MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
575 
and the sphero-conic most be given by 
fa~ = 2/3y. 
As a particular case, we may suppose (y, z) to be the pair of great circles which can 
be drawn from the pole of (w) to have double contact with the curve, and then it 
follows that the centre of the sphero-conic coincides with the centre of the circle 
which passes through the points of contact of these double tangents : i.e., the centre 
of the sphero-conic coincides with the centre of the polar circle of the centre of w 
with respect to the splieri-quadric. 
198. Taking for circles of reference any three bitangent circles orthogonal to w, the 
equation of the curve takes the form 
a vx-\-b\/ y-\- c\/z—0, 
and the equation of the sphero-conic becomes 
ct 2 
OL 
7 2 „2 
+l+-=0. 
P 7 
Hence, if P be any point on the curve, A, B, C three foci on the same principal 
circle, then 
a. sin -gAP + 6. sin ^BP + c. sin -gCP = 0.(209) 
Or, again, if the curve is a sphero-Cartesian, so that the sphero-conic becomes a circle, 
then A, B, C being the centres of any three bitangent circles of the system, P a point 
on the curve ; 2p, 2 q, 2 r the tangents from P to these circles ; we have 
sin \a. sin \p-\- sin \b. sin \q-\- sin \e. sin iO’=0, .... (210) 
a, b, c being the sides of the triangle ABC. 
PART III.—SYSTEMS OP SPHERES. 
Chapter I.—General Systems of Spheres. 
The Power of two Spheres .—§§ 199-201. 
199. The power of two spheres is the square of the distance between their centres 
less the sum of the squares of their radii. 
Thus if any two spheres be denoted by (1, 2) we shall have 
tt 1i2 =# 1 i3 —r-L 3 —r 2 2 =2r 1 r 2 cos <u La : 
