ON POLYZOMAL CURVES. 
566 
[414 
SljDi = 31*2), SSjGq = 33S (No. 65) and referring to the formulae, ante, Nos. 100 et seq., 
it appears that we can find l x , m 1 , n x , p x such that identically 
— 1% + 3 + —p'J) = — /jSij + 771^! + n x (E i —p&i, 
and moreover that lp = liPi, mn = m 1 n 1 . 
The equation of the curve gives 
— £21 + 777.23 +?d£ — jp3) — 2 V/p s 2l2) + 2 Vmn^Qi, =0, 
which may consequently be written 
— ZjSlj + m^ + n^ - pi2)! — 2 VlipM^i + 2 V777 1 ?q23 1 C 1 = 0; 
viz., this is 
V^Sli + V7?7 1 55 1 + V77^ + Vp 1 2) I = 0 ; 
that is, the two trizomals expressed by the original tetrazomal equation involving the 
set of concyclic foci (A, B, C, D) are thus expressed by a new tetrazomal equation 
involving the different set of concyclic foci (J. 1( B l} C x , D x ); and we might of course 
in like manner express the equation in terms of the other two sets of concyclic foci 
{A^ B,, C. 2 , D. 2 ) and (d. 3 , B 3 , C 3 , D 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, M r —0, of the same order r, then writing 
J(U V if) _ d ( U ’ v > W ) _ d x U, d x V, d x W , 
d ( x >y> z ) d y U, d y V, d y W 
d z U, d z V, d z W 
we have the Jacobian curve J(U, V, IT) = 0, of the order 3?-— 3. 
A fundamental property is that if the curves U— 0, 7=0, 17 = 0 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 = 0, and we have also 
d x J = 0, dyJ = 0, d z J = 0. 
It follows that for the three curves 10 + Zd> = 0, = 0, 77® + A r <h = 0 
(0 = 0 of the order r — s', = 0 of the order r — s, 1 = 0, m = 0, n = 0 each of the 
order s', L = 0, M — 0, N = 0 each of the order s) which have in common the