52 
ME. A. CAYLEY ON THE EQUATION FOE A FUNCTION 
The equation in & is thus (^— 6)^(^+12)(^^+108)=0, or multiplying out it is 
(1, 0, 0, +432, -11664, +46656^;^, 1)^=0, 
which in fact (observing that \/ □=36) is what the preceding formula becomes for 
the equation (1, 1, 1, 1, 1, l%v, 1)®=0, 
The analogous verifications for the cubic and the quartic equations are as follows ; 
16. For the cubic, if the assumed equation is ?j^+v^+'y+l = 0, the roots whereof are 
— 1, i, — i (i^= — 1), then we should have (3, 7)=\/ — □ =4i, which will be the 
case if a, )3, y=— 1, ^, — respectively, and the roots (3 — y, y— a, a — (3 of the equation 
in 0 then are 2^, — ^+l, — 1, so that the equation in ^ is (^^+2^^— 2)(^— 2^)=0, or 
( 1 , 0 , 2 , 4 ^ 1 )^= 0 , 
which (observing that \/ □ =4^) is what the formula for the equation in ^ becomes for 
the equation (1, 1, 1, l^v, 1)'*=0. 
17. For the quartic equation, taking this to be 'y‘‘+t;^+i)^+'y+l:=0, the roots are 
( 0 ^, o;®, a*, where is an imaginary fifth root of unity (a;^+(y^+ft;®+4;+l = 0), and 
putting cc, (3, y, ^ equal to <y, a* respectively, we have 
— \/ □ =^^(a, (3, y, 
giving, as it should do, □ =125. The equation in 6 is therefore by the formula 
(1, 0, 0, -25a;+^^-«"-a;^ 125^^, 1)^ = 0. 
But the roots are 
6- ^^(/3,y,a)= 2- ^y+ co‘^-2cJ'z= 2-X, 
&.2— — ^^(y^ «)= — [co^ — — 0) — CO )= — l + 3«y+2<y^+ co^— — 1+Y, 
^ 3 = ^^(^,a,/3)= (co* — co)(co* — co^)(co — co^)= — 4 — 3co — 2<y’* — co^~ — 4 — Y, 
— ^^(a,j3,y)= — (co — co^){co — co^)(co^ — oo^)= 3+ co — co^-j-2co^= 3+X, 
if, for shortness, 
X=iy — «y^+2<y®, Y=3co-\-2co‘^-\-co^. 
The equation in d is therefore 
(^~2+X)(^+l-Y)(^+4+YX^--3-X)=:0, 
where the left-hand side is the product of the factors 
(^-2+XX^-3-X)=^^-5^+6- X-X^=^^-5^+10-5(^y + 0 ;“) 
and 
(^+l_YX^+4 + Y)=^^+5^+4-3Y-Y^=^^+5^+10-5(^y^+ft;^); 
and the equation in 6 is, therefore, as it should be, 
(1, 0, 0, -2dco+co^~co^-co\ 125p, 1X=0. 
Passing from the denumerate to the standard forms ; 
