“274 ME. A. CAYLEY ON A NEW AUXILIAEY EQUATION IN 
26. Passing now to the standard form (a, 5, c, d, 1)®=0, the equation in (p is 
where □, = &c., denotes the Discriminant for the Standard form. 
27. And if we put 
(*Xa-, l)^=20(2ah Sab, 22ac-l0b\ 18ad~10bc, 7ae~10bd+bc^%a^, ly, 
tlien we have 
®^(^l'?'6+?2?’4 + 'p3'p5) = (*X''^’l5 l)h 
+ = l)h 
«'(®l<p5 + <P.<?>3 + <p4<p6)=(*fe, l)h 
«'('?1<P3+ ^2‘P6+ ^3<P5) = (*X'*^4, l)h 
l)h 
which lead to rational expressions for the roots x^, aq, .^’ 3 , x^, x^ in terms of the combina- 
tions <P,<p 6 +<P' 2^4 + ^^ 3 <p 65 &c- respectively. 
28. Consider now the quintic function 
\J = (a, b, c, d, e,fjx, yy=a{x—ay){x—(iy){x~yy){x~ly){x—zii); 
and treating the numbers 1, 2, 3, 4, 5 as corresponding to a, /3, 7 , s respectively, write 
0 = 12345 
where . ^ ^ ^ ^ ^ 
12345 = 12 + 23-1-34+45+51, &c., 
in which 12 , &c. denote respectively 
1 
X — uy x — ^y 
, (&C. 
