PROFESSOR CAYLEY ON POLYZOMAL CURVES. 27 



showing how the single series of the variable zomal is selected out of the double 

 series of the curves aZ7+bF+cJF=0in involution with the given curves. 

 Such a geometrical interpretation of the condition may be sought for as follows, 

 but it is only in a particular case, as afterwards mentioned, that a convenient 

 geometrical interpretation is thereby obtained. 



54. Consider the fixed line Q = px + qy + rz = 0, and let it be proposed to 

 find the locus of the (r — - 1) 2 poles of the line = in regard to the series of curves 



a (j + hV + cW = 0, where — + ^ + n = 0. Take (x, y, z) as the co-ordinates 



a b c 



of any one of the poles in question, then in order that (x, y, z) may belong to one 



of the (r — l) 2 poles of the line Q = px + qy + rz = in regard to the curve 



all + bV + cW — 0, we must have 



d x (s.U + hV + cW) : d 9 (&U + hV + cW) : d,(aU + bV + cW) -p-.q-.7- ; 



or, what is the same thing — 



= d^n : dpil : d z SI 



and these equations give without difficulty 



a :b :c = J{V, W,n)-.J(W, U,n) :J(U, V, n; , 



whence, substituting in the equation — I- -r- + — = 0, we have 



a o c 



I . m n 



+ T,1T XT ~N =0 



J(V, W, a) T J(W, U, xi) T J {IT, V, a) 



as the locus of the (r— I) 2 poles in question. Each of the Jacobians is a func- 

 tion of the order lr — 2, and the order of the locus is thus = 4r — 4. As the 

 given curves £7=0, F = 0, W = belong to the single series of curves, it is clear 

 that the locus passes through the 3(r — l) 2 points which are the (r — l) 2 poles of 

 the fixed line in regard to the curves U = 0, V= 0, W= respectively. 

 55. In the case where the given trizomal is 



Jl(0 + Z*) -f >/m(0 + Mb) + Jn(® + N<P = , 



s. = r — 1, that is, Avhere the zomals e + L$ = 0, + M§> — 0, 9 + iV(j> = 

 are each of them curves of the order r, passing through the r intersections of the 

 line $ = with the curve = 0, then, taking this line <$> = for the fixed line 

 Q = 0, we have 



J{ V, W,a) = J(0+ M&, © + m, *) = *{ M, N] , 



if, for shortness, {M, N\ = J{M-N, Q, <J>) + $ J{M, N, $), and the like as to the 

 other two Jacobians, so that, attaching the analogous significations to {N, L) and 

 {L, M] , the equation of the locus is 



/ m n n 



{M t N}^ {N,L} ^ {L,M} 

 where observe that each of the curves {M, N\ = 0, {N, L\ = 0, {Z, J/} = is 



