1921-22.] Concomitants of Quadratic Differential Forms. 201 
pair is given by five of the six equations in Pg for the respective 
values of the principal curvatures. 
Spaces of Constant (and Zero) Curvature. 
39. Two special examples may be noted. 
In the first, we take the amplitude of § 36, given by /(P, Q, R, S, T) = 0 
to be fiat, say 
T=.0; 
then the element of arc in that amplitude is 
But now we can take 
P = iTj , Q = a’2 , R = ^3 , 8 = 2^4; 
so that 
C?s2 ^ (^2‘ 2 _j_ 
This is a quadratic form for which 
a — h = c = d= I , 
f = g~h = l = m=^n = 0. 
For this form, all the quantities 
D, M, . . ., J, A 
vanish, because they involve first and second derivatives of the constant 
coefficients a, . . n. 
Hence all the line-covariants vanish except only A(P), which is 
A(P) = P,2 + P/ + P 32 + p;2 + p^2 p^2 ^ 
manifestly not vanishing. Consequently, all the line-covariants of any 
form, which arises from 
dx-^^ -}- dx^^ -h dx^^ -1- dx^^ 
by transformation of the variables, also vanish. In particular, we must 
have 
u (P) = 0 
for all values of the variables P^^ , . . ., Pg ; and therefore every coefficient 
in '?^(P) must vanish ; that is, we have the relations 
D =- 0 , M = 0 , . . . , J = 0 , A = 0 . 
Moreover, when these quantities vanish, all the remaining line-covariants 
vanish. Further, all the invariants will then vanish except only the 
discriminant A. 
