PKOFESSOR CAYLEY ON POLYZOMAL CURVES. 69 



or, what is the same thing, if 



a, 



a', 



1 



i, 



V, 



1 



c, 



c, 



1 



the centres A, B, C are in a line; taking it as the axis of x, we have 

 a = a = a, j8 = j3' = b, y = y = c; and the conditions for the cusps at /, J respectively 

 reduce themselves to the single condition 



I m n n 



+ + i = : 



b — c c — a a — b 

 so that this condition being satisfied, the curve 



Jl{(x-az) 2 + y 2 -a" 2 z 2 } + Jm{{x - l%f +y 2 - 6V} + Jn(x-cz) 2 + y 2 -c" 2 z 2 } = 



is a Cartesian ; viz., given any three circles with their centres on a line, there are 

 a singly infinite series of Cartesians, each touched by the three circles respectively ; 

 the line of centres is the axis of the curve, but the centres A, B, C are not the 

 foci, except in the case a — 0, b" — 0, c" = 0, where the circles vanish. The con- 

 dition for I, m, n is satisfied \il:m\n — (b — c) 2 : (c — a) 2 :(a — b) 2 ; these values 

 writing J l : Jm'- Jn = b — c : c — a : a — b, give not only J I + Jm + Jn = 0, 

 but also a J I + bjm + c*/n = ; these are the conditions for a branch contain- 

 ing (z 2 = 0) the line infinity twice ; the equation 



(b- c) y/(x-az) 2 +y 2 -a"h 2 + (c-a) J(x-bz) 2 + y 2 -b" s z 2 + (a-b) s/(x-cz) 2 + y 2 -c" 2 z 2 = , 



is thus that of a conic, and if a" = 0, b" = 0, c" = 0, then the curve reduces itself 

 to y 2 — 0, the axis twice. 



165. If & is not = 0, then we have 



I : m : n = (/3 - 7 ) (/3- y) : ( 7 - «)(/- </) : (a - /3)(a' - 0) , 



viz., /, m, n are as the squared distances BC\ CA 2 , AB 2 , say as f 2 : g 2 : h 2 ; or 

 when the centres of the given circles A, B, Care not in a line, then/, g, h being 

 the distances BC, CA, AB of these centres from each other, we have, touching 

 each of the given circles twice, the single Cartesian 



fjK + 9s/& + h s/& = ° . 



which, in the particular case where the radii a", b", c" are each = 0, becomes 



/x/A + 9+/B + hJC = , 

 viz., this is the circle through the points A, B, C, say the circle ABC, twice. 



VOL. XXV. PART I. S 



