570 
MR. R, LACHLAN ON SYSTEMS OE CIRCLES AND SPHERES. 
183. From the form of the equations (192), it follows that all curves given by 
ar y* 
I r\ o 
W~ 
a 3 + K (3 2 + tc 7 2 + k b~ + k 
must be confocal with the curve 
OOO 0 
^qX + ~'' + -= 0 
o 1 n q I q I io 
ctr p " 7 “ o~ 
l-rr—= 0 , . . 
(193) 
Subtracting these equations, we have 
x 2 y 2 z 2 w- _ 
« 2 (« 2 + /c) yS 3 (/S 3 + «) 7 2 (7 2 + «) t 2 (t 2 + /c) 
Hence the curves cut orthogonally at their common points. 
And since equation (193) may be regarded as a quadratic in k, we infer that 
through any point on a sphere two spheri-quadrics can be drawn confocal with a 
given spheri-quadric, and these two cut orthogonally. 
184. We may prove exactly, as in § 123, that the coordinates of the osculating 
circle at any point ( x'y'zw ') on the curve 
ax 2 + hi/ + cz~-\- dur = 0 , 
will be proportional to 
(a — h)(a — c)(a—d)x' 3 , (b — a)(b—c)(b — d)y ' 3 , (c — a)(c — b)(c — d)z ' 3 , 
(d—a)(d—b)(d—c)w' 3 ; .(194) 
and if It be the radius of curvature at the point we shall have by § 154, 
(cdx~ + by 2 +C- V 2 + dV )\cot R 
= (a — b)(a.—c)(a—d) cot i\x' iJ r{b — a)(b — c){b — d) cot r„y' 3 
-f ~(c — a)(c — b)(c — d) cot 7' s z s -\-(d — a)(d—b)(d — c) cot r 3 w' 3 . . (195) 
1 84'". If one of the principal circles is a great circle, the corresponding foci may 
coincide with the centres of one of the other principal circles; in this case the curve 
has been called by Casey a “sphero-Cartesian.” Thus suppose ?q=—, and let one of 
the foci on w= 0 coincide with the centre of the circle whose radius is r l5 the coordi¬ 
nates of this point are — cosec r x , cot r 2 , cot r 3 ; and the necessary condition that the 
curve 
ax 2 + by 3 + cz~ + di v : = 0 
