PEOFESSOE A. E. FOESYTH OX THE 
:^,02 
where 
P' = 
Q' 
Tl' = G(f>\oi) + F(/)'qio 
+ G'^VjOI 
+ ^^'ooi 
> . 
These quaiitities bear to (ji' the same relation as the quantities a, b, c, f, S’, h bear to (f). 
The differential invariants up to the order siDecified are the simultaneous invariants 
and contravariants of the simultaneous ternary quantics 
100’ ^ 010’ 4 * OOlX^’ ’ ^)’ 
{a, h, c, f, g, hJX, Y, Zf, 
(a, b, c, f, g, hXX, Y, Zf, 
(a', b', c',f, g',h'XX, Y, Zf, 
tlie variables of tlie contravariants being The total number of 
algebraically indejiendent relative differential invariants is IG, being the number of 
independent solutions of five ecjuations (as in § 18 ) which are a complete Jacobian set 
and involve 21 variables. Hence the total number of algebraically indejjendent 
alisolute dltferential invariants of the specified order- involving derivatives of two 
functions (f) and cfy' is 15 . 
As regards the indices of the respective relative invariants, they are given as 
before. Let the invariant be homogeneous in (f)^QQ, of degree n ; in (j)\oo> 
(p'ovh 4 ^'m degree rd ; in a, h, c, /, g, h of degree m,; in a, b, c, f, g, h of degree /; 
and in a', b', c', L, g', h' of degree A Then its index g is given liy 
8 (/ + /') + 2 m fi- n + n = 3 /x. 
21. Tlie algebraically conq:)lete aggregate of sixteen i-elative invai-iants can be 
expressed in various forms that are equivalent to one another. One such aggregate 
of invariants of the second order can be obtained as follows. AYe write 
2l) — (f) ^ 
rffl 
+ 4 *'m) ^ 
100 
4*0 
+ 4 *\ 
001 
CIO 
? 4 ), 
001 
and we use the results of § 16 . 
If there wei-e only one surface (^, the aggregate of invariants of the second order 
would be 
-^1’ ^1:1’ ^21’ ^0’ ^b’ 
As tliere is a second surface 4 >', introducing a third ternary quadratic, the aggregate 
of invariants of the second order must include concomitants that may be denoted by 
“^IS’ ^3> ®13’ ^3 
(or their ecpiivalent), these concomitants having the same relation to the third 
