28 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



a curve of the order 2i — 3 ; the order of the locus is thus = 4r — 6, and (as 

 before) this locus passes through the 3 (r — I) 2 points which are the (r — l) 2 poles 

 of the line $ = in regard to the curves + L<]> = 0, 6 + Mq> — 0, 6 + iV$ = 

 respectively. 



56. In the case r = 2, the trizomal is 



Jl(® + LQ) + «/m(® + Jf*) + >/n(@ + N*) = , 



where the zomals are the conies + L<& — 0, + Mfy — 0, + N§> — 0, each 

 passing through the same two points = 0, <£ = ; the locus of the pole of the 

 line <£ = 0, in regard to the variable zomal, is the conic 



I m n 



+ Txr-r, + n—n-i = , 



viz., {.¥, N\ = 0, {N,L} = 0, \L,M\ = i), are here the lines passing through 

 the poles of the line <J> = in regard to the second and third, the third and first, 

 and the first and second of the given conies respectively: treating /, m, n as arbi- 

 trary, the locus is clearly any conic through the poles of the line <J> = in regard 

 to the three conies respectively. The Jacobian of the three given conies is a conic 

 related in a special manner to the three given conies, 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, <£ = are the circular points at infinity, 

 then the theorem can be enunciated under a more simple form — viz., if A = 0, 

 B° = 0, C° = are the equations of any three circles, then in the trizomal 



JW> + N / wB u + N //TC° = , 



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 co-efficients /, wz, n are not given in the first instance, but are regarded as 

 arbitrary, then the last-mentioned conic is any conic whatever through the three 

 centres, and there belongs to such conic and the series of zomals derived there- 

 from as above, a trizomal curve JJA + *JmB° + s/tiC = 0. This is obviously 

 the theorem, that if a variable circle has its centre on a given conic, and cuts at 

 right angles a given circle, then the envelope of the variable circle is a trizomal 

 curve JJK° + y/mB° + JvG~\ where A = 0, B° = 0, C° = are any three 

 circles, positions of the variable circle, and /, m, n are constant quantities depend- 

 ing on the selected three circles. 



