,770	0,359	5221	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.
Lead . . . . . .	3,786	3,801	3,794	0,113	16620	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.
Pine . . . . . .	3,262	3,276	3,269	0,084	22342	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.
Steel, for the 2nd time	8,114	8,133	8,123	0,518	3630	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.
Steel, for the 3rd time	8,156	8,186	8,171	0,524	3578	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.
Steel after repeated use at various, sometimes very great (up to 2 m), heights of fall of the load	8,317	8,252	8,285	0,537	3500	The diameter of the ball was \(D = 25\) mm, and the height of fall of the load \(h = 50\) cm; thus the kinetic energy \(L = 1874,5\) kgcm. The test plate was steel.

The table just presented clearly shows what enormous significance the “backing plate” has in these experiments. This is why it is very important that the “backing plate” for balls of each definite size should be one and the same, so that the indentation fully corresponds to the given size of ball and so that it has been brought, so to speak, into a “settled state,” i.e. into a state that is not altered by subsequent blows; for this it was necessary, before carrying out the experiments, to make an indentation with the corresponding ball, causing the ram to fall several times from a height at least twice as great as the height of fall in the experiments. For comparison of experiments with different balls, the “backing plates” must be of one material and brought into the same state.

The results obtained from experiments of this kind will be only relative, valid only for the given type of “backing plate”; although, however, as also followed from my experiments, the very law determining the relation between kinetic energy and deformation remains unchanged, independent of the type of “backing plate”; only the coefficients will change from the use of different kinds of “backing plates.”

To establish such a dependence of the results of the experiments on the material of the “support,” I was interested in finding out whether, for a given definite material of the “support,” its thickness also affects these results; in other words, whether the action of the vis viva changes through the process of its passage through the support, depending on whether it must traverse a greater or a lesser path. To clarify this influence of the thickness of the support, I ordered copper cylinders 30 mm in diameter and of various heights, from 5 mm to 90 mm, and, placing them in accordance with the above on the ball, caused a weight to fall upon them; from the magnitude of the indentations obtained on the body being tested (steel), as well as on the copper cylinders, it was possible to obtain an answer to the question posed above. The following Tables No. 2 and No. 3 give the results of these experiments, the first of them giving the results for the body being tested, and the second for the copper cylinders.

Table 2.

Height of cylinder, in mm	Diameter of indentation \(a_1\), in mm	Diameter of indentation \(a_2\), in mm	Diameter of indentation \(\frac{a_1+a_2}{2}\), in mm	Mean diameter from several experiments, in mm	Area of indentation \(\omega\), in cm	\(\frac{L}{\omega}\)	Note
5,55	5,52	5,53	5,525	5,530	0,241	3889	Diameter of ball \(D=25\) mm. Height of fall of the weight \(h=25\) cm; consequently, vis viva \(L=937,25\) kgcm.
5,55	5,54	5,46	5,550	5,530	0,241	3889	Diameter of ball \(D=25\) mm. Height of fall of the weight \(h=25\) cm; consequently, vis viva \(L=937,25\) kgcm.
5,20	5,51	5,52	5,515	5,530	0,241	3889	Diameter of ball \(D=25\) mm. Height of fall of the weight \(h=25\) cm; consequently, vis viva \(L=937,25\) kgcm.
10,45	5,57	5,56	5,565	5,57	0,243	3857
9,73	5,57	5,57	5,57	5,57	0,243	3857
10,45	5,57	5,57	5,57	5,57	0,243	3857
20,34	5,64	5,74	5,69	5,69	0,253	3704
30,60	5,66	5,56	5,61	5,61	0,247	3795
58,20	5,54	5,54	5,54	5,54	0,242	3873
88,00	5,78	5,78	5,78	5,78	0,262	3600

Table 3.

Height of the copper cylinder, in mm	Diameter of the indentation \(a_1\), in mm	Diameter of the indentation \(a_2\), in mm	Diameter of the indentation \(\dfrac{a_1+a_2}{2}\), in mm	Mean diam. from several experiments	Area of indentation \(\omega\), in cm	\(\dfrac{L}{\omega}\)	Note
5.55	10.76	10.76	10.76	10.72	0.900	1041	Diameter of ball: \(D=25\) mm. Height of fall of load: \(h=25\) cm. \(L=937.25\) kgcm.
5.55	10.67	10.60	10.64	10.72	0.900	1041	Diameter of ball: \(D=25\) mm. Height of fall of load: \(h=25\) cm. \(L=937.25\) kgcm.
5.20	10.76	10.74	10.75	10.72	0.900	1041	Diameter of ball: \(D=25\) mm. Height of fall of load: \(h=25\) cm. \(L=937.25\) kgcm.
10.45	10.62	10.57	10.60	10.50	0.864	1085
9.73	10.39	10.25	10.32	10.50	0.864	1085
10.45	10.62	10.57	10.59	10.50	0.864	1085
20.34	10.74	10.74	10.74	10.74	0.905	1036
30.60	10.62	10.64	10.63	10.63	0.886	1058
58.20	10.48	10.45	10.46	10.46	0.858	1092
88.00	10.73	10.77	10.75	10.75	0.906	1034

From the tables presented, there is no possibility of discerning any influence of the height of the cylinders on the “hardness” of the bodies tested—steel and copper cylinders—since the fluctuations in the numbers characterizing the measure of hardness do not exceed the errors possible in carrying out the experiments. It is possible, of course, that this influence is very insignificant and therefore eludes observation in the present experiments.

Before now proceeding to the presentation of the results of the experiments themselves, it remains for me to say a few words about the possible accuracy of these experiments and about the measurement of the diameters of the circumferences bounding the indentations on the surface of the bodies tested.

The Kopér apparatus on which I had to work did not have those devices which exist in the newest constructions of this kind and which make it possible to determine with perfect accuracy the height to which the “ram” is raised above the surface struck. We had to—

one had to determine this height directly by eye, having first made a mark on the crosshead sliding along the rails—the uprights of the drop weight—and determining the position of this mark on the scale located on one of the rails; with such a method of reading off the height, of course, it was impossible to attain great accuracy, especially at higher positions of the “drop weight.” Then, of course, I cannot take precise account of the sliding friction of the crosshead along the rails; this friction, depending on the position of the crosshead relative to the rails (this position, admittedly, could not vary very greatly), could manifest itself to a greater or lesser degree and thus unequally change the magnitude of the kinetic energy. In particular, this difference in friction could show itself strongly when the load was varied. To show what influence the friction of the crosshead against the rails has on the magnitude of the deformation, I give the results of two series of experiments: one with the sliding surfaces of the crosshead on the rails not lubricated, and the other with the surfaces lubricated (see Table 4).

Table 4.

Kinetic energy in kg·cm	$\dfrac{L}{\omega}$ Sliding surfaces not lubricated	$\dfrac{L}{\omega}$ Sliding surfaces lubricated	Remarks
374,9	1240	—	Material: nickel steel; ball diameter $D = 25$ m/m.
1124,7	2160	1990	Material: nickel steel; ball diameter $D = 25$ m/m.
1874,5	2760	2530	Material: nickel steel; ball diameter $D = 25$ m/m.
2624,3	3260	3040	Material: nickel steel; ball diameter $D = 25$ m/m.
3374,1	—	3460	Material: nickel steel; ball diameter $D = 25$ m/m.

Drawing No. 1.

As can be seen, the difference in the values of $\dfrac{L}{\omega}$, for example, for a drop height of the load of 70 cm, is

\[ \frac{3260 - 3040}{3040}\cdot 100 = 7{,}25\%, \]

for a height of 30 cm—

\[ \frac{2160 - 1990}{1990}\cdot 100 = 8{,}54\%. \]

The circumstance could not fail to influence the results of the experiments that, when the “drop weight” was released, the hook from which it slid off-

raises the weight, it is lowered somewhat when the trigger cord is pressed and, possibly, not always by one and the same amount; finally, the position of the “base” may have deviated somewhat from the horizontal, for whose exact position there could, of course, be no guarantee. All this introduced an error into the experimental results—an error that must be considered as by no means less than 2–3%, and perhaps somewhat greater.

