414] 
ON POLYZOMAL CURVES. 
543 
Article No. 178. Case of Double Contact, Casey s Equation in the Problem of Tactions. 
178. In the case where the conic has double contact with the orthotomic circle, 
then (as we have seen) the envelope of the variable circle is a pair of circles, each 
touching the variable circle; or, if we start with three given circles and a conic 
through their centres, then the envelope is a pair of circles, each of them touching 
each of the three given circles; that is, we have a solution of the problem of 
tactions. Multiplying by 2, the equation found ante, No. 173, for the variable circle, 
and then for the moment representing it by (a, b, c, f, g, h]£w, v, wf = 0; then 
attributing any signs at pleasure to the radicals Va, Vb, Vc, the equation of a conic 
through the centres of the given circles, and having double contact with the ortho 
tomic circle, will be 
(a, b, c, f, g, h][w, v, w) 2 — (u Va + v Vb + w Vc) 2 = 0, 
viz., representing this equation as before by 
we have 
Ivw + mwu + nuv = 0, 
l : m : n = f— Vbc : g — Vca : h —Vab, 
that is, substituting for a, b, c, f, g, h their values, and taking, for instance, a, b, c 
= a"V 2, b"f 2, c"V2, we find 
l : m : n= (b" — c"f — (b — cf — (b r — c'f 
: (c" — a") 2 — (c — a) 2 — (c' — a) 2 
: {a" - b"f -(a-b) 2 -(a'~ b') 2 , 
that is, l, m, n are as the squares of the tangential distances (direct) of the three 
circles taken in pairs, and this being so, the equation of a pair of circles touching 
each of the three given circles is flW + + V?i(£° = 0. It is clear that, instead 
of taking the three direct tangential distances, we may take one direct tangential 
distance and two inverse tangential distances, viz., the tangential distances corresponding 
to any three centres of similitude which lie in a line; we have thus in all the 
equations of four pairs of circles, viz., of the eight circles which touch the three 
given circles. This is Casey’s theorem in the problem of tactions. 
Article No. 179. The Intersections of the Conic and Orthotomic Circle are a set of four 
Concyclic Foci. 
179. The conic of centres intersects the orthotomic circle in four points, and for 
each of these the radius of the variable circle is = 0, that is, the points in question 
are a set of four concyclic foci (A, B, C. D) of the curve. Regarding the foci as given, 
the circle which contains them is of course the orthotomic circle; and there are a 
singly infinite series of curves, viz., these correspond to the singly infinite series of 
conics which can be drawn through the given foci. As for a given curve there are