496 
ON POLYZOMAL CURVES. 
[414 
55. In the case where the given trizomal is 
VJ (0 + L<£>) + Vra (® + M<$) + V?i(® + iV4>) = 0, 
s = r — 1, that is, where the zomals 0 + L<$> = 0, 0 + M<P = 0, 0 + N<S> = 0 are each of 
them curves of the order r, passing through the r intersections of the line d> = 0 
with the curve 0 = 0, then, taking this line = 0 for the fixed line 12 = 0, we have 
J(V, W, n) = /(© + №, 0+I^<I), <!>) = <& [M, N], 
if, for shortness, [M, N)=J(M—N, ®, d>) + <I> J (M, N, d>), 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 
l t m | n A 
[M, N\ + \nTl) + [L7M] = °’ 
where observe that each of the curves [M, IV} = 0, {N, L}= 0, {L, M} = 0 is a curve 
of the order 2r — 3; the order of the locus is thus = 4r — 6, and (as before) this 
locus passes through the 3(r—l) 2 points which are the (r — l) 2 poles of the line <I> = 0 
in regard to the curves ® + X4>=0, ® + Jfd> = 0, ®+IV’<I> = 0 respectively. 
56. In the case r = 2, the trizomal is 
Vi (0 + L<$>) + Vm (0 + M<i>) + Vn (0 + N<t>) = 0, 
where the zomals are the conics 0 + = 0, 0 + M<S> = 0, 0 + = 0, each passing- 
through the same two points 0 = 0, = 0; the locus of the pole of the line <I> = 0, 
in regard to the variable zomal, is the conic 
l m i\ - 
{mTn} + \n7l\ + {L7r\ = 0 - 
viz., {M, N] = 0, {iV, L) = 0, [L, M) = 0, are here the lines passing through the poles 
of the line 4? = 0 in regard to the second and third, the third and first, and the first 
and second of the given conics respectively: treating l, m, n as arbitrary, the locus is 
clearly any conic through the poles of the line $ = 0 in regard to the three conics 
respectively. The Jacobian of the three given conics is a conic related in a special 
manner to the three given conics, and which might be called the Jacobian conic 
thereof, and it would be easy to give a complete enunciation of the theorem for the 
case in hand. (See as to this, Annex No. I, above referred to.) But if, in accordance 
with the plan adopted in the remainder of the memoir, we at once assume that the 
points 0 = 0, = 0 are the circular points at infinity, then the theorem can be 
enunciated under a more simple form—viz., if 21° = 0, 33° = 0, (£° = 0 are the equations 
of any three circles, then in the trizomal 
V/r + Vmf + V?JT = 0, 
the variable zomal is any circle whatever of the series of circles cutting at right 
angles the orthotomic circle of the three given circles, and having its centre on a 
certain conic which passes through the centres of the given circles. Moreover, if the