716 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
347. If W, W' be two cyclides, S=Aa+^j3+i/y+ g<5=0 any sphere, and it be required 
to find the condition that the binodal cyclides formed by the generating spheres which 
touch W, W' along their curves of intersection with S may intersect along a third 
cy elide W", the three cyclides being given by the equations 
W ^aod -i-6/3 2 +cf +dh 2 =0, 
W' a!a? -f Z//3 2 -\-c'f+d'h 2 =0, 
W ' W V + b"\ 3 2 + c"f + d"l 2 = 0, 
then we have 
a -bedbd + edaud - \-dabv 2 -\-ahcf, A —abed, 
d == Vc’d'vd dda’tjd -f d'a'b'v 2 + d Vco\ A' ~ ^ a'b'c'd 
Hence (see art. 326) the equations of the binodals which circumscribe W and W' along 
the sphero-quartics (WS) and (W'S) are 
(7+T+7+$) w - S, = 0 ’ 
(v+v+-aS) w '- s ’= 0 - 
Hence we must have, by the given conditions, 
where k is some constant ; and equating coefficients, we get 
$ {d~d!) + X ' 2 (a - ?) =/£C "’ 
A 2 ^ v 2 f 
multiplying these equations by — „ J— „ and adding, the left side vanishes identi- 
cally. Hence the required condition is 
II ,,2,7/ ^rlll 
(230) 
a y 
ad 
. , a %" , . „ „ „ A 
jJr bU +77 + ^— 
VC 
cc 
q°~d" 
dd! 
348. The envelope of the sphere S=xa-}-|U,j3-}-i'y-f g&=0 is the cyclide 
^ W I CC> 2| dd ' 
a' ' - ■ W -f-Ji 
^ 2 T bb_ fi 2 I 2 I u ' Jj V2 A /noi \ 
n a + fji l~ J +7T 7~ J r~W ° — 
the tangential equation of its focal quadric being (230); I shall denote the cyclide (231) 
by W[. 
