564 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Substituting for (x, y, z , w) in equation (182), we see that the curve in question is 
the curve of intersection of the surfaces 
X 3 +Y 2 +Z 2 =R 3 , 
and 
(X 3 +Y 2 +Z 3 ) 2 +U 1 (X 3 +Y 3 +Z 3 )+U 2 =0, 
Uj and U 3 being homogeneous expressions of the first and second degree respectively 
in (X, Y, Z). 
So that the curve is the complete intersection of a sphere with a quadric surface, 
and therefore may be called a “ spheri-quadric.” These curves have been extensively 
studied. Casey calls them sphero-quartics (“ Cyclides and Sphero-quartics,” (1871). 
e Phil. Trans.,’ vol. 161). Darboux calls them spherical cyclics (‘ Sur une Classe 
remarquable de Courbes et de Surfaces Algebriques,’ 1873). Mr. H. M. Jeffery 
(‘London Math. Soc. Proc.,’ vol. 20, 1885, p. 102) has proposed to call them sphero- 
cyclides. The name spheri-quadric is due to Professor Cayley. 
Equation to Tangent at any Point .—§§ 166-171. 
166. Let be the coordinates of any circle touching the spheri-quadric 
q>(xyziv) — 0 at the point ( x'y'z'w ), then \p(xyziv)=0 being the equation of the absolute, 
we must have 
d ±M+ d Sh'+^z = 0 , 
dy’ J ^0/ 
^Sx’+^Sy'+^Sz +^>' = 0 . 
dw 
3- 
dw 
And hence we must have 
bip drjr 
dr] 0f 
0 ip 
dor 
d^r 
dg ______ } 
dy> d\fr d(f) dyjr d(f> dyjr dcf> 0ip 
5? + a? dH ,+ %’ a? +i ’a7 S/ +I W 
(183) 
where h is indeterminate. 
Hence every circle which touches the curve at the point {x’y'z'w') has its equation 
of the form 
0 0 . 0 . 0 
i-'vv+ +V ■+" V-) W'+^ • - 
(184) 