Thus, since the experimental results could not be more accurate than 2–3%, it was unnecessary to measure the diameters more accurately than 1%; and since the diameters were measured in mm, there was no need for me to make the measurements more accurately than hundredths of a mm. Therefore, since I made these diameter measurements with the aid of a Zeiss comparator, which made it possible to read ten-thousandths of a millimeter, or with a Zeiss micrometer, which made it possible to read thousandths of a millimeter, in the first case I discarded the last two decimal places, and in the second, one decimal place.

First of all, steel specimens measuring \(12 \times 25 \times 70\) mm were investigated; these specimens were of the best English steel (Sheffield—best grade). In experiments with these specimens I used balls of diameters 10, 12, 15, 20, and 25 mm. Since the plates were not always of identical quality, it was necessary, with balls of each given size, to carry out experiments on several plates—3–4, and sometimes more—and to take the mean results. In addition, to avoid the influence of nonuniformity of the plate, blows of one and the same kinetic energy were applied at different places on the plate.

The following tables give the results of these experiments; moreover, for each value of the kinetic energy we calculate the ratios \(\frac{L}{\omega}\) and \(\frac{L}{\omega^2}\), where \(L\) is the kinetic energy of the falling weight, and \(\omega\) is the area of the depression produced by the ball; \(d_m\) is the mean diameter of the circumference of the depression from several experiments with one and the same kinetic energy.

Specimen No. 1. Ball diameter \(D=10\) mm.  Table 5.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	4,58	4,56	4,57	4,58	0,165	2270	13800
—	—	4,59	4,58	4,59	4,58	0,165	2270	13800
30	1124,7	6,07	6,09	6,08	6,11	0,293	3840	13100
—	—	6,17	6,21	6,19	6,11	0,293	3840	13100
—	—	6,04	6,08	6,06	6,11	0,293	3840	13100
50	1874,5	6,93	6,93	6,93	6,95	0,379	4950	13000
—	—	6,95	6,97	6,96	6,95	0,379	4950	13000
70	2624,3	7,55	7,57	7,56	7,56	0,448	5858	13060
90	3374,1	8,02	8,04	8,03	8,03	0,506	6680	13100
Average	—	—	—	—	—	—	—	13210

Specimen No. 2. Ball diameter \(D=10\) mm.  Table 6.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	4,57	4,62	4,60	4,59	0,166	2260	13730
—	—	4,56	4,60	4,58	4,59	0,166	2260	13730
30	1124,7	5,96	6,04	6,00	6,10	0,292	3850	13200
—	—	6,17	6,25	6,21	6,10	0,292	3850	13200
50	1874,5	7,00	7,09	7,04	7,07	0,392	4780	12200
—	—	7,14	7,12	7,13	7,07	0,392	4780	12200
—	—	7,06	7,00	7,03	7,07	0,392	4780	12200
70	2624,3	7,70	7,63	7,66	7,68	0,463	5670	12270
—	—	7,71	7,72	7,71	7,68	0,463	5670	12270
90	3374,1	8,25	8,31	8,28	8,27	0,537	6300	11720
—	—	8,27	8,26	8,26	8,27	0,537	6300	11720
Average	—	—	—	—	—	—	—	12624

Specimen No. 3. Diameter of the ball \(D=10\) mm.

Table 7.

Height of fall of the load, in cm	Kinetic energy \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374.9	4.90	4.85	4.87	4.88	0.186	2013	10800
—	—	4.87	4.88	4.88	4.88	0.186	2013	10800
30	1124.7	6.20	6.20	6.20	6.38	0.326	3510	10920
—	—	6.57	6.56	6.56	6.38	0.326	3510	10920
50	1874.5	7.10	7.18	7.14	7.09	0.395	4750	12010
—	—	7.06	7.02	7.04	7.09	0.395	4750	12010
70	2624.3	7.78	7.73	7.76	7.2	0.468	5620	11980
—	—	7.69	7.67	7.68	7.2	0.468	5620	11980
90	3374.1	8.10	8.10	8.10	8.10	0.515	6550	12730
Average	—	—	—	—	—	—	—	11590

From the data of these tables the following mean results have been derived:

Table 8.

Kinetic energy, in kg·cm	\(\dfrac{L}{\omega}\) (5)	\(\dfrac{L}{\omega}\) (6)	\(\dfrac{L}{\omega}\) (7)	Average	\(\dfrac{L}{\omega^2}\) (5)	\(\dfrac{L}{\omega^2}\) (6)	\(\dfrac{L}{\omega^2}\) (7)	Average	Note
374.9	2270	2260	2010	2180	13800	13730	10800	12780	The figures (5), (6), (7) under \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) indicate from which table the corresponding values of \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) were taken.
1124.7	3840	3850	3510	3730	13100	13200	10920	12410	The figures (5), (6), (7) under \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) indicate from which table the corresponding values of \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) were taken.
1874.5	4950	4780	4750	4830	13000	12200	12010	12400	The figures (5), (6), (7) under \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) indicate from which table the corresponding values of \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) were taken.
2624.3	5858	5670	5620	5718	13060	12270	11980	12440	The figures (5), (6), (7) under \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) indicate from which table the corresponding values of \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) were taken.
3374.1	6680	6300	6550	6510	13100	11720	12730	12520	The figures (5), (6), (7) under \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) indicate from which table the corresponding values of \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) were taken.
Average	—	—	—	—	13210	12620	11390	12510	The figures (5), (6), (7) under \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) indicate from which table the corresponding values of \(\dfrac{L}{\omega}\) and \(\dfrac{L}{\omega^2}\) were taken.

Hence it is seen that the greatest difference of the mean values of \(\dfrac{L}{\omega^2}\) is

\[ \frac{12780-12400}{12400}\cdot 100=3.1\%. \]

Specimen No. 4. Diameter of the ball \(D = 12.5\) mm.  Table 9.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	6,38	6,44	6,41	6,43	0,325	3460	10710
—	—	6,42	6,46	6,44	6,43	0,325	3460	10710
50	1874,5	7,38	7,35	7,36	7,36	0,425	4410	10380
70	2624,3	8,03	7,85	7,94	8,00	0,503	5210	10370
—	—	8,07	8,06	8,06	8,00	0,503	5210	10370
90	3374,1	8,56	8,56	8,56	8,56	0,575	5880	10190
Average	—	—	—	—		—	—	10410

Here the difference between the values of \(\dfrac{L}{\omega^2}\) is

\[ \frac{10710-10190}{10190}=5.1\%. \]

Specimen No. 5. Diameter of the ball \(D = 15\) mm.  Table 10.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	6,84	6,90	6,87	0,371	3040	8150
50	1874,5	7,60	8,08	7,84	0,483	3890	8050
70	2624,3	8,52	8,48	8,50	0,567	4640	8180
90	3374,1	9,12	9,03	9,08	0,647	5220	8050
Average	—	—	—	—	—	—	8110

Specimen No. 6. Diameter of the ball \(D=15\) mm. Table 11.

Height of fall of the load, cm	Kinetic energy \(L\), kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\dfrac{a_1+a_2}{2}\), mm	\(d_m\), mm	\(\omega\), cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	6,68	6,69	6,69	6,66	0,348	3240	9300
—	—	6,60	6,65	6,63	6,66	0,348	3240	9300
50	1874,5	7,55	7,55	7,55	7,55	0,448	4190	9330
70	2624,3	8,20	8,30	8,25	8,25	0,535	4900	9180
90	3374,1	8,76	8,70	8,73	8,73	0,599	5640	9400
Average	—	—	—	—	—	—	—	9300

Specimen No. 7. Diameter of the ball \(D=15\) mm. Table 12.

Height of fall of the load, cm	Kinetic energy \(L\), kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\dfrac{a_1+a_2}{2}\), mm	\(d_m\), mm	\(\omega\), cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	5,01	5,08	5,05	5,04	0,200	1870	9370
—	—	5,02	5,05	5,04	5,04	0,200	1870	9370
30	1124,7	6,72	6,73	6,73	6,73	0,356	3160	8860
50	1874,5	7,61	7,62	7,62	7,62	0,456	4120	9010
70	2624,3	8,32	8,23	8,27	8,27	0,537	4890	9110
90	3374,1	8,88	8,79	8,83	8,83	0,612	5510	9020
Average	—	—	—	—	—	—	—	9070

Hence the average results are derived:

Table 13.

