230 
ME. A. CAYLEY ON THE POEISM OF 
But by Abel’s theorem, this transcendental equation is equivalent to an algebraical 
one. 
In fact, calling the radical s/ Dcr, then if <px, xx are rational and integral functions 
of X with arbitrary coefficients, and if 
<p^x — x^^ nx=A{x—ki){x—k2)-..{x—k^), 
(this implies that is of a degree not exceeding n and x^^ of a degree not exceeding 
n — 3 ; that is, for 7i even the degrees of ipx, x^ are \[n—A) ; and for n odd they are 
^(n — 1), ^(n — 3)), then the algebraical equation is that obtained by the elimination of 
the arbitrary coefficients from the system of equations 
d” ^ ^2 0 » 
or, what is the same thing, for n odd =2^—1, it is 
and for n even, =2j?, it is 
{1, ^, ... x/"^, 6^/WO, ..^^-"x/d^}=0. 
where the expressions in { } denote respectively the determinants, of 2j? — 1 lines, and 
lines, formed by substituting for 6 the values ... Jc„ respectively. Thus for w = 3, 
the equation is 
1, ^1, \/ 
1 , n k^, 
and for w=4, it is 
and so on. 
Suppose 
1, k\, \/ Uk^ 
1 5 ^ 2 ? ^25 ^2 
1, ^3, ^3, X/^ d ^3 
1, k^, k\, x/d^4 
x/n|=A+BH-Cr+Df+Er+...; 
then substituting the corresponding expressions for \/ Hk^, \/ Hk^, &c., the determinant 
will divide by {1, 6, and it may be seen without difficulty that the resulting 
equation on putting therein ki=k^... =k^=0, will, according as n—S, 4, 5, 6, &c., be 
C=0, D=0, 
C, D 
= 0, 
D, E 
D, E 
E, F 
C, D, E 
D, E, F 
E, F, G 
=0, &c., 
which is the theorem above referred to. 
