1921-22.] Concomitants of Quadratic Differential Forms. 175 
the combinations of the second derivatives of ct, h, . . n which occur in 
the symbols usually associated with the names of both Riemann and 
Christoffel. It should further be noted that very many of the coefficients 
of the derivatives of (p in the preceding forty equations are the combina- 
tions often called Christoffiers symbols, which, in the case of our four 
independent variables, are as follows : — 
11 
_2 
12 
2 
— 2 ’ 
— ) 
11 
3 
12 
3 
hi 92 + fi-h) 
12 ' 
4 
[V] = K- 
[7] = K. ['2^]=i('»3+/l-^2). [3^] = ^. 
[*2*^]=i(*4+™i-^2). [3*]=i(f'4 +«i-y. [ 7 ]=k; 
= = [f]=i( 2 / 3 -y 
13 
4 
— ^(2Zi a^) ; 
— 2 (^2 4" ~ ^ 
= 2 (^h + h- 9 i); 
A), 
1 1 
i ! 
— 2^3> 
1 1 
CO 
CO 
1 1 
“24 
II 
"24 
“i) , 
_2^ 
3 
— » 
23' 
4 
+ fi) ) 
'24' 
4 
= lr7 • 
2"2 J 
33 
3 
- Ir 
~ 2 ^'3 ’ 
P] = i(2 
“341 1 
r34"| 
3 
L4 J 
— — fZg) 
[Y]=i ('‘4 + 
[^/]=i(^3 + !/4~«l). ^2 =i(™3+/4-»2). 
the general term being 
r J 2 V dxq 3 j‘^ dxy 
with ^„=a, gi3=g, g.,^=f, gu^h, g^^^c, g^^ = l, g^^ = m, g^^ = n, 
ga=d. 
We also require the first minors of the determinant A where 
— 1(27^3 cq) ; 
P] = K; 
a, h, g , I 
h, b, f, m 
9, f, ^ 
I, m, n, d 
abed 
- adp - bd(f - cdli^ - beV- - cam- - abn'^ 
-f 2qfm7i + 2bgnl + 2chlm + 2dfgh 
+fH‘^ + + b9‘7i^ - 2fglm - 2ghmn - 21ifnl . 