Live force in kg. cm.	$\dfrac{L}{\omega}$ (10)	$\dfrac{L}{\omega}$ (11)	$\dfrac{L}{\omega}$ (12)	Average	$\dfrac{L}{\omega^2}$ (10)	$\dfrac{L}{\omega^2}$ (11)	$\dfrac{L}{\omega^2}$ (12)	Average
1124,7	3040	3240	3160	3150	8150	9330	8860	8780
1874,5	3890	4190	4120	4070	8050	9330	9010	8800
2624,3	4640	4900	4890	4810	8080	9180	9110	8790
3374,1	5220	5640	5510	5460	8050	9400	9020	8820
Average	—	—	—	—	8110	9300	9070	8800

It is seen from this that the greatest difference of the average values of $\dfrac{L}{\omega^2}$ amounts to

\[ \frac{8820-8780}{8780}\cdot 100 = 0,46\%. \]

Specimen No. 8. Diameter of the ball $D = 20$ mm. Table 14.

Height of fall of the load in cm.	Live force $L$ in kg. cm.	$a_1$ in mm.	$a_2$ in mm.	$d=\dfrac{a_1+a_2}{2}$ in mm.	$d_m$ in mm.	$\omega$ in cm.	$\dfrac{L}{\omega}$	$\dfrac{L}{\omega^2}$
30	1124,7	7,14	7,11	7,12	7,14	0,400	2810	7030
—	—	7,19	7,13	7,16	7,14	0,400	2810	7030
50	1874,5	8,06	7,99	8,03		0,506	3710	7320
70	2624,3	8,93	8,85	8,89		0,621	4240	6800
90	3374,1	9,44	9,37	9,41		0,695	4860	6990
130	4873,7	10,35	10,33	10,34		0,839	5809	6920
Average	—	—	—	—		—	—	7012

Specimen No. 9. Ball diameter \(D = 20\) mm.
Table 15.

Height of fall of the load, cm	Live force in kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\dfrac{a_1+a_2}{2}\), mm	\(d_m\), mm	\(\omega\), cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	5,41	5,50	5,46	5,44	0,232	1620	6970
—	—	5,63	5,50	5,56	5,44	0,232	1620	6970
—	—	5,33	5,30	5,31	5,44	0,232	1620	6970
50	1874,5	8,11	8,21	8,16	8,21	0,529	3550	6700
—	—	8,23	8,28	8,26	8,21	0,529	3550	6700
90	3374,1	9,56	9,52	9,54	9,54	0,715	4730	6600
130	4873,7	10,47	10,42	10,44	10,44	0,856	5694	6660
Average	—	—	—	—	—	—	—	6732

From the data of Tables 14 and 15 the following average results have been derived:

Table 16.

Live force	\(\dfrac{L}{\omega}\) (14)	\(\dfrac{L}{\omega}\) (15)	Average	\(\dfrac{L}{\omega^2}\) (14)	\(\dfrac{L}{\omega^2}\) (15)	Average
374,9	—	1620	1620	—	6970	6970
1124,7	2810	—	2810	7030	—	7030
1874,5	3710	3550	3630	7320	6700	7010
2624,3	4240	—	4240	6800	—	6800
3374,1	4860	4680	4720	6990	6600	6795
4873,7	5810	5694	5750	6920	6660	6790
Average	—	—	—	7012	6730	6870

Whence it is seen that the greatest difference of the average values of \(\dfrac{L}{\omega^2}\) is

\[ \frac{7030-6790}{6790}\cdot 100 = 3{,}55\%. \]

Specimen No. 10. Ball diameter \(D = 25\) mm.  Table 17.

Height of fall of the load, cm	Live force \(L\), kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\dfrac{a_1+a_2}{2}\), mm	\(d_m\), mm	\(\omega\), cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	7,43	7,65	7,54	7,54	0,447	2520	5620
50	1874,5	8,77	8,76	8,76	8,76	0,603	3110	5150
70	2624,3	9,54	9,51	9,52	9,52	0,712	3680	5180
90	3374,1	9,90	9,92	9,91	9,94	0,776	4350	5600
—	—	9,94	10,00	9,97	9,94	0,776	4350	5600
Average	—	—	—	—	—	—	—	5390

Specimen No. 11. Ball diameter \(D = 25\) mm.  Table 18.

Height of fall of the load, cm	Live force \(L\), kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\dfrac{a_1+a_2}{2}\), mm	\(\omega\), cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	5,80	5,81	5,80	0,264	1420	5380
50	1874,5	8,59	8,59	8,59	0,580	3240	5580
70	2624,3	9,39	9,37	9,38	0,691	3800	5500
90	3374,1	10,20	10,16	10,18	0,814	4150	5090
Average	—	—	—	—	—	—	5388

Specimen No. 12. Ball diameter \(D = 25\) mm.  Table 19.

Height of fall of the load, cm	Live force \(L\), kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\dfrac{a_1+a_2}{2}\), mm	\(\omega\), cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	7,67	7,77	7,72	0,468	2400	5140
50	1874,5	8,61	8,64	8,62	0,584	3210	5500
70	2624,3	9,24	9,29	9,27	0,675	3900	5760
90	3374,1	10,19	10,23	10,21	0,818	4130	5040
Average	—	—	—	—	—	—	5360

Finally, Tables Nos. 17, 18, and 19 give the following mean results:

Table 20.

Live force.	$\dfrac{L}{\omega}$ (17)	$\dfrac{L}{\omega}$ (18)	$\dfrac{L}{\omega}$ (19)	Mean	$\dfrac{L}{\omega^2}$ (17)	$\dfrac{L}{\omega^2}$ (18)	$\dfrac{L}{\omega^2}$ (19)	Mean
374,9	—	1420	—	1420	—	5380	—	5380
1124,7	2520	—	2400	2460	5620	—	5140	5380
1874,5	3110	3240	3210	3187	5150	5580	5500	5410
2624,3	3630	3800	3900	3773	5180	5500	5760	5480
3374,1	4350	4150	4130	4210	5600	5090	5040	5240
Mean	—	—	—	—	5390	5388	5360	5380

It is seen from this that the greatest difference of the mean values of $\dfrac{L}{\omega^2}$ is

$$ \frac{5480-5240}{5240}\cdot 100=4{,}58\%. $$

The data of the tables cited above undoubtedly indicate proportionality between the live force and the square of the area of the indentation produced by the ball; in each summary table (§§ 9, 13, 16, 20) calculations were made of the greatest differences between the individual values of $\dfrac{L}{\omega^2}$ in percentage ratios; it turned out that this difference varies from $0{,}46\%$ to $5{,}1\%$; such a difference is not very large, and thus the proportionality just indicated between $L$ and $\omega^2$ may be accepted, at least, as a first approximation to the true dependence existing between $L$ and $\omega$.

Thus, one obtains the relation $\dfrac{L}{\omega^2}=C_c$, or

$$ L=cd^4 \ldots\ldots (1) $$

In this formula, for each size of ball the coefficient of proportionality has a different magnitude, decreasing with increase in the diameter of the ball; it was therefore very interesting to introduce into the relation found above between $L$ and $\omega^2$ also the diameter of the ball $D$. To this end some calculations were made, the results of which are given in the following table:

Table 21.

