212 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
the last being in no case a loss of generality. Hence hnahy, with the 
single condition that has been imposed, the solution of the initial 
differential equations gives the form 
ds^ = — ~ r'^dO'^ — sin^ 6d(f)- + . 
r 
It will be noted that, in deriving the form, there has been no necessity 
to use the property that the functional determinant should be taken equal 
to — 1 ; it was a property taken by Einstein mainly for convenience.* 
Further, the condition of comparative symmetry as regards the polar 
coordinates, imposed by Einstein f and adopted by Schwarzschild,| is 
superfluous ; it is actually, but implicitly, carried by the condition as 
to the form of ds^ in the degenerate case. All that has been required 
is the set of five fundamental equations applied to the type of ds'^ that 
was assumed, together with the single condition. 
45. In passing, it may be pointed out that, in a non-gravitation field, 
the seven non-evanescent equations 
A = 0, B-0, C = 0, D-0, E = 0, F-0, M = 0 
are satisfied. Einsteins mathematical definition (§ 41) of the gravitation 
field rests on the five non-evanescent equations 
Bn = 0, B,^ = 0, B33 = 0, B44 = 0, = 
It is, however, easy to verify that the coefficients of the foregoing form, 
as deduced from these five equations, actually satisfy the six equations 
Bo2 = 0 , B^4 = 0 , Bj2 = 0 , 
B_C 
b c 
0 
a d 
?-?= 0 ; 
a d 
so that it would appear as if the imposition of the limiting condition, 
under degeneration, has the effect of bringing the number of the 
equations of the gravitation field one unit nearer the number for the 
non-gravitation field. 
* Ann. d. Physik, Bd, xlix, pp. 789, 801. f Sitzungsb. Berlin (1915), j). 801. 
+ Ih. (1916), p. 192. 
31 si December 1921. 
{Issued separately July 4, 1922.) 
