205 
1921-22.] Concomitants of Quadratic Differential Forms. 
notation of this paper, and having regard 
we have 
_F E A 
F . D . B 
Bq2 = — + — + 
a c Cl 
V 
a 
to the limiting assumptions, 
B 
12 “ 
M 
Bj3 = 0 , Bg3 = 0 , Bj^ = 0 , B24 = 0 , Bg^ = 0 , 
the last five vanishing identically. Thus the differential equations of the 
limited field of gravitation are 
Bii = 0, B22 = 0, B33 = 0, B4, = 0, B^2 = 0, 
which must be satisfied by the coefficients a, b, c, cl. 
42. I now proceed to the integration of these equations without all the 
limiting conditions assumed (of course justifiably, in postulating his problem) 
by Einstein and followed by Schwarzschild. Beyond satisfying these 
equations, the only conditions that need to be imposed upon the coefficients 
a, b, c, d of the form 
a dx-^^ + b + c dx^ + d dx^ , 
other than those already imposed upon the occurrence of the variables, are 
that the values must be such as to allow the quadratic differential form to 
degenerate into 
- dr^ — _ ^.2 gq^2 ^ ^ 
when there is no field of gravitation; that is, when r tends to an infinite 
value. The supposition as to polar symmetry seems superfluous ; and the 
supposed complete non-occurrence of (or t) in the coefficients is a very 
definite limitation. 
With our former notation, we have 
2B = 
2a ’ 
20 = , 
2a ’ 