Height of fall of the load \(H\), cm	Live force \(L\), kgcm	Diameter of ball \(D\), cm	Diameter of circumference of indentation \(d_m\), cm	\(d_m^4\)	\(\dfrac{d_m^4}{D}\)	Mean \((\eta)\) from \(\dfrac{d_m^4}{D}\) for each separate height of fall of the load	\(\dfrac{L}{\eta}\)
10	374,9	1,0	0,468	0,0480	0,0480	0,0450	8330
—	—	1,5	0,504	0,0645	0,0430	0,0450	8330
—	—	2,0	0,544	0,0876	0,0438	0,0450	8330
—	—	2,5	0,580	0,1129	0,0452	0,0450	8330
30	1124,7	1,0	0,620	0,1475	0,1475	0,1376	8170
—	—	1,25	0,643	0,1707	0,1366	0,1376	8170
—	—	1,5	0,675	0,2079	0,1386	0,1376	8170
—	—	2,0	0,714	0,2600	0,1300	0,1376	8170
—	—	2,5	0,763	0,3387	0,1355	0,1376	8170
50	1874,5	1,0	0,704	0,2460	0,2460	0,2313	8100
—	—	1,25	0,736	0,2938	0,2350	0,2313	8100
—	—	1,5	0,770	0,3516	0,2344	0,2313	8100
—	—	2,0	0,812	0,4343	0,2172	0,2313	8100
—	—	2,5	0,865	0,5595	0,2238	0,2313	8100
70	2624,3	1,0	0,765	0,3422	0,3422	0,3232	8120
—	—	1,25	0,800	0,4096	0,3277	0,3232	8120
—	—	1,5	0,834	0,4844	0,3229	0,3232	8120
—	—	2,0	0,889	0,6241	0,3120	0,3232	8120
—	—	2,5	0,939	0,7779	0,3112	0,3232	8120
90	3374,1	1,0	0,813	0,4369	0,4369	0,4199	8035
—	—	1,25	0,856	0,5372	0,4297	0,4199	8035
—	—	1,5	0,888	0,6225	0,4150	0,4199	8035
—	—	2,0	0,948	0,8082	0,4041	0,4199	8035
—	—	2,5	1,011	1,0447	0,4139	0,4199	8035
130	4873,7	2,0	1,044	1,1880	0,5940	—	8205
Mean	—	—	—	—	—	—	8160

From this table it is evident that, with great accuracy (the greatest difference between the values of \(\frac{L}{\eta}\) is

\[ \frac{8330-8035}{8035}\cdot 100 = 3.7\%), \]

\[ \frac{L}{\eta}=Const=C \]

or

\[ L=C\,\frac{d^4}{D}\ldots\ldots (2) \]

where the coefficient of proportionality is already the same for balls of different diameters.

All the preceding experiments were carried out with steel, and steel plates were used as the support. In order to be convinced that the dependencies found between \(L\), \(\omega\), and \(D\) for steel do not depend on the material of the support, it was necessary to repeat the experiments with steel, taking plates of other metals as the support. Copper plates, previously compacted on the same anvil, were chosen for this purpose. The following tables give the results of experiments again performed with steel, but with copper supports.

Specimen No. 13. Ball diameter \(D=25\) mm.  Table 22.

Height of fall of the load, cm	Live force \(L\), kg·cm	\(a_1\), mm	\(a_2\), mm	\(d=\frac{a_1+a_2}{2}\), mm	\(\omega\), cm	\(\frac{L}{\omega}\)	\(\frac{L}{\omega^2}\)
30	1124.7	7.51	7.52	7.51	0.443	2540	5740
50	1874.5	8.56	8.44	8.50	0.567	3310	5840
70	2624.3	9.38	9.13	9.26	0.673	3900	5790
90	3374.1	9.92	9.75	9.84	0.760	4440	5840
110	4123.9	10.50	10.25	10.37	0.842	4900	5810
Average	—	—	—	—	—	—	5804

Already from this table it is evident that the law

\[ \frac{L}{\omega^2}=\mathrm{const.} \]

is entirely valid here as well; the only difference from the data of the preceding experiments lies in the numerical value of the coefficient \((Const.)\).

Specimen No. 14. Diameter of the ball \(D = 25\) mm.  Table 23.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	5,75	5,76	5,75	0,260	1440	5550
30	1124,7	7,60	7,60	7,60	0,454	2480	5460
50	1874,5	8,58	8,63	8,60	0,581	3230	5550
90	3374,1	9,93	9,95	9,94	0,776	4348	5600
130	4873,7	11,00	10,84	10,92	0,935	5212	5580
150	5623,5	11,30	11,30	11,30	1,002	5612	5600
Average	—	—	—	—	—	—	5557

Specimen No. 15. Diameter of the ball \(D = 25\) mm.  Table 24.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	7,39	7,41	7,40	7,53	0,445	2530	5690
—	—	7,64	7,70	7,67	7,53	0,445	2530	5690
50	1874,5	8,71	8,36	8,53	8,54	0,573	3270	5720
—	—	8,54	8,54	8,54	8,54	0,573	3270	5720
—	—	8,56	8,56	8,56	8,54	0,573	3270	5720
70	2624,3	9,31	9,29	9,30	—	0,678	3870	5720
90	3374,1	9,90	9,92	9,91	—	0,771	−4376	5680
Average	—	—	—	—	—	—	—	5700

Tables Nos. 22, 23, and 24 give the following mean results:

Table 25.

Live force \(L\), in kg. cm.	\(\dfrac{L}{\omega}\) (22)	\(\dfrac{L}{\omega}\) (23)	\(\dfrac{L}{\omega}\) (24)	Mean	\(\dfrac{L}{\omega^2}\) (22)	\(\dfrac{L}{\omega^2}\) (23)	\(\dfrac{L}{\omega^2}\) (24)	Mean
374,9	—	1440	—	1440	—	5550	—	5550
1124,7	2540	2480	2530	2517	5740	5460	5690	5630
1874,5	3306	3230	3270	3268	5840	5550	5720	5700
2624,3	3900	—	3870	3885	5790	—	5720	5760
3374,1	4440	4348	4376	4386	5840	5604	5630	5710
4123,9	4900	—	—	4900	5810	—	—	5810
4873,7	—	5212	—	5212	—	5580	—	5580
5623,5	—	5612	—	5612	—	5600	—	5600
Mean	—	—	—	—	—	—	—	5670

As is evident from this table, despite the incompleteness of the data for all the stated values of the live force, nevertheless the greatest difference between the individual mean values of \(\dfrac{L}{\omega^2}\) does not exceed

\[ \frac{5810-5550}{5550}\cdot 100=4{,}6\%, \]

and the difference between the mean of all and the value of \(\dfrac{L}{\omega^2}\) deviating most from it does not exceed \(2{,}5\%\).

Specimen No. 16. Diameter of the ball, \(D=20\) mm.  Table 26.

Height of fall of the load, in cm.	Live force \(L\), in kg. cm.	\(a_1\), in mm.	\(a_2\), in mm.	\(d=\dfrac{a_1+a_2}{2}\), in mm.	\(d_m\), in mm.	\(\omega\), in cm.	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	6,85	6,97	6,91	7,05	0,390	2890	7400
—	—	7,25	7,14	7,19	7,05	0,390	2890	7400
50	1874,5	7,97	7,90	7,94	7,94	0,495	3790	7650
70	2624,3	8,90	8,84	8,87	8,74	0,600	4370	7290
—	—	8,69	8,56	8,62	8,74	0,600	4370	7290
90	3374,1	9,36	9,26	9,31	9,31	0,681	4950	7270
Mean	—	—	—	—	—	—	—	7400

Specimen No. 17. Ball diameter \(D = 10\) mm.
Table 27.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	7,28	7,33	7,25	0,413	2730	6580
50	1874,5	8,31	8,30	8,30	0,541	3465	6400
70	2624,3	9,03	9,03	9,03	0,640	4100	6400
90	3374,1	9,46	9,49	9,48	0,706	4780	6775
Average	—	—	—	—	—	—	6540

Specimen No. 18. Ball diameter \(D = 25\) mm.
Table 28.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
30	1124,7	7,27	7,31	7,29	7,29	0,417	2700	6460
50	1874,5	8,25	8,26	8,25	8,25	0,535	3500	6550
—	—	8,27	8,22	8,24	8,25	0,535	3500	6550
70	2624,3	8,87	8,87	8,87	8,87	0,618	4250	6870
110	4123,9	9,91	9,89	9,90	9,90	0,770	5360	6950
130	4873,7	10,40	10,41	10,40	10,40	0,849	5740	6770
Average	—	—	—	—	—	—	—	6720

Tables Nos. 26, 27, and 28 in turn give: Table 29.

