203 
1921-22.] Concomitants of Quadratic Differential Forms. 
where W is a constant ; and this relation must be satisfied for all directions 
P. Hence we must have the relations 
D = Wd 
M = Wm 
J = Wj 
A= Wa, 
necessary and sufficient to secure the constancy of the curvature.* Various 
forms for the element of arc in such a space have been given, the simplest 
being Riemann’s form f 
Einstein’s Law of Gravitation: Integration of the 
Defining Equations. 
41. It may be convenient, at the beginning of this next stage-, to recall 
the difference between the Riemann-Christoffel symbols {rh, ih) and the 
symbols {rh, ih} due to the same authors. The latter represent the elements 
of the Riemann-Christoffel tensor in Einstein’s theory of relativity. In the 
notation of § 23, the relation between them is expressed in the form 
Y 1 ' 2 , 3 , 4 
{/xp, Tcr} = - 2 av), 
k 
while the symbol on the left-hand side is denoted by 
B'’ 
/XVC7 
by Einstein, f 
Consequently, for a form — dx^ — dy^ — dz^ + all the tensors 
vanish ; and their identical evanescence for any form 
(a, b, c, d, f, g, h, I, m, n \ dx-^, dx^, dx^, dx^ 
suffices to secure that it is reducible to the earlier four-square form with 
constant coefficients. 
* The notion of amplitudes with a constant (or variable) measure of curvature 
originated with Riemann. The literature dealing with amplitudes having a constant 
measure of curvature is copious, and the developments really belong to the domain of 
differential geometry. Some account is given by Bianchi, Lezioni di geometria differenziale, 
t. i, cap. xi. 
t He obtained it for a space of n dimensions. 
1 Bianchi, Lezioni di geometria differenziale, t. i, p. 72 ; Einstein, An.n. d. Phys., 
Bd. xlix, p. 800. 
