PROFITS,SOK A. R. FORSYTH OX THE 
been derived from tlieii' use and there would have been the disadvantao-e that 
modification of the formal expressions, while changing their original form without any 
obvious necessity or obvious benefit, i-emoves that form in which they naturally 
arise. 
‘ 25 . The third set of equations consists of nine numbers. By modifving them in 
the same way as the corresponding nine equations in § 10, we obtain similar results. 
Tims with the old notation for the operator Aj but with the extended significance due 
to the occurrence of derivatives of (f,of the third order and derivatives of a. h, c, /i r/, ]i 
of the second order, we find 
Aja" = + (,)0,, - 2 
AI If' = 2b" -h PWo — 21105 , 
A,b" = k", 
A,k" = 0, 
t> _ Of'' 
-j1 5 
// _ 
2B@,i + Q@i, 
A,r = r. 
All" = 0, 
A^c" = m" 
A|m" = P@i, 
A^n" = 0 ; 
Aj©! = 0, AfiO. = 206, Ai03 
A,P = 0, A,Q = — P, AjB 
0, Ai0, 
- @ 5 , Ai @5 0, Ai©6 = ©1; 
= 0 . 
When we insert in these equations the zero values of the quantities 0, the first ten 
of them become 
A^a' 
Alb" r 
ib" = 
= 3b", 
^ 2b", 
= k". 
A,k" = 
0 , 
Aig" = 2f", 
Aif" == 1", 
A,l" = 0, 
AiC" = m' 
A,ni" = 0, 
Aiu" = 0 ; 
and so for the equations of the other sets. 
Proceeding now exactly as in § IG, we find ultimately that the differential invariants 
involving the quantities n, 6, c, /’ (j. h and theii' derivatives, as Avell as the derivatives 
of (ji, iqj to the respective specified orders, are the invariants and contravariants of the 
simultaneous ternary forms 