Live force \(L\), in kg·cm	\(\dfrac{L}{\omega}\) (26)	\(\dfrac{L}{\omega}\) (27)	\(\dfrac{L}{\omega}\) (28)	Average	\(\dfrac{L}{\omega^2}\) (26)	\(\dfrac{L}{\omega^2}\) (27)	\(\dfrac{L}{\omega^2}\) (28)	Average
1124,7	2890	2730	2700	2770	7400	6580	6460	6810
1874,5	3790	3465	3500	3585	7650	6400	6550	6870
2624,3	4370	4100	4250	4240	7290	6400	6570	6850
3374,1	4950	4780	—	4865	7270	6775	—	7020
4123,9	—	—	5360	5360	—	—	6950	6950
4873,7	—	—	5740	5740	—	—	6770	6770
Average	—	—	—	—	7400	6540	6720	6880

From this table it is evident that here too the greatest difference between the individual values of \(\dfrac{L}{\omega^2}\) is only

\[ \frac{7020-6770}{6770}\cdot 100 = 3{,}7\%. \]

Specimen No. 19. Diameter of the ball \(D=15\) mm. Table 30.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	4,98	4,97	4,98	0,195	1920	9870
30	1124,7	6,59	6,57	6,58	0,340	3310	9700
50	1874,5	7,47	7,51	7,49	0,441	4250	9660
70	2624,3	8,13	8,10	8,12	0,518	5080	9790
Average	—	—	—	—	—	—	9755

Specimen No. 20. Diameter of the ball \(D = 15\) mm.
Table 31.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(\omega\), in cm	\(\dfrac{L}{\omega}\)	\(\dfrac{L}{\omega^2}\)
10	374,9	5,14	5,16	5,15	0,208	1800	8660
30	1124,7	6,81	6,83	6,82	0,365	3080	8460
50	1874,5	7,56	7,61	7,53	0,451	4160	9230
70	2624,3	8,30	8,36	8,33	0,545	4820	8840
Average	—	—	—	—	—	—	8797

As the average from the two following tables, we obtain:

Table 32.

Height of fall of the load, in cm	Live force \(L\), in kg·cm	Mean diameter \(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374,9	5,06	0,201	1860	9280
30	1124,7	6,70	0,353	3190	9000
50	1874,5	7,54	0,447	4200	9370
70	2624,3	8,22	0,531	4950	9310
Average	—	—	—	—	9240

The greatest difference between the mean separate values of \(\dfrac{L}{\omega^2}\) here is

\[ \frac{9370-9000}{9000}\cdot 100 = 4{,}1\%. \]

For the dependence of the live force, the diameters of the balls, and the depth, the data of the preceding tables give:

Table 33.

Height of fall of the load $H$, in cm	Live force $L$, in kg·cm	Diameter of the ball $D$, in cm	Diameter of the circular indentation $d_m$, in cm	$d_m^4$	$\dfrac{d_m^4}{D}$	Mean $(\eta)$ of $\dfrac{d_m^4}{D}$ for each separate height of fall of the load	$\dfrac{L}{\eta}$
10	374,9	1,5	0,506	0,0635	0,0437	0,0437	8580
—	—	2,5	0,575	0,1093	0,0437	0,0437	8580
30	1124,7	1,5	0,670	0,2015	0,1343	0,1324	8490
—	—	2,0	0,718	0,2657	0,1329	0,1324	8490
—	—	2,5	0,755	0,3249	0,1300	0,1324	8490
50	1874,5	1,5	0,754	0,3231	0,2154	0,2170	8640
—	—	2,0	0,816	0,4434	0,2217	0,2170	8640
—	—	2,5	0,855	0,5344	0,2138	0,2170	8640
70	2624,3	1,5	0,822	0,4564	0,3043	0,3048	8610
—	—	2,0	0,890	0,6273	0,3137	0,3048	8610
—	—	2,5	0,928	0,7413	0,2965	0,3048	8610
90	3374,1	2,0	0,940	0,7806	0,3903	0,3873	8710
—	—	2,5	0,990	0,9604	0,3842	0,3873	8710
110	4123,9	2,0	0,990	0,9604	0,4802	0,4705	8760
—	—	2,5	1,037	1,1520	0,4608	0,4705	8760
130	4873,7	2,0	1,040	1,1664	0,5832	0,5760	8460
—	—	2,5	1,092	1,4220	0,5688	0,5760	8460
150	5623,5	2,5	1,130	1,6290	0,6516	0,6516	8630
Mean	—	—	—	—	—	—	8610

The greatest difference is

\[ \frac{8760-8460}{8460}\cdot 100=3.55\%. \]

As we see, the previously found dependence between the live force, the diameter of the ball, and the diameter of the indentation is confirmed here with high accuracy; the intermediate body between the ball and the falling load affects only the magnitude of the coefficient, and not the very form of the dependence.

It was then very important to verify the dependence found also for other metals. To this end, further experiments were carried out with copper, cast iron, and nickel steel. These experiments were carried out in the same way as before; in all subsequent experiments the “pads” were steel. The results of the experiments are presented in the following tables.

Cast iron. The cast-iron specimens had dimensions of \(250 \times 50 \times 30\) mm and were completely freed from scale; the cast iron was ordinary gray cast iron.

Diameter of the ball: 25 mm.  Table 34.

Height of fall of the load	Live force, \(L\), in kg. cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\frac{a_1+a_2}{2}\), in mm	Mean of several tests, diam. \(d_m\), in mm	\(\omega_m\), in cm	\(\frac{L}{\omega_m}\)	\(\frac{L}{\omega_m^2}\)
30	1124,7	9,45	9,54	9,49	9,42	0,697	1620	2320
—	—	9,38	9,47	9,43	9,42	0,697	1620	2320
—	—	9,23	9,41	9,35	9,42	0,697	1620	2320
50	1874,5	10,78	10,84	10,81	10,63	0,887	2120	2390
—	—	10,54	10,51	10,53	10,63	0,887	2120	2390
—	—	10,50	10,59	10,55	10,63	0,887	2120	2390
70	2624,3	11,75	11,80	11,77	11,67	1,066	2460	2310
—	—	11,77	11,84	11,80	11,67	1,066	2460	2310
—	—	11,43	11,44	11,44	11,67	1,066	2460	2310
90	3374,1	12,52	12,51	12,52	12,43	1,219	2770	2280
—	—	12,51	12,52	12,52	12,43	1,219	2770	2280
—	—	12,25	12,34	12,29	12,43	1,219	2770	2280
Mean	—	—	—	—	—	—	—	2325

Here the greatest difference is

\[ \frac{2390-2280}{2280}\cdot 100=4.9\%. \]

Diameter of the ball \(D=20\) mm.
Table 35.

Height of fall of the load, in cm	Live force \(L\), in cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\frac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\frac{L}{\omega_m}\)	\(\frac{L}{\omega_m^2}\)
30	1124,7	8,78	8,76	8,77	8,72	0,597	1890	3160
—	—	8,82	8,88	8,85	8,72	0,597	1890	3160
—	—	8,60	8,60	8,60	8,72	0,597	1890	3160
—	—	8,65	8,65	8,65	8,72	0,597	1890	3160
50	1874,5	9,90	9,94	9,92	9,89	0,768	2450	3180
—	—	9,96	9,90	9,93	9,89	0,768	2450	3180
—	—	9,81	9,83	9,82	9,89	0,768	2450	3180
70	2624,3	10,87	10,92	10,90	10,91	0,934	2810	3010
—	—	11,04	11,10	11,07	10,91	0,934	2810	3010
—	—	10,77	10,74	10,76	10,91	0,934	2810	3010
90	3374,1	11,60	11,59	11,59	11,57	1,052	3210	3050
—	—	11,73	11,75	11,74	11,57	1,052	3210	3050
—	—	11,37	11,41	11,39	11,57	1,052	3210	3050
Average	—	—	—	—	—	—	—	3100

The greatest difference between the values of \(\frac{L}{\omega_m^2}\) is

\[ \frac{3180-3010}{3010}\cdot 100=5.6\%. \]

Diameter of the ball \(D=15\) mm.      Table 36.

Height of fall of the load, in cm	Live force \(L\), in kgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374,9	6,13	6,12	6,12	6,15	0,297	1260	4250
—	—	6,10	6,13	6,12	6,15	0,297	1260	4250
—	—	6,19	6,24	6,22	6,15	0,297	1260	4250
30	1124,7	7,96	7,99	7,98	8,08	0,513	2190	4270
—	—	8,38	8,36	8,37	8,08	0,513	2190	4270
—	—	7,91	7,90	7,91	8,08	0,513	2190	4270
50	1874,5	9,17	9,22	9,19	9,22	0,668	2810	4210
—	—	9,24	9,35	9,30	9,22	0,668	2810	4210
—	—	9,15	9,21	9,18	9,22	0,668	2810	4210
70	2624,3	10,07	10,07	10,07	10,014	0,787	3340	4230
—	—	10,06	10,05	10,06	10,014	0,787	3340	4230
—	—	9,89	9,94	9,92	10,014	0,787	3340	4230
90	3374,1	10,71	10,70	10,70	10,676	0,895	3770	4220
—	—	10,63	10,68	10,65	10,676	0,895	3770	4220
—	—	10,69	10,66	10,68	10,676	0,895	3770	4220
Average	—	—	—	—	—	—	—	4236

The greatest difference in the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{4270-4210}{4210}\cdot 100=1,4\%. \]

The last three tables again make it possible to establish the dependence between the live force, the diameter of the indentation, and the diameter of the ball;

Table 37.

