PROFESSOR CAYLEY ON THE TRANSFORMATION 
400 
(In fact if 
g =a-\-cx 2 . . . 4 -qx n ~ z -\-sx n ~\ 
35 =b-{-dx 2 . . . -\-rx n ~*-\-tx n ~\ 
then 
Q* =s-\-qk 2 x 2 . . . + ck n ~ 3 x n ~ 3 -J- ak n ~ 1 x n ~\ 
35*=i t+rJfx 2 . . . + dk n ~ 3 x n ~ 3 + bk n ~ 1 x n ~\ 
and the assumed equation gives 
_ 1 , _ 
a — ,OM(n-D C — 
fc 2 
k n ~ 
■l=nW^» d ’ s= 
frn-1 
b; 
that is. 
b= 
kUn- 
n 
d= 
m g 
ytK»-D T 
C ’ 
m »- 2 
t —W^o 
and therefore 35= ^i n z T) $*.) 
a 
From these equations =Q 2 , that is = as it should be ; so that O signifying as 
above, the required condition will be satisfied if only > or substituting for 
91, 35 their values, if only 
(P 2 + 2PCb 2 + QV)*=OF”- 1) (P 2 + 2PQ + Q V), 
where each side is a function of x 2 of the order \{n— 1), or the number of terms is 
-|(w- f-1), the several coefficients being obviously homogeneous quadric functions of the 
1) coefficients of P, Q. We have thus ^(n-{- 1) equations, each of the form 
U = OV, where U, V are given quadric functions of the coefficients of P, Q, say of the 
•1(^+1) coefficients a, 0, y, S, &c., and where O is indeterminate. 
6. We may from the 1) equations eliminate the ^(n— 1) ratios a : 0 : y . . ., thus 
obtaining an equation in O (involving of course the parameter Jc) which is the Q#-mo- 
dular equation above referred to ; and then fl denoting any root of this equation, the 
■|(w + 1) equations give a single value for the set of ratios a : 0 : y : & . . so that the ratio 
of the functions P, Q is determined, and consequently the value of y as given by the 
equation 
I—? / (1 — a?)(P — Q#) 2 #(P 2 + 2PQ + Q 2 ,z 2 ) 
lT^ = (l+tf)(P + Q*) 2 ’ or y~ P 2 +2PQa? 9 +Q 8 a? 2 ’ 
The entire problem thus depends on the solution of the system of 1) equations, 
(P 2 +2PQ^ 2 + QV)*=Q^- )) (P 2 +2PQ+QV). 
The ^Ik-Modular Equations , n— 3, 5, 7, 11. — Article No. 7. 
7. For convenience of reference, and to fix the ideas, I give these results, calculated, 
as above explained, from the standard or wv-forms. 
