306 
ON THE THEORY OF INVOLUTION. 
[348 
and we have 
(a + k x a', . R^X, Y, Z) = ^8^- (a + a!,. .][X, Y, Z)' 2 = (\X + /xY + vZ) 2 = X (XX + /xY + vZ) : 
and similarly the values of (h + kji,. .j£X, Y, Z) and (g + kgg,. .\X, Y, Z) are 
= /i (XX + /xY + vZ) and = v (XX + g,Y + vZ) respectively. 
Hence the equation of the tangent to the diacritic curve is 
XX + g Y + vZ = 0, 
that is, the tangent being independent of the values of (a, /3, 7) is the same for all 
the diacritic curves, and is the tangent at the cusp of the cuspidal curve U + k x V= 0. 
31. The conditions in order that the curve W= U-\-kV=0 may have a cusp are 
given by a plexus equivalent to three relations between the coefficients a + ha!,.. of 
W, and using for a moment ¡3 to denote the degree of the plexus or system, then 
eliminating k between the equations of the plexus we find between the coefficients 
a,... of U and the coefficients a', ... of V an equation Q = 0 of the degree /3 in 
regard to the two sets of coefficients respectively. Conversely, given the equation 
Q = 0, we may find the plexus between the coefficients a + ha',... of W. The value 
of /3, as will be shewn post, Annex, is 
= 12 0-1) 0-2). 
32. Thirdly, when the functions U, V are such that there exists a curve U+kV=0 
{suppose the curve U+h 1 V= 0) which has a pair of nodes; each of these nodes is a 
critic centre, and (by means of the equation —k : l — h x U : 8 X V) gives the value k x 
of k, that is, k\ is a twofold root of the equation in k. 
33. The conditions in order that the curve W= U+kV= 0 may have a pair of 
nodes are given by a plexus of the degree a; then the coefficients being a + ka!,... if we 
eliminate k between the equations of the plexus, we find between the coefficients a,... 
of U and a',... of V an equation P = 0 of the degree a. in the two sets of coefficients 
respectively. And conversely, given the equation P = 0, we may find the plexus between 
the coefficients a + ka',... of W. I have not succeeded in finding directly the value 
of a, but only derive it from the equation □ = RQ 3 P 2 , which if a had been found 
independently, would have been verified by means of such value of a; the value is 
a = %.3(n-l)(n-2)(3n*-3n-ll). 
34. The equation □ = 0 is satisfied if R = 0, or if Q — 0, or if P = 0, and it may 
be seen that it is not satisfied in any other case. Hence □ is made up of the factors 
P, Q, R, and I assume that its form is the same as in the case of a binary quantic, 
that is, that we have