Live force $L$, in kgcm	Diameter of ball $D$, in cm	Diameter of circular indentation $d_m$, in cm	$d_m^4$	$\dfrac{d_m^4}{D}$	Mean $(\eta)$ from $\dfrac{d_m^4}{D}$ for each separate value of live force	$\dfrac{L}{\eta}$
1124,7	2,5	0,942	0,7868	0,3147	0,2959	3800
—	2,0	0,872	0,5776	0,2888	0,2959	3800
—	1,5	0,808	0,4264	0,2843	0,2959	3800
1874,5	2,5	1,063	1,2769	0,5108	0,4903	3820
—	2,0	0,989	0,9565	0,4783	0,4903	3820
—	1,5	0,922	0,7225	0,4817	0,4903	3820
2624,3	2,5	1,167	1,8547	0,7419	0,7065	3720
—	2,0	1,091	1,4168	0,7084	0,7065	3720
—	1,5	1,001	1,0040	0,6693	0,7065	3720
3374,1	2,5	1,245	2,4025	0,9610	0,9084	3710
—	2,0	1,157	1,7926	0,8968	0,9084	3710
—	1,5	1,068	1,3010	0,8673	0,9084	3710
Mean	—	—	—	—	—	3760

The greatest difference between the separate values of $\dfrac{L}{\eta}$ is, here,

\[ \frac{3820-3710}{3710}\cdot 100 = 2,96\%. \]

Lead. The specimens of lead had dimensions $200 \times 35 \times 15$ mm.

Ball diameter \(D = 25\) mm.
Table 38.

Height of fall of the load	Live force \(L\), in kgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374,9	8,43	8,48	8,46	8,44	0,559	671	1200
—	—	8,42	8,41	8,42	8,44	0,559	671	1200
30	1124,7	11,07	11,05	11,06	11,05	0,959	1170	1220
—	—	11,01	11,07	11,04	11,05	0,959	1170	1220
50	1874,5	12,68	12,63	12,66	12,65	1,256	1500	1190
—	—	12,63	12,66	12,64	12,65	1,256	1500	1190
70	2624,3	13,80	13,80	13,80	13,75	1,484	1770	1190
—	—	13,71	13,74	13,71	13,75	1,484	1770	1190
Average	—	—	—	—	—	—	—	1200

The greatest difference between the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{1220-1190}{1190}\cdot 100 = 2{,}5\%. \]

Ball diameter \(D = 20\) mm.
Table 39.

Height of fall of the load, in cm	Live force \(L\), in kgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374,9	7,99	7,97	7,98	8,03	0,506	740	1460
—	—	8,08	8,08	8,08	8,03	0,506	740	1460
30	1124,7	10,77	10,81	10,79	10,56	0,875	1290	1470
—	—	10,56	10,57	10,56	10,56	0,875	1290	1470
—	—	10,33	10,33	10,33	10,56	0,875	1290	1470
50	1874,5	11,98	12,08	12,03	12,05	1,140	1640	1440
—	—	12,04	12,10	12,07	12,05	1,140	1640	1440
70	2624,3	13,10	13,11	13,11	13,07	1,341	1960	1460
—	—	13,04	13,03	13,04	13,07	1,341	1960	1460
Average	—	—	—	—	—	—	—	1458

The greatest difference between the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{1470-1440}{1440}\cdot 100 = 2.1\%. \]

Diameter of the ball \(D = 15\) mm.

Table 40.

Height of fall of the load, in cm	Live force \(L\), in klgr. cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374.9	7.51	7.56	7.54	7.50	0.441	850	1920
—	—	7.47	7.44	7.46	7.50	0.441	850	1920
30	1124.7	9.88	9.86	9.87	9.87	0.765	1470	1920
—	—	9.86	9.88	9.87	9.87	0.765	1470	1920
50	1874.5	11.22	11.27	11.25	11.25	0.994	1890	1900
—	—	11.22	11.28	11.25	11.25	0.994	1890	1900
70	2624.3	12.26	12.30	12.28	12.24	1.176	2240	1900
—	—	12.26	12.28	12.27	12.24	1.176	2240	1900
—	—	12.16	12.18	12.17	12.24	1.176	2240	1900
Average	—	—	—	—	—	—	—	1910

The greatest difference between the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{1920-1900}{1900}\cdot 100 = 1.1\%. \]

As a result of the preceding tables, we have:

Investigation of deformations produced by steel balls.

Table 41.

Live force \(L\) in kg·cm	Diameter of the ball \(D\) in cm	Diameter of the circumference of the indentation \(d_m\), in cm	\(d_m^4\)	\(\dfrac{d_m^4}{D}\)	Mean \((\eta)\) from \(\dfrac{d_m^4}{D}\) for each separate value of live force	\(\dfrac{L}{\eta}\)
374.9	2.5	0.844	0.507	0.203	0.207	1810
—	2.0	0.803	0.416	0.208	0.207	1810
—	1.5	0.750	0.316	0.211	0.207	1810
1124.7	2.5	1.105	1.491	0.596	0.617	1820
—	2.0	1.056	1.243	0.622	0.617	1820
—	1.5	0.987	0.949	0.633	0.617	1820
1874.5	2.5	1.265	2.560	1.024	1.049	1790
—	2.0	1.205	2.108	1.054	1.049	1790
—	1.5	1.125	1.603	1.068	1.049	1790
2624.3	2.5	1.375	3.576	1.430	1.462	1800
—	2.0	1.307	2.917	1.459	1.462	1800
—	1.5	1.224	2.244	1.496	1.462	1800
Mean	—	—	—	—	—	1805

The greatest difference in the values of \(\dfrac{L}{\eta}\) here is

\[ \frac{1820-1790}{1790}\cdot 100 = 1.7\%. \]

Nickel steel proved, in comparison with ordinary steel, to be not at all harder; its deformations are almost identical with the deformations of ordinary steel under the same live forces.

The nickel-steel specimens had dimensions \(140 \times 25 \times 10\) mm.

V. Pinegin.

Diameter of the ball \(D = 15\) mm.
Table 42.

Height of fall of the load \(H\), in cm	Live force \(L\), in kgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374,9	5,39	5,36	5,37	5,46	0,234	1600	6840
—	—	5,51	5,57	5,54	5,46	0,234	1600	6840
30	1124,7	7,12	7,15	7,14	7,14	0,400	2810	7030
—	—	7,13	7,15	7,14	7,14	0,400	2810	7030
50	1874,5	8,16	8,18	8,17	8,18	0,525	3570	6790
—	—	8,18	8,18	8,18	8,18	0,525	3570	6790
70	2624,5	8,71	8,66	8,69	8,90	0,622	4220	6780
—	—	9,12	9,10	9,11	8,90	0,622	4220	6780
Average	—	—	—	—	—	—	—	6860

The greatest difference in the values of \(\dfrac{L}{\omega_m}\) is

\[ \frac{7030-6780}{6780}=3,7\%. \]

Diameter of the ball \(D = 20\) mm.
Table 43.

Height of fall of the load \(H\), in cm	Live force \(L\), in kgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
10	374,9	5,87	5,88	5,88	5,85	0,269	1400	5190
—	—	5,82	5,81	5,81	5,85	0,269	1400	5190
30	1124,7	7,75	7,78	7,76	7,70	0,466	2410	5180
—	—	7,62	7,65	7,64	7,70	0,466	2410	5180
50	1874,5	8,85	8,87	8,86	8,79	0,607	3100	5090
—	—	8,73	8,69	8,71	8,79	0,607	3100	5090
70	2624,5	9,62	9,57	9,60	9,55	0,716	3670	5120
—	—	9,50	9,50	9,50	9,55	0,716	3670	5120
Average	—	—	—	—	—	—	—	5145

The greatest difference is

\[ \frac{5190-5090}{5090}\cdot 100=1.97\%. \]

Diameter of the ball \(D=25\) mm.
Table 44.

