302 
ON THE THEORY OF INVOLUTION. 
[348 
a plexus of three curves, each of them of the order 2 (n — 1); any two of the three 
curves intersect in 4(n — l)' 2 points; but (n — 1)- of these do not lie on the third 
curve; the remaining 3 (n — l) 2 of them lie on all three of the curves, and they are 
the critic centres of the involution. 
21. More generally the critic centres lie on any curve whatever of the form 
№ 
s x v, 
/3, 
7 
= 0, 
8 y U, 8 Z U 
ByV, 8 Z V 
and any such curve, viz. any curve of the order 2(?i — 1) passing through the 3 (n — 1 ) 2 
critic centres, may be termed a diacritic curve. 
22. For any one of the critic centres we have 
8 X U : SyU : 8 Z U=8 X V : 8 y V : 8 z V=k : -1, 
Avhich gives the value of k corresponding to the point in question. 
23. The condition in order that the equation in k may have a twofold root is 
Disc 1 . Disc 1 . (U+ k V) = 0, 
or say 
□ = 0, 
where Disc 1 . Disc 1 . (U + kV), =□, is a function of the degree 2.3 (n — l) 2 {3 (n — l) 2 — 1} 
in regard to the coefficients of TJ, V conjointly; but it is separately homogeneous, and 
therefore of the degree 3 (n — l) 2 {3 (n — l) 2 — 1} in regard to each set of coefficients. 
24. To each of the critic centres there corresponds a value of k. Hence if two 
of the critic centres coincide, or say if there is a twofold critic centre, the equation 
in k will have a twofold root. Now first if the curves U= 0, F=0 touch each 
other (have a point of contact or twofold intersection) then the diacritic curves will 
all touch (have a point of twofold intersection) at the point in question, which is 
therefore a twofold critic centre. It may be remarked in passing that the diacritic 
curves do not at the twofold critic centre touch the curves U— 0, F= 0. But 
secondly the diacritic curves may touch at a point which is not a point of contact 
of the curves TJ = 0, F= 0. Such a point is a twofold critic centre. In each of 
these two cases the equation in k has a twofold root. Moreover, in the first case 
the curve U+kV—0 corresponding to the twofold root has a node at the point of 
contact of the two curves U— 0, F=0; in the second case the curve TJ + kV=tt 
corresponding to the twofold root has the twofold centre (not a mere node but) a 
cusp. And thirdly, without any twofold critic centre, two distinct critic centres may 
give by the equations 
8 X U : S y U : 8 Z U=8 X V : 8 y V : 8 z V=k : -1 
the same value of k, and then the curve U+kV—0 corresponding to such value of 
k is a curve having a node at each of the critic centres in question, that is, it has 
two nodes.