179 
1921-22.] Concomitants of Quadratic Differential Forms. 
Thus there are forty-five quantities in all. The complete system of fifteen 
equations will therefore possess thirty independent integrals, which can be 
taken in a variety of ways ; and, among these, we shall have 
+ U2X2 4 - U3X3 -h U4X, 
as a mixed concomitant, 
as a (vanishing) line-co variant, and 
N + 0 + V 
as a (vanishing) differential invariant. 
25 . The initial forms of the coefficients of 0 .^ involve derivatives 
of (X , . . n of the first and second orders ; and the terms must be trans- 
formed so as to introduce the necessary combinations. When the rather 
laborious (but otherwise simple) transformations have been effected, we 
find the following as the aggregate of the equations, viz. the first twelve, 
arising out of the coefficients of , ^3 , ; '>71 , '>73 , '>74 ; > ^2 > ^4 j 5 ^2 ’ ^3 ’ 
in tlie form 
\\T ^ X + P 
W =X ^ + P ^ 
W =X -^ + P 
w _ X ^ 4- P ^ 
W =X -^ 4 -P — 
^ ^ax, "ap^ 
_ p ^ XJ 
lap, 
a 
au. 
H, = 0, 
P — 
=0P. 
^'eu. ° ’ 
— + p--. 
*0X3^ 'iePi 
Wg + p 
W —X ^ I p ® 
ax^ ' ^ap^ 
w.=x,4 + P 34 
w.=x 3 ^ 4 -+p ,4 
-p 
p 
p 
p 
p 
p 
^^3 
d_ 
^au, 
U,^-H, = 0, 
2 aP 3 
_d_ 
^aPi 
^ap ~ 
Us^^-^4 = 0, 
-U3Jr-Z2 = 0, 
^ap, 
'^ap. 
p — 
'0P. 
20p: 