Height of fall of the load \(H\), in cm	Live force \(I_1\), in kg/cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{I}{\omega_m}\)	\(\dfrac{I}{\omega_m^2}\)
10	374.9	6.21	6.18	6.20	6.21	0.303	1240	4080
—	—	6.22	6.21	6.21	6.21	0.303	1240	4080
30	1124.7	8.21	8.25	8.23	8.18	0.526	2140	4070
—	—	8.14	8.12	8.13	8.18	0.526	2140	4070
50	1874.5	9.37	9.32	9.34	9.28	0.676	2770	4100
—	—	9.24	9.19	9.21	9.28	0.676	2770	4100
70	2624.3	10.15	10.14	10.15	10.11	0.803	3270	4070
—	—	10.07	10.09	10.08	10.11	0.803	3270	4070
Average	—	—	—	—	—	—	—	4080

The greatest difference is

\[ \frac{4100-4070}{4070}\cdot 100=0.74\%. \]

Finally, as follows from the data of the preceding tables, we have.

Table 45.

Height of fall of the load \(H\), in cm	Live force \(L\), in kgcm	Diameter of the ball \(D\), in cm	Diameter of the circumference of the indentation \(d_m\), in cm	\(d_m^4\)	\(\dfrac{d_m^4}{D}\)	Mean \((\eta)\) from \(\dfrac{d_m^4}{D}\) for each separate value of the live force	\(\dfrac{L}{\eta}\)
10	374,9	2,5	0,621	0,1487	0,0595	0,0591	6340
—	—	2,0	0,585	0,1171	0,0586	0,0591	6340
—	—	1,5	0,546	0,0889	0,0592	0,0591	6340
30	1124,7	2,5	0,818	0,4477	0,1791	0,1761	6390
—	—	2,0	0,770	0,3516	0,1758	0,1761	6390
—	—	1,5	0,714	0,2599	0,1733	0,1761	6390
50	1874,5	2,5	0,928	0,7413	0,2965	0,2978	6290
—	—	2,0	0,879	0,5969	0,2985	0,2978	6290
—	—	1,5	0,818	0,4477	0,2985	0,2978	6290
70	2624,3	2,5	1,011	1,0445	0,4178	0,4173	6290
—	—	2,0	0,955	0,8317	0,4159	0,4173	6290
—	—	1,5	0,890	0,6274	0,4183	0,4173	6290
Mean	—	—	—	—	—	—	6328

The greatest difference is

\[ \frac{6390-6290}{6290}\cdot 100 = 1{,}6\%. \]

After formula (2) had proved valid for various metals, it became necessary to determine what significance, for a given work \(L\), is to be attached to the change in velocity at the end of the fall of the load—a question which (as applied to the investigation of the effect, on the compression of lead cylinders, of changing the velocity of fall of the load upon them) had already been studied by Kick, but which still remains unresolved (see Martens, Materialienkunde, p. 170).

If it should turn out that a change in velocity, for one and the same work of the falling load, has no significance, or has so slight a significance that the changes in the coefficients due to the change in velocity do not exceed the errors of observation, then from formula (2), applying it to two cases in which \(H\) remains constant while \(P\) varies, we should obtain the following two relations:

\[ P_1 H_1 = C \frac{d_1^4}{D} \tag{3} \]

\[ P_2 H_1 = C \frac{d_2^4}{D}, \tag{4} \]

whence

\[ \frac{P_1}{d_1^4}=\frac{P_2}{d_2^4} \]

or

\[ \frac{P_1}{\omega_1^2}=\frac{P_2}{\omega_2^2}=\text{const.} \tag{5} \]

We now turn to the clarification of this question. For this purpose it was necessary for us to continue our experiments with other magnitudes of the falling load and heights of fall, but with the same vis viva \(L\) as before.

The new loads with which the further experiments were carried out weighed 56.87 and 129.06 klgr. From these it was already easy to calculate the heights of fall of the loads, wishing to preserve the former vis viva.

In these experiments the material tested was again steel and cast iron, but, apart from the former grades, of a different quality than in the experiments described above; in view of this, the results of the new experiments cannot be compared with the results of the earlier experiments. But since the loads (56.87 and 129.06 klgr) differ from one another by more than a factor of two, the new experiments are quite sufficient for determining the effect of changing the velocity of fall of the load. The experiments themselves were carried out by the previous method. The results of the experiments are set forth in the following tables.

Ball diameter \(D=20\) mm. Material—steel (\(130 \times 27 \times 12\) mm.)
Weight of the falling load \(P=56.87\) klgr.

Table 46.

Height of fall of the load, in cm.	Live force \(L\), in klgcm.	\(a_1\), in mm.	\(a_2\), in mm.	\(d=\dfrac{a_1+a_2}{2}\), in mm.	\(d_m\), in mm.	\(\omega_m\), in cm.	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
6.6	375.3	6.08	6.14	6.11	6.17	0.299	1260	4200
—	—	6.23	6.25	6.24	6.17	0.299	1260	4200
19.8	1126.0	8.19	8.17	8.18	8.20	0.528	2130	4040
—	—	8.21	8.22	8.21	8.20	0.528	2130	4040
33.0	1876.7	9.27	9.27	9.27	9.24	0.670	2800	4180
—	—	9.22	9.21	9.21	9.24	0.670	2800	4180
46.2	2627.4	10.11	10.07	10.09	10.09	0.799	3300	4120
—	—	10.10	10.08	10.09	10.09	0.799	3300	4120
Mean	—	—	—	—	—	—	—	4135

The greatest difference of the values \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{4200-4040}{4040}\cdot 100=3.96\%. \]

Material—steel; weight of the load \(P=56.87\) klgr. Ball diameter \(D=20\) mm.

Table 47.

Height of fall of the load, in cm.	Live force \(L\), in klgcm.	\(a_1\), in mm.	\(a_2\), in mm.	\(d=\dfrac{a_1+a_2}{2}\), in mm.	\(d_m\), in mm.	\(\omega_m\), in cm.	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
6.6	375.3	6.15	6.19	6.17	6.14	0.296	1270	4290
—	—	6.10	6.11	6.10	6.14	0.296	1270	4290
19.8	1126.0	8.18	8.17	8.17	8.10	0.516	2180	4240
—	—	8.04	8.03	8.03	8.10	0.516	2180	4240
33.0	1876.7	9.21	9.26	9.23	9.17	0.661	2840	4300
—	—	9.12	9.11	9.11	9.17	0.661	2840	4300
46.2	2627.4	10.00	10.00	10.00	9.96	0.779	3380	4330
—	—	9.93	9.91	9.92	9.96	0.779	3380	4330
Mean	—	—	—	—	—	—	—	4290

The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{4330-4240}{4240}\cdot 100 = 2.1\%. \]

As the mean from the two preceding tables, we obtain the following values of \(\dfrac{L}{\omega_m^2}\) for various magnitudes of kinetic energy:

For the kinetic energy \(L=375.3\) kg·cm, we have \(\dfrac{L}{\omega_m^2}=4249\)

\[ \begin{aligned} &=1126.0 &&\text{— }=4142\\ &=1876.7 &&\text{— }=4240\\ &=2627.4 &&\text{— }=4228 \end{aligned} \]

\[ \text{mean}=4214 \]

at the greatest difference

\[ \frac{4249-4142}{4142}\cdot 100 = 2.6\%. \]

Material—steel; weight of the load \(P=56.87\) klgr. Diameter of the ball \(D=15\) mm,

Table 48.

