PROFESSOR CAYLEY ON POLYZOMAL CURVES. 99 



viz., the points A, B,C,B lie here in a circle, and we have ^ + Y + c + d = ^ " 



Taking (A^ D x ) the anti-points of (A, D) ; (B v C x ) the anti-points of (B, C) ; 

 then A 1 D x = AD, B x C x = BC (No. 65) and referring to the formulae, ante, 

 Nos. 100 et seq., it appears that we can find l v m lt n lt p x such that identically 



— ZA + mB + nC — pD = - l^ + m 1 B 1 + n 1 C 1 — p 1 D 1 , 



and moreover that Ip = l 1 p 1 , mn = 7n 1 ?i 1 . 

 The equation of the curve gives 



-lA + mB+nO-pD -2 JjfiKD + 2 J^BC = , 



which may consequently be written 



- l^ + m l B 1 + ^Cj - ^Dj - 2 Jl^A^ + 2 Jv^a^C, = ; 



viz., this is 



that is, the two trizomals expressed by the original tetrazomal equation involving 

 the set of coney clic foci (A, B, C, D) are thus expressed by a new tetrazomal 

 equation involving the different set of concyclic foci (A V B V C^DJ; and we 

 might of course in like manner express the equation in terms of the other two 

 sets of concyclic foci (A 2 , B 2 , C 2 , D. 2 ) and (A 3 , B 3 , C 3 , B 3 ) respectively. It might 

 have been anticipated that such a transformation existed, for we could as regards 

 each of the component trizomals separately pass from the original set to a 

 different set of concyclic foci, and the two trizomal equations thus obtained would, 

 it might be presumed, be capable of composition into a single tetrazomal equation ; 

 but the direct transformation of the tetrazomal equation is not on this account 

 less interesting. 



Annex I. — On the Theory of the Jacobian. 



Consider any three curves U = 0, V = 0, W = 0, of the same order r, then 

 writing 



d x U, d x V, d x W 



d(x,y,z) 



d v U, d y V, d y W 

 d z U, d z V, d z W 



we have the Jacobian curve J( U, V, W) = , of the order 3r — 3 . 



A fundamental property is that if the curves U = 0, V = 0, W— have any 

 common point, this is a point on the Jacobian, and not only so, but it is a node, 

 or double point, that is, for the point in question we have J — , and also 

 d x J= 0, d y J = 0, d z j= 0. 



It follows that for the three curves lQ + Z<£ = 0, mQ + Mq> = 0, nQ + iV$ = 

 (q = of the order r—sf, $ — of the order r—s. £ = 0, m = 0, n = Q each of 



