414] ON POLYZOMAL CURVES. 541 
or, what is the same thing, it is 
(au + bv + cw) x + (a'u + b'v + cw) y — (a y u + b'v + dw) z =0, 
viz., it is 
(au + bv + cw) 2 + (a'u + b'v + c'w) 2 — (a''u + b y v + c'w)(u + v + w) = 0, 
that is, substituting for a', b\ c' their values, it is 
a''hi 2 + b" 2 v 2 + c"hu 2 
+ (b" 2 + c" 2 — (b — c) 2 — (b' — c') 2 ) vw 
+ (c" 2 + a” 2 — (c — a) 2 — (c' — a') 2 ) wu 
+ (a" 2 + b" 2 — (a — b) 2 — (a' — b') 2 ) uv = 0, 
and it may be observed that using for a moment a, /3, 7 to denote the angles at 
which the three circles taken in pairs respectively intersect, then we have 2b"c" cos a. 
= b" 2 + c" 2 — (b — c) 2 — (b' — c ) 2 , &c., and the equation of the orthotomic circle thus is 
(1, 1, 1, cosa, cos/3, cos7\a"u, b"v, c"w) 2 —0. 
174. We have in the foregoing enunciation of the theorem made use of the 
three given circles A, B, G, but it is clear that these are in fact any three circles 
in the series of the variable circle, and that the theorem may be otherwise stated 
thus : 
The envelope of a variable circle which has its centre in a given conic, and cuts 
at right angles a given circle, is a bicircular quartic, such that its nodo-foci are the 
foci of the conic. 
Article Nos. 175 to 177. Properties depending on the relation between the Conic and Circle. 
175. I refer to the conic of the theorem simply as the conic, and to the fixed 
•circle simply as the circle, or when any ambiguity might otherwise arise, then as the 
orthotomic circle. This being so, I consider the effect in regard to the trizomal curve, 
of the various special relations which may exist between the circle and the conic. 
If the conic touch the circle, the curve has a node at the point of contact. 
If the conic has with the circle a contact of the second order, the curve has a 
•cusp at the point of contact. 
If the centre of the circle lie on an axis of the conic, then the four intersections 
lie in pairs symmetrically in regard to this axis, or the curve has this axis as an 
axis of symmetry. 
If the conic has double contact with the circle (this implies that the centre of 
the circle is situate on an axis of the conic) the curve has a node at each of the 
points of contact, viz., it breaks up into two circles intersecting in these two points.