525] AN EXAMPLE OF THE HIGHER TRANSFORMATION OF A BINARY FORM, 
and the result of the elimination therefore is 
399 
a, 
46, 
6c, 
47, e 
a, 46, 
6c , 
47, 
e 
P, 
2Q, R 
P, 
2Q, 
R 
P, 
2Q, 
£ 
P, 2 Q, 
R 
viz. this determinant is the transformed quartic (cq, 6 ls c 1} d u e l )(x 1 , y^) 4 . 
The developed expression of the determinant is 
a?R 4 — 8 abQR 3 
+(; \l t) PR2 - 24 mQ,R2+ (- 4't) PQRs - 32 
+ ^- 32 bdj P’№ + (“ PQ‘R + 16 cwQ 1 
+ (- 48 rf) p W - 32 beP «* + C “ #) P2R + 24 ^ 
— 8 deP 3 Q + e 2 P 4 , 
so that writing for P, Q, R their values, we have the transformed function 
(oq, b 1} c 1} d 1} e 2 ) (x 1 , i/0 4 , the coefficients being of the forms 
0! = (a, b, c, d, ef (a, ¡3, y) 4 
&i=( „ ) 2 («, 0, 7) 3 ( a '> y) 
0i = ( » ) 2 • • • («. 7) 4 - 
Writing 7, J for the invariants of the quartic (1), and 
A = 4 (a/3' - a'/3) (#/ “ £'7) “ (7 a ' — 7'«) 2 > 
£ = (e, c, a, b, c, d) (a/3' - a'/3, 7 a ' — 7 a > — /^V) 2 » 
we have /, 7, ri, B simultaneous invariants of the forms (1) and (2). Putting more 
over V = 7 3 — 27/ 2 , and writing I 1 , J 1} V 1; for the like invariants of the form (3), I find 
4 = 4 (4 ZB 2 + 12 74P +£ 7 2 ri 2 ), 
7 X = 8 {8 7P 3 + f P-AP 2 + 2 774 2 P + (27 2 - ^7 3 ) A 3 }, 
and thence 
Vj = 256 (4B 3 — IA 2 B — JA 3 ) V.