56 
ME. A. CAYLEY ON THE EQUATION EOE A EUNCTION 
and therefore 
= -V =- ( 3)=^ 
+ (21) 
— 3a/3y — 3(13) 
3(2^— 3«5e+15^ 
-{■oa^d—labc 
= ^{^a^d~-iabc-\-l¥). 
23 ^ 1 ^ 2 =— ^{^a^d‘^—4iabcd-{-lJfd). 
And lastly, 
1,6, &A, or tlj^2^3=a^/3V(a— ^)(f3— y)(y— a) 
= — 05"/3y^^(a, f3, y) 
So that the equation is the one given above, No. 7. 
Annex No. 2, containing the calculation of the equation 11^(6— 6, ) = 0, where 
7, ^), ^2=— y^aC^(y, a), a, f3), and 6^= —cc^y^^(oc, (3, y) 
a, (3, 7, ^ being the roots of the quartic equation (a, b, c, d, e%v, 1)^=0. 
lJ, = ^2Ayl^^{(3, 7, ^)=— (a— /3)(a — 7)(a— ^)(/3— 7)0— ^)(7— S) 
— 7^ ^) 
= yz. 
where Z = 25Qa^e^-\- &c. is the discriminant of the quartic. 
26^1^2= 26^3^4= 26^0^^(^5 (^)x —oc(3y!^^(tx, (3, y) 
= le^cc(3(h—oi)(^—(3)(a—(3) X —al3y(tx—(3)(a~y)(j3 — y) 
= —a(3y})lea(3(ci~-(3y(ci—y)(a — 'b)((3 — y)((3 — h), 
where a(3yh= — -, and 
* The signs of ^2> ®3> are taken account of implicitly. 
