348] 
ON THE THEORY OF INVOLUTION. 
297 
which is an equation of the order 2 (n — 1) as regards k, and of the same order as 
regards the coefficients of U and V conjointly. And to each of the 2 (n — 1) values 
of k there corresponds a point (x, y) satisfying the required conditions; that is, a 
point which is a twofold point of the range U + k V = 0. The points in question may 
be termed the ‘ critic centres ’ of the involution. 
3. The elimination of k from the before-mentioned two equations gives 
B X U, SyU = 0, 
B x v, ByV 
where the determinant, which for shortness I call J, is the Jacobian of the two 
functions U, V. The equation J= 0 gives a range of 2 (n — 1) points which are in 
fact the critic centres; and for each of these points we have 
B X U : B X V —B y TJ : S y V=-k : 1, 
which gives the value of k corresponding to the point in question. 
4. The condition in order that the equation in k may have a twofold root is 
Disc*. Disc 1 . (U+kV) = 0, 
or say 
□ =0, 
where Disc 1 . Disc 1 . (U + kV), = □, is a function of the degree 2 (2n — 2) (2n — 3) in 
regard to the coefficients of U, V conjointly; but it is separately homogeneous, and 
therefore of the degree (2n — 2) (2n — 3) in regard to each of the two sets of coefficients. 
o. To each point of the range J— 0, there corresponds a value of k ; hence if 
the range J have a twofold point, then the equation in k will have a twofold root. 
Now first if the ranges U = 0, V — 0 have a common point, then this is a twofold 
point of the range V= 0. But secondly, without a common point in the ranges 
U = 0, V — 0, the range J = 0 may have a twofold point ; and in this case also we 
have a twofold root of the equation in k. And thirdly, without a twofold point in 
the range V = 0, there may be in this range two onefold points giving each of them 
the same value of k, and so giving a twofold root of the equation in k. And the 
three suppositions correspond respectively to the cases of there being a range U+kV = 0 
having a twofold point at a common point of the ranges U = 0, V— 0, having a 
threefold point, and having a pair of twofold points. 
38