146] A MEMOIR ON CURVES OF THE THIRD ORDER. 407 
Now considering the equation 
F . U - II = (£k + yy + (Ç'æ + y y + Ç'z), 
in order to find f', y, £' it will be sufficient to find the coefficients of a?, y 3 , z 3 in 
the function on the left-hand side of the equation. The coefficient of x 3 in II is 
Oj 2 + 2 ly x z^) (¿r 2 2 + 2 Iz&t) (x 3 2 + 2 lx 3 y 3 ) 
—— /y> 2 /y» 2/y» 2 
- u/j t^2 ^3 
+ 2£ (xi l x?y 3 z 3 + &c.) 
+ 4£ 2 {x?y 3 z 3 y 3 z 3 + &c.) 
+ 8£ 3 2/iMsWs ; 
and it is easy to see that representing the function 
(1* 1> M$/3£-7?7, yf-a£ 
b y 
(a, b, c, f, g, h, i, j, k, l$a, /3, 7) 3 , 
the symmetrical functions can be expressed in terms of the quantities a, b, &c., and 
that the preceding value of the coefficient of oc 3 in II is 
a 2 
+ 21 (9hj — 6al) 
+ 4Z 2 (6gk - 3fj - 3hi + 31 2 ) 
+ 81 3 be ; 
and substituting for a, &c. their values, this becomes 
+ 21 {-9(Çy* + 2ly?)(?Ç+2lÇy 3 )} 
+ 4Z 2 {- 6 (№ + 2lÇrf) (p77 + 2IÇÇ 2 ) 
+ S(yp + 2l&y(& + 2lfr) 
+ 3 (%rf + 2lyÇ 2 ) (rfÇ + 2ly!?)} 
+ 81 3 (?-y 3 ) (?-?), 
and reducing, we obtain for the coefficient of x 3 in II the following expression, 
W- £ 3 ) 2 
-18J frç 2 ^ 
- 2U 2 (Ç 3 + y 3 +Ç 3 )%yÇ 
- 2U 3 {y 3 ? + + Ç 3 r) 
+ 8 P(f-if) (£»-?).