1920-21.]/ The Equations of Motion of a Single Particle. 33 
and in place of (25) we get 
(29) . . ^ y r *Y dr 2 + r 2 d6 2 + r 2 sin 2 6d<f) 2 + 2Y ^y'l . 
The values of A and y 4 are given by 
Here Y 1 is completely undetermined except that (i) it is a function of r 
alone, (ii) it agrees with the Newtonian potential to a first approximation 
apart from the additive constant a. 
If, in addition, we impose Kottler’s * condition that the determinant of 
the form (29) equals — c 2 when expressed in rectangular coordinates, we 
readily deduce that Y 1 = — J(a + /3/r ), a being constant and /3 = — 2c/ JZ- 2y, 
and then (29) becomes 
c 2 
(31) . . . — -^ydr 2 + ? ,2 d0 2 + r 2 sin 2 6d(j> 2 + 2Y 1 di/f , 
which is equivalent to the Einstein-Schwarzschild form, 
A similar discussion can be easily made if the second form of Laplace’s 
equation is adopted, the result being that Y has the form given in (19) and 
that (29) is replaced by 
~( d ~Y dr 2 + rW 2 + r 2 sin 2 0dd>‘ i + 2V, A/j. 
?n z ' dr * 1 
* F. Kottler, Ann. d. Phys., lvi (1918), p. 401. 
( Issued separately January 24, 1921.) 
VOL. XLI. 
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