PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



73 



the orthotomic circle by means of the areal co-ordinates (w, v, w). Writing for 

 shortness a 2 + a 2 — a" 2 = a, &c, and therefore 



A° = x 2 + y 2 — 2axz — 2a'yz — a'z 2 , &c, 



then if as before 



u : v : to = 



x,y, z 





x,y, z 





x, y, z 



b, V, 1 





c, c, 1 





a, a, 1 



c, c, 1 





a, a', 1 





b, V, 1 



and therefore 



x :y :z = cm + bv + cw : a'u + b'v + c'w : it + v + w, 



the equation of the orthotomic circle is 



x — az, y — a'z, ax + a'y — cCz 

 x — bz, y — b'z, bx + b'y — b'z 

 x — cz, x — c'z, ex + c'y — c'z 



viz., throwing out the factor z, this is 



u(ax + a'y — a'z) + v(bx + b'y — b'z) + vj(cx + c'y — c'z) = , 



or what is the same thing, it is 



(au + bv + civ) x + (a'u + b'v + c'w) y — (a'u + b'v + c'w)z = 0, 

 viz., it is 



(au + bv + cv)) 2 + (au + b'v + c'w) 2 — (a'u + b'v + civ) (u + v + w) = 0, 



that is, substituting for a\ b\ c' their values, it is 



a" 2 ic 2 + b" 2 o 2 + c"ho 2 

 + (h" 2 + c" 2 - (b - c) 2 - (V - c'f) vw 

 + (c" 2 + a" 2 - (c - a) 2 - (c' - a') 2 ) wu 

 + (a" 2 + b" 2 -(a- b) 2 - (a - b') 2 ) wv = 0, 



and it may be observed that using for a moment a, /3, y to denote the angles at 

 which the three circles taken in pairs respectively intersect, then we have 

 2 b" c" cos a — b" 2 + c" 2 — (b — cf — (b' — c') 2 , &c, and the equation of the ortho- 

 tomic circle thus is 



(1, 1, 1, cos a, cos j8, cos y) (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, C, 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 bi- circular quartic, such that its nodofoci 

 are the foci of the conic. 



VOL. XXV. PART I. T 



