181 
1921 - 22 . 1 Concomitants of Quadratic Differential Forms. 
*■ - + ™s-E ^ 4 - 4 * ■■ 4 
I /-\7 r\\^ ^ ^ ^ \ I 7^ I 07^ I ^1 
= A^ + 2Q 2T^ + + 
aQ 
aF ax 
aD an as aK 
aM 
+ <“ - '’4 + “4 - 4)+''li + "’4.+ 'I. + 
df' 
= 4 + ^4 - 4 - ^4 - " 4 - 4 
+ ( 0 -N)— +jf— -—') + d ^- + 2 n — + l ^ + 
^ 4 m V8N 00/ dn 8c 8 a 
8g ■ 
In addition to these twelve equations, W^ — 0, . . ., Wj 2 = 0, there are the 
remaining three equations which arise by equating to one another the four 
coefficients of , >?2 > ^3 ^ ^4 • ^^^d it is in virtue of these three equations 
that we satisfy the non-evanescent Jacobi conditions, viz. : 
(W,,W,) = 0, (W2,W,) = 0, (W3,w^,)=.o, 
(W,,W3) = 0, (W,,W,,)^0, (W,,w,,) = 0, 
in order that the system Wj = 0, . . ., — ^ may be a complete Jacobian 
system. We thus have the fifteen equations which constitute the system ; 
and we have to find a full set of algebraically independent integrals, thirty 
in number. 
26. Now, an inspection of these fifteen equations shows * that the fifteen 
equations are the equations characteristic of the complete system of con- 
comitants, under linear point-transformations, of the two quantics 
aX,2 + 6X32 + cXg 2 -1- dx ;- 4- 2/X2X3 4- 277X3X1 4 - 2/7X1X2 4 - 2 /XiX^ 4- 2777X3X4 -h 271X3X4 
(which is a point-quantic), and 
DPi^ 4- 2MP4P2 4- 2LP4P3 4- 2TP4P4 4- 2SP1P5 + 2 NPiPs 
4 - EP32 4 - 2KP2P3 4- 2UP3P4 4- 2OP3P5 4- 2PP2P6 
4 - FP32 4 - 2VP3P4 4 - 2RP3P3 4 - 2QP3P, 
4-CP/4-2GP4P54-2HP4P3 
4-BP32 + 2JP3P3 
4- AP32 
* See a memoir of my own, “ Systems of Quaternariaiits that are algebraically complete,” 
Gamh. Phil. Trans., vol. xiv (1889), pp. 409-466, specially pp. 430 et seq., 460 et seq. The 
subsequent analysis is modified from the analysis in that memoir. 
