MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
563 
(ii.) If the system (1, 2, 3, 4) he a semi-orthogonal system, we may take cot r 1 , 
jL= cot r 2 , h 3 =h 4 = Gosec e, where r l5 r 2 are the radii of the circles, 2e their common 
chord. We shall have by § 149, 
xp(ccyzw )= x z + y" — ziv , 
and also 
K= — 2, 
■4 / '(a6cc?) = rt 3 + 6 2 —; 
cot 2 r x + cot 2 r 2 — cot 2 e= 0. 
164. By § 128, if two circles S, S' he inverted with respect to any point (O) on the 
sphere, then the expression 
_^S,_S'_ 
V 7 7Tq, S-TTo, S' 
is unaltered. Hence, if ( xyzw ) be the coordinates of any point referred to a system 
(1, 2, 3, 4), and (XYZW) be the coordinates of the inverse point, with respect to any 
point (0), referred to the inverse system with respect to the same point, we must have 
cc=aX, y=/3Y, z=yZ, w=yW; 
and if £ y, £, co be the coordinates of any circle, the coordinates of the corresponding 
circle referred to the new system will be aif, /3y, Soo : a/3yS being some constants. 
Chapter V.—General Equation of the Second Degree in Power- 
Coordinates. 
Nature of the Curve. —§ 165. 
165. The most general form of the equation of the second degree may be written 
y, z, w) = ax 2 -f by" -f cz 2 -f- dw 3 -f-2 fyz + 2 gzx +2 hxy + 2 lxw-\- 2myw-\- 2 nzw= 0, (182) 
(xyzw) being the coordinates of a point, and therefore satisfying the equation of the 
absolute i {/, which is also of the second degree ; it follows, then, that the general 
equation of the second degree contains only eight arbitrary constants. 
Let P be any point on the curve, and let the Cartesian coordinates of P referred to 
rectangular axes through the centre of the sphere be X, Y, Z ; and let B be the radius 
ot the sphere, then by § 125 we see that we may put 
x=N+Y*+Z*-a l X-b 1 Y-c 1 Z, 
v=X 3 +Y 3 +Z 3 —n 2 X—6 3 Y—c 2 Z, 
z=X 2 +Y 2 +Z 3 — a 3 X —6 3 Y—c 3 Z, 
w =X 2 -f Y 2 +Z 2 —«,X - bfi-cfZ. 
4 C 2 
