1921 - 22 .] Concomitants of Quadratic Differential Forms. 165 
from those of Ii 34. ^134. ^134. ^134 W6 haV6 
2 ^+ M-+ _# = 0 ' 
^^13 
^i_+ M + ^.^=0 
^^^34 14 ^^13 
11 + 2 ^+ 
^i/34 ^^14 ^^^3 
+ A_^ 2 A. ^.0 
^^34 ^^14 
and from those of ^234 > '^234 > ^234 ’ ^234 have 
d(f) 
+ 
d<f> 
+ 
dcf) 
= 0" 
^^34 
^92i 
^*^23 
> A 
-i- 
dcf) 
+ 
dcf) 
= 0 
CD 
CO 
^/24 
dm^^ 
d(j) 
+ 
2^ 
+ 
d(f> 
= 0 
%4 
d(‘2i 
dcf) 
-t- 
dcj) 
-1- 
2^ 
= 0 
^^^34 
^-^23 
18 . Now these eighty partial differential equations constitute a com- 
plete Jacobian linear system * by themselves ; it is, indeed, a feature of 
Lie’s theory of the infinitesimal transformations of a continuous group that 
the aggregate of the partial difierential equations should be of this nature. 
The quantities that occur in them are one hundred in number; hence f 
the system of eighty equations possesses twenty functionally independent 
integrals. 
In the eighty equations, there are twenty-eight which are single- 
termed ; they require that none of the quantities » <^13 ’ ^i4 j A > 
^22 ’ 5^11 ’ 9 s 3 5 ^11 ’ ^44 ? ^12 > ^22 ’ ^23 ’ ^24 5 ./*22 ’ fs 3 ’ '^^22 > ^44 i ^13 > ^23 ’ ^33 ’ ^34 > 
%3 ’ '^^4 5 ^14 ’ ^^24 ’ ^^34 ’ ^44 occur ill any of the integrals. Thus the 
integrals must be proper combinations, the simpler the better, of the 
remaining seventy-two double suffix quantities. There are twelve sets 
of two-term equations, each set consisting of three members; and there 
are four sets of three-term equations, each set consisting of four members. 
Thus there remain fifty-two equations in all, involving seventy-two 
quantities. 
A complete set of twenty functionally independent integrals (or, as 
* For the characteristics and the properties of such a system, see my Theory of Differential 
Equations, vol. v, §§ 37-46. 
t Loc. cit., p. 86. 
