1921-22.] Concomitants of Quadratic Differential Forms. 
173 
(x) From the coefficients of , 6^^ , 
dS j 8(1) dd) nj86 
- + (*1 - 27,,) II + (c'l - 2g,) 
S/3 
8 y 
8(j) defy 
('^^s + 92 /i) ~ 
dej) 
1"./ ^ 1" 
OTH^ CTi^ vd^ 
+ («2- + 7-2II + (C2- 2/3) II 
Ba "S/3 ' •= ■'^'Sy 
8J> 
'^K 
8(f) 8(f) 
8(f> 8(f> , o 
^sTj "*■ ® ^ "'“Srf, 
+ (% - 2yi)|| + (7,3 - 2/,) - <^3 
- (!7o +Ji- ■ ‘'1 sf ^ 
,5^ 8(f) 8(f) 8(f) ^ j8(f) 
or + o + + 2a^ 
81 
8m ^ 8n^ ' 
8d, 
+ (a^- 2/d^^ + (^4- 27?io)|^ +C4- 2/?3)^^ 
8(f) 
(I, + - //d^ - ih + ^1 -9,)-^- -fd^ - ^• 
8(f> 
’8^ 
'd$ 
20. This set of equations, forty in number, satisfies all the conditions 
for a complete linear Jacobian system of the first order. 
The number of independent quantities that occur in the system are : — 
20, being the magnitudes a, /5, . • . from the previous sub-group ; 
4-40, . . . first derivatives of a, . . .,n; 
4- 10, . . . quantities a, . . . , n‘, 
that is, 70 in all. 
Hence under the Jacobian theory the system possesses thirty (70 — 40) 
independent integrals. As derivatives with regard to a, . . ., 71 do not 
occur, it is clear that ten of these integrals are provided by 
a, b, c, d, f, g, h, Z, m, n. 
We therefore require twenty other integrals, independent of these and 
independent of one another. 
