PROFESSOR CAYLEY ON POLYZOMAL CURVES. 101 



but from the equations of the two conies multiplying by \H, \h and adding, we 

 have 



\{cE + hC)z 2 + (JiF + fH)yz + (gH + Aff)m + 2hHxy = ; 



viz., the condition is thus reduced to 



cH + hC-2(/G + gF) = 0, 



so that being satisfied for one of the points in question, it will be satisfied for the 

 other of them. Now for the three conies 



cz 2 + 2fyz + 2gzx + 2hxy = , 

 cz 2 + 2fyz + 2g'zx + 2h'xy = , 

 c"z 2 + 2f"yz + 2g"zx + 2h"xy = , 



forming the Jacobian, and throwing out the factor z, we may write the equation 



in the form 



Cz 2 + 2Fyz + 2Gzx + 2Hxy = , 



where the values are 



= gift - f'd) + ft (ft -fc") + g"(fc' - f'c) , 



H- g(h'f" - h"f) + g (h"f - hf") + g"(hf - h'f) , 



2F = h(fc" -f'c') + h\f"c-fc") + h"(JV -fc) , 



2G = h{c'g" - c"g) + h' (c"g - eg") + h"(cg' - eg) ; 



and we thence obtain 



cH +hC = - (fg' - fg) (c"h - oh") + (f'g - fg") (ch' - c'h) 

 = 2(JQ + ff F), 



viz., the condition is satisfied in regard to the Jacobian and the first of the three 

 conies ; and it is therefore also satisfied in regard to the Jacobian and the other 

 two conies respectively. 



I do not know any general theorem in regard to the Jacobian which gives the 

 foregoing theorem of the orthotomic circle. It may be remarked that the use in 

 the Memoir of the theorem of the orthotomic circle is not so great as would at 

 first sight appear : it fixes the ideas to speak of the orthotomic circle of three 

 given circles rather than of their Jacobian, but we are concerned with the ortho- 

 tomic circle less as the circle which cuts at right angles the given circles than as 

 a circle standing in a known relation to the given circles. 



Annex II. — On Casey's Theorem for the Circle which touches three given Circles. 



The following two problems are identical : — 



1. To find a circle touching three given circles. 



2. To find a cone-sphere (sphere the radius of which is = 0) passing through 

 three given points in space. 



VOL. XXV. PART I. 2 C 