Height of fall of the load, in cm	Kinetic energy \(L\), in kgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
6.6	375.3	5.99	5.94	5.96	5.67	0.252	1490	5900
—	—	5.60	5.54	5.57	5.67	0.252	1490	5900
—	—	5.53	5.61	5.57	5.67	0.252	1490	5900
—	—	5.75	5.77	5.76	5.67	0.252	1490	5900
19.8	1126	7.58	7.57	7.58	7.49	0.440	2560	5820
—	—	7.37	7.42	7.40	7.49	0.440	2560	5820
33.0	1876.7	8.54	8.58	8.56	8.48	0.564	3330	5890
—	—	8.39	8.40	8.39	8.48	0.564	3330	5890
46.2	2627.4	9.39	9.38	9.39	9.20	0.665	3950	5960
—	—	8.98	9.06	9.02	9.20	0.665	3950	5960
Mean	—	—	—	—	—	—	—	5890

The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{5960-5820}{5820}\cdot 100 = 2.4\%. \]

Material—steel. Weight of the falling load \(P=129.06\) klgr. Diameter of the ball \(D=20\) mm.

Table 49.

Height of fall of the load, in cm	Live force \(L\), in klgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_r\), in mm	\(\omega m\), in cm	\(\dfrac{L}{\omega m}\)	\(\dfrac{L}{\omega m^2}\)
2.9	374.3	6.19	6.12	6.15	6.18	0.300	1250	4160
—	—	6.10	6.06	6.08	6.18	0.300	1250	4160
—	—	6.33	6.31	6.32	6.18	0.300	1250	4160
8.7	1122.8	8.15	8.18	8.16	8.12	0.518	2180	4190
—	—	7.98	8.02	8.00	8.12	0.518	2180	4190
—	—	8.14	8.21	8.17	8.12	0.518	2180	4190
14.5	1874.4	9.22	9.12	9.17	9.16	0.659	2850	4310
—	—	9.07	9.04	9.05	9.16	0.659	2850	4310
—	—	9.26	9.26	9.26	9.16	0.659	2850	4310
20.3	2620.0	9.91	9.88	9.90	9.96	0.779	3370	4320
—	—	9.89	9.88	9.89	9.96	0.779	3370	4320
—	—	10.09	10.07	10.08	9.96	0.779	3370	4320
Average	—	—	—	—	—	—	—	4245

The greatest difference of the values \(\dfrac{L}{\omega m^2}\) is

\[ \frac{4320-4160}{4160}\cdot 100=3.8\%, \]

Material—steel. Weight of the load, \(P=129.06\) klgr. Diameter of the ball \(D=15\) mm.

Table 50.

Height of fall of the load, in cm	Live force \(L\), in klgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
2.9	374.3	5.75	5.74	5.75	5.67	0.252	1480	5890
—	—	5.59	5.58	5.59	5.67	0.252	1480	5890
8.7	1122.8	7.53	7.57	7.55	7.50	0.442	2550	5760
—	—	7.43	7.48	7.46	7.50	0.442	2550	5760
14.5	1871.4	8.53	8.61	8.57	8.46	0.563	3320	5900
—	—	8.34	8.38	8.36	8.46	0.563	3320	5900
20.3	2620.0	9.24	9.28	9.26	9.22	0.667	3940	5880
—	—	9.16	9.22	9.19	9.22	0.667	3940	5880
Average	—	—	—	—	—	—	—	5858

The greatest difference between the values of \(\dfrac{L}{\omega_m^2}\) is

\[ \frac{5900-5760}{5760}\cdot 100 = 2.4\%. \]

As is seen from the data of these tables, the difference for the mean values of \(\dfrac{L}{\omega_m^2}\), both for a ball of 20 mm diameter and for a ball of 15 mm diameter, for loads of 129.06 klgr. and 56.87 klgr., is very insignificant; in the first case it amounts to only

\[ \frac{4245-4214}{4214}\cdot 100 = 0.71\%, \]

and in the second,

\[ \frac{5890-5858}{5858}\cdot 100 = 0.55\%. \]

And thus the validity of formula (5) is confirmed with great accuracy. The same confirmation of formula (5), although not in such a categorical form, is given to us by our experiments with cast iron, as is seen from the data of the following tables.

Material—cast iron. Weight of the falling load \(P=56.87\). Diameter of the ball \(D=20\ \mathrm{mm}\).

Table 51.

Height of fall of the load, in cm	Live force \(L\), in kgf·cm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
6,6	375,3	7,25	7,20	7,22	7,27	0,415	905	2180
—	—	7,33	7,34	7,34	7,27	0,415	905	2180
—	—	7,12	7,13	7,13	7,27	0,415	905	2180
—	—	7,26	7,25	7,26	7,27	0,415	905	2180
—	—	7,34	7,33	7,34	7,27	0,415	905	2180
—	—	7,34	7,33	7,34	7,27	0,415	905	2180
19,8	1126	9,49	9,47	9,48	9,55	0,717	1570	2190
—	—	9,57	9,56	9,57	9,55	0,717	1570	2190
—	—	9,56	9,62	9,59	9,55	0,717	1570	2190
—	—	9,54	9,62	9,58	9,55	0,717	1570	2190
33,0	1876,7	10,79	10,86	10,83	10,80	0,916	2050	2240
—	—	10,71	10,78	10,75	10,80	0,916	2050	2240
—	—	10,78	10,87	10,82	10,80	0,916	2050	2240
—	—	10,77	10,85	10,81	10,80	0,916	2050	2240
46,2	2627,4	11,68	11,70	11,69	11,73	1,079	2440	2250
—	—	11,64	11,67	11,66	11,73	1,079	2440	2250
—	—	11,73	11,78	11,75	11,73	1,079	2440	2250
—	—	11,76	11,74	11,75	11,73	1,079	2440	2250
—	—	11,74	11,76	11,75	11,73	1,079	2440	2250
Average	—	—	—	—	—	—	—	2215

The greatest difference of the values \(\dfrac{L}{\omega_m^2}\) is, here,

\[ \frac{2250-2180}{2180}\cdot 100=3.2\%. \]

Material—cast iron. Weight of the falling load \(P = 129.06\) klgr. Diameter of the ball \(D = 20\) mm.

Table 52.

Height of fall of the load, in cm	Live force \(L\), in klgcm	\(a_1\), in mm	\(a_2\), in mm	\(d=\dfrac{a_1+a_2}{2}\), in mm	\(d_m\), in mm	\(\omega_m\), in cm	\(\dfrac{L}{\omega_m}\)	\(\dfrac{L}{\omega_m^2}\)
2.9	374.3	7.17	7.20	7.18	7.20	0.407	940	2250
—	—	7.22	7.21	7.22	7.20	0.407	940	2250
8.7	1122.8	9.43	9.46	9.45	9.49	0.707	1590	2240
—	—	9.52	9.56	9.54	9.49	0.707	1590	2240
14.5	1871.4	10.67	10.68	10.68	10.69	0.897	2090	2330
—	—	10.69	10.71	10.70	10.69	0.897	2090	2330
20.3	2619.9	11.58	11.63	11.60	11.64	1.064	2460	2310
—	—	11.58	11.76	11.67	11.64	1.064	2460	2310
26.1	3368.5	12.36	12.30	12.33	12.38	1.203	2800	2330
—	—	12.42	12.44	12.43	12.38	1.203	2800	2330
Average	—	—	—	—	—	—	—	2292

The greatest difference in the values of \(\dfrac{L}{\omega_m^2}\) here is

\[ \frac{2330-2240}{2240}\cdot 100 = 4\%. \]

The difference in the values of the averages of \(\dfrac{L}{\omega_m^2}\) for different falling loads is here

\[ \frac{2292-2215}{2215}\cdot 100 = 3.5\%. \]

Thus, summarizing the foregoing, we may present the results of our experiments in the form of the following propositions, stipulating, however, that, first, we ourselves regard these propositions only as a first approximation to the true dependencies existing between the quantities under consideration, and that, second, they are applicable only within the limits of the experiments performed:

1) The ratio of the kinetic energy acting on a steel ball to the square of the area of the indentation produced by the ball in any metal is a constant quantity, depending only on the diameter of the ball and on the kind of metal.

2) The product of the kinetic energy acting on a steel ball, divided by the square of the area of the indentation produced by the steel ball in any metal, by the diameter of the ball is a constant quantity, depending only on the kind of metal.

3) The ratio of the weight of a load falling onto a steel ball to the square of the area of the indentation produced by the ball in any metal is, for a given height of fall of the load, a constant quantity, depending on the kind of metal, but not on the magnitude of the load.

In conclusion, I cannot refrain from expressing my deep gratitude to Prof. Meyer, who kindly placed at my disposal, entirely free of charge, the material and everything necessary for the experiments.

V. Pinegin.