348] 
ON THE THEORY OF INVOLUTION. 
30 7 
35. Comparing the degrees of the two sides we have 
3 (n — l) 2 (3w 2 — 6n + 2) = 3n(n — 1) 
+ 36 (n — l)(w — 2) 
+ 3 (n — 1) (w — 2) (3n 2 — 3n — 11); 
or, what is the same thing, 
(n — 1) (3w 2 — Qn + 2) = n 
+ 12 (n — 2) 
+ (n — 2) (3w 2 — 3n — 11), 
which is true, but, as just remarked, this equation itself was used to find the value 
a. = |. 3 (n — 1) {n — 2) (3n 2 — 3w — 11). 
36. Recapitulating, the equation in h will have a twofold root 
1°, if R- 0, that is, if the curves U = 0, F=0 touch each other, and in this 
case there is a twofold critic centre at the point of contact: 
2°, if Q — 0, that is, if there be a curve U + kV = 0 having a cusp, and in this 
case the cusp is a twofold critic centre: 
3°, if P — 0, that is, if there is a curve U+kV=0 having a pair of nodes. 
37. The three cases may be geometrically illustrated by supposing that the curves 
U — 0, V = 0 are in the first instance nearly, but not exactly, in the several relations 
in question. 
First, if the curves U= 0, V = 0 are about to touch each other, that is, if there 
are two points of intersection about to coincide with each other. There are here 
two critic centres in the neighbourhood of the two points of intersection, and which, 
when the two points of intersection become a point of contact, coincide each with the 
point of contact. 
39—2