264 ]VIE. A. CAYLEY ON A NEW AUXILIAEY EQUATION IN 
(where on the right-hand side 12, 23, &c. stand for x^x^^ &c.), then the auxiliary 
equation, say 
1)«=0, 
has for its roots 
^5, =12345-24135, <p,=21435-13245, 
9,=13425-32145, <^5= 31245-14325, 
^ 53 = 14235 - 43125 , ?) 6 = 41325 - 12435 , 
and, it follows therefrom, is of the form 
(1, 0, C, 0, E, F, GI®, 1)^=0, 
where C, E, G are rational and integral functions of the coefficients of the given equa- 
tion, being in fact seminvariants, and F is a mere numerical multiple of the square root 
of the discriminant. 
The roots of the given quintic equation are each of them rational functions of the roots 
of the auxiliary equation, so that the theory of the solution of an equation of the fifth 
order appears to be now carried to its extreme limit. We have in fact 
?’1<?3 + <P2<P6+?>3‘P5=(*I^4, 1)', 
^l<p4 + ?>2<p5 + ^3'p6 = (*X-3?5, 1)\ 
where (*y^Xi, 1)\ &c. are the values, corresponding to the roots Xi, See. of the given 
equation, of a given quartic function. And combining these equations respectively with 
the quintic equations satisfied by the roots Xi, Sec. respectively, it follows that, con- 
versely, the roots Xi, x^, See. are rational functions of the combinations ?>iip6+?’2'p4+?’395i 
‘Pi?’2+?’3?’4+?’5‘p65 respectively, of the roots of the auxiliary equation. 
It is proper to notice that, combining together in every possible manner the six roots of 
the auxiliary equation, there are in all fifteen combinations of the form ?5i'p2-l-?>3?>4+?>5®6- 
But the combinations occurring in the above-mentioned equations are a completely 
determinate set of five combinations : the equation of the order 15, whereon depend 
the combinations is not rationally decomposable into three quintic 
equations, but only into a quintic equation having for its roots the above-mentioned five 
combinations, and into an equation of the tenth order, having for its roots the other 
ten combinations, and being an irreducible equation. Suppose that the auxiliary equa- 
tion and its roots are known ; the method of ascertaining what combinations of roots 
correspond to the roots of the quintic equation would be to find the rational quintic 
factor of the equation of the fifth order, and observe what combinations of the roots of 
the auxiliary equation are also roots of this quintic factor. The direct calculation of 
the auxiliary equation by the method of symmetric functions would, I imagine, be very 
laborious. But the coefficients are seminvariants, and the process explained in my 
