PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QU ANTICS. 
521 
or, as this may be written, 
1, 2a , a 2 
1 , 0 + 7 , (3y 
1, b -J-£ , &g 
= 0 . 
Hence writing x-\-a! for x , the last-mentioned equation is the condition in order that 
the equation 
(x—a)(x—(3)(x—y)(x—l)(x—s) = 0 
may be transformable into 
x(x —fi')(x— y')(x — £')(# — g') = 0, 
where /3'+y' = 0, h' 4-s' = 0, that is, into the form x(x 2 — (3 ,2 )(x 2 —d ,2 )=0. Or replacing y , 
if we have 
(a, b, c, d, e,fjx, yy=a{x-uy)(x-^y){x-yy){x-ly){x-zy\ 
then the equation in question is, the condition in order that this may he transformable 
into the form {a!, 0, c', 0, e', Ofx, y) 5 , that is, in order that the 18-thic invariant I may 
vanish. Hence observing that there are 15 determinants of the form in question, and 
that any root, for instance a, enters as a 2 in 3 of them and in the simple power a in the 
remaining 12, we see that the product 
a ,8 n 
1, 2a , a 2 
1, /3 + y, (3y 
1, S -fig , he 
contains each root in the power 18, and is consequently a rational and integral function 
of the coefficients of the degree 18, viz. save as to a numerical factor it is equal to the 
invariant I. And considering the equation (a, . ,l[x, yf — 0 as representing a range of 
points, the signification of the equation 1 = 0 is that, the pairs (f 3 , y) and (5, g) being 
properly selected, the fifth point a is a focus or sibiconjugate point of the involution 
formed by the pairs (f 3 , y) and (S, g). 
Article Nos. 257 to 267. — Theory of the determination of the Character of an Equation ; 
Auxi’liars ; Facultative and Non-facultative space. 
257. The equation ( a , b, c . . ,fx, y) n = 0 is a real equation if the ratios a:b:c, . . of 
the coefficients are all real. In considering a given real equation, there is no loss of 
generality in considering the coefficients ( a , b, c . .) as being themselves real, or in taking 
the coefficient a to be =1 ; and it is also for the most part convenient to write y— 1, 
and thus to consider the equation under the form (1, i, c . .fx, l) n =0. It will there- 
fore (unless the contrary is expressed) be throughout assumed that the coefficients 
(including the coefficient a when it is not put =1) are all of them real ; and, in speaking 
of any functions of the coefficients, it is assumed that these are rational and integral real 
functions, and that any values attributed to these functions are also real. 
mdccclxvii. 4 B 
