MR. A. CAYLEY ON A NEW AUXILIARY EQUATION IN 
272 
I proceed now to form the expression for <Pi<P 4 +<p 2 <P 6 +<p 3 (P 6 - 
23. Writing for convenience x in the place of the root 6 , we have 
<p,=A--B+ {23-14+^(14-4-2-3)} 
<P4=A-B-{23-14+.2;(1 + 4-2-3)}, 
or 
<p.<p,=(A-B)^-{23-14+^(l+4-2-3)}*. 
The terms without x are, as before, (A— B)®— C^+ 4 . 1234, or — aj3+4. 1234, and we have 
4,04= —a/3+4.1234 
+2^-(l+4-2-3)(14-23) 
-a^*(l + 4-2-3)^ 
and in like manner 
<p2<P6 = -/3y+4.1234 
+241+2-3-4)(12-34) 
-^^(1+2-3- 4)^ 
and 
<p3<P6=‘-y«+4-1234 
+2^(l + 3-4-2)(13-42) 
-^^(l + 3-4-2)^ 
24. The roots 1, 2, 3, 4, contained in these expressions explicitly, and in a, (3, y, are 
the roots of the equation (a, b, c, d, e,f'Jv, 1 )®= 0 , or, what is the same thing, 
(«', V, c\ d', e’Jv, 1)^=0, 
where 
a’ a, 
h'=ax +5, 
c'=ax^-\-bx +c, 
d'=zaaf^-\-bx^-\-cx +cZ, 
e'-=ax*-{-hx^-\-cx'^-\-dx-\-e. 
Omitting, as before, a power of a, which is ultimately restored, we have 
4i 44+^245+4346= — 2a/3+12aV 
+2^2(l + 4-2-3)(14-23) 
-a:^2(l + 4-2-3)^ 
where the 2 ’s in the second and third lines denote each of them the sum of the three 
terms obtained by the cyclical permutations of 2, 3, 4. 
The first line is 
(16aV-4^'(^'+c'^)+12«V 
=2^a!d-ih’d’+lc'\ 
The second line is 2x into 21^2 — 32123, 
= {-h'd-^?>a!d')+3a!d'-. 
