DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
714 
then 
K=sum of the six terms y,ab(yf — dg) 2 , 
k --- the sum H(aV -\-a'b)('fi — v'ff. 
Therefore 
*r 2 - 4RR'^ 2 (ah' - dl)\v£ -v\y 
-j- 1(aV — ■ a!b)(ac' — ct!c)(gjf — gfiy(f — 
+ 2 2 { (ad' - a'd)(cb' - c'b) + (ad - a'c) (db' - d'b) } 
x (xrjj'—\'(jjy(v%'-~v'%y= o. 
. . ( 226 ) 
343. The surface generated by the circles which have double contact with the curve of 
intersection ofW and. AY' may be generated as the envelope of a variable sphere which 
cuts U orthogonally and whose centre moves along the cuspidal edge of the developable 
circumscribed about the focal quadrics of AY, AY' which correspond to their common sphere 
of inversion U. For let us consider any circle having double contact with the curve 
(AVAV'). Then, since a circle having double contact with AVW' is orthogonal to the 
sphere U, it is plain that a line through its centre perpendicular to its plane is a line of 
the developable circumscribed to the two focal quadrics, and therefore the sphere con- 
taining two consecutive circles will have its centre at the point of intersection of two 
consecutive lines of the developable that is on the cuspidal edge. Hence the theorem 
is proved. 
344. The curve (AY AY') is a cuspidal edge on the surface generated by the circles 
having double contact with (AY AY'). This is evident ; for any circle having double contact 
is the characteristic of the surface (see Monge, ‘Application de 1’ Analyse a la Geometrie,’ 
p. 53), and the points of intersection of each characteristic with the consecutive one 
form the cuspidal edge. Hence the proposition is proved. 
Cor. The cuspidal edge is an anallagmatic. 
345. To find the equation of the surface generated by the circles which have double 
contact with the curve (AY AY'). 
Let us consider any pair of inverse points on any circle which has double contact with 
(AY AA r '). The polar sphere of this pair, with respect either to AA r or AY', passes evidently 
through the two points of contact of the circle under consideration with the curve 
(AY AY'). The circle of intersection, therefore, of the two polar spheres intersects the 
curve (AY AY') in two points; therefore the equation of the required surface is found by 
substituting in the equation (226) for A, p, v, § 
dW dW dW dW 
da ’ d(Z ’ dy ’ t?8 
This surface is of the sixteenth degree, being of the eighth degree in a, (3, y, b; when 
we use the canonical forms «a 2 + fy3 2 -f-c y 2 4- db 2 , a'u 2 +b' (3 2 + c'y- + dV for AY, AY', the 
