Page:Die Kaufmannschen Messungen.djvu/5

This page has been proofread, but needs to be validated.

contains the magnetic deflection ${\bar {z}}$ according to Kaufmann's table VI (l. c. p. 524), the second column the corresponding values of the angle φ₁ as calculated from (10), the third column the value of u in degrees following from (26), where the following value of the ratio of charge ε to mass μ₀ [extrapolated by Kaufmann (l.c. p. 551) on the basis of Simon's number 1,865.10⁷ valid for all theories] is:

{\frac {\epsilon }{\mu _{0}}}=1,878\cdot 10^{7}

27)

The fourth and sixth column contain the values of $\beta ={\frac {q}{c}}$ calculated from u according to (24) and (25), the fifth and seventh column contain the values of ${\bar {y}}$ following from (18), where the required φ' and φ₂ are taken from (17) and (11); and finally the eighth column contain the "observed" values of ${\bar {y}}$ according to Kaufmann's table VI.

Observed ${\bar {z}}$	φ₁	u	Sphere theory		Relative theory		Observed ${\bar {y}}$
Observed ${\bar {z}}$	φ₁	u	β	${\bar {y}}$	β	${\bar {y}}$	Observed ${\bar {y}}$
0,1354 0,1930 0,2423 0,2930 0,3423 0,3930 0,4446 0,4926 0,5522	1,977° 2,810 3,517 4,231 4,925 5,623 6,325 6,692 7,735	0,3871 (0,3870) 0,5502 (0,5502) 0,6883 (0,6881) 0,8290 (0,8286) 0,9634 (0,9630) 1,100 (1,099) 1,23 (1,234) 1,360 (1,358) 1,510 (1,506)	0,9747 0,9238 0,8689 0,8096 0,7542 0,7013 0,6526 0,6124 0,5685	0,0262 (0,0262) 0,0394 (0,0394) 0,0526 (0,0526) 0,0682 (0,0682) 0,0853 (0,0855) 0,1054 (0,1055) 0,1280 (0,1281) 0,1511 (0,1512) 0,1823 (0,1822)	0,9326 0,8762 0,8237 0,7699 0,7202 0,6728 0,6289 0,5924 0,5521	0,0273 (0,0274) 0,0415 (0,0415) 0,0555 (0,0554) 0,0717 (0,0717) 0,0893 (0,0895) 0,1099 (0,1099) 0,1328 (0,1328) 0,1562 (0,1561) 0,1878 (0,1874)	0,0247 0,0378 0,0506 0,0653 0,0825 0,1025 0,1242 0,1457 0,1746

First, in order to enable a comparison of my method of calculation with that of Kaufmann, I have put under the values of u as well as under the theoretical values of ${\bar {y}}$ , those numbers in brackets resulting from the same quantities, when we (like Kaufmann) rely, not on the observed values ${\bar {z}}$ , but on the values "reduced to infinitely small deflection" z' (l.c. Table VII, p. 529), and from which u is calculated by using Kaufmann's equations (14) and (17 ), determining the corresponding $v={\frac {u}{\beta }}$ according to each of the two theories, and then pass to y' by using Kaufmann's equation (18). Then ${\bar {y}}$ is given by Kaufmann's equation (12). For this calculation, Kaufmann's constants A and B are of course not the "constants of curve" but the "constants of apparatus", which were measured independently of the deflection experiments. The comparison of the bracketed numbers with the numbers stated above shows, that the results of Kaufmann's method of calculation differ from those of mine only very marginally, so each of the two methods supports the other in some way.

As regards the comparison of theoretical values of ${\bar {y}}$ with the observed ones, it can be seen that the latter are closer to the sphere theory than to the relative theory. However, in my opinion this can not be interpreted as a final confirmation of the first and a refutation of the second theory. Because for that it would be necessary that the deviations of the theoretical numbers from those observed, are small for the sphere theory against those of the relative theory. But this is not at all the case: on the contrary, the deviations of the theoretical numbers from each other are throughout smaller than the deviations of any theoretical number from the observed ones.

One might think now, perhaps, that the lack of agreement is caused by the employed value (27) for the ratio ε:μ₀, and that by a suitable amendment of this value a sufficient correspondence can be obtained for one of the two theories. This can be easily tested in the following way. Equation (18) gives, if one substitutes for ${\bar {y}}$ any observed value, the corresponding value of the velocity q = βc regardless of any special theory, and the corresponding value of u is derived separately for each theory by (24) or by (25), and then from (26) the ratio ε:μ₀ can be calculated. This procedure gives not only for none of the two theories constant values for ε:μ₀, but even for β it gives numbers that are unacceptable from the outset for each theory. The same is found, of course, in Kaufmann's method of calculation. Kaufmann^[1] gives two equations for the deflections y' and z' , which combined have the form:

\beta ={\frac {E}{cM}}\cdot {\frac {z'}{y'}}

.

Where

{\frac {E}{cM}}=0,1884

is a constant of apparatus, independent of the value of ε:μ₀ and independent of any specific

↑ l. c. p. 529, equations (14) and (15).

[1] . c. p. 529, equations (14) and (15).

[1]