MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
51.9 
63. In the case of the orthogonal system it is most convenient to take the constants 
1c i, k. 2 , /i- 3 , Zq equal to the reciprocals of the four radii, so that the equation to the 
absolute is 
and 
y, 2 , R’) = ar + 7/ 3 +s 3 +«d=(r 
K= — 4 
4>(cq b, c, d)—cr-\-lr J r c i - s r d l 
(91) 
64. In the case of the semi-orthogonal system (see § 29), if r 1} r 3 be the radii of 
the circles, e the distance between their points of intersection, it is convenient to take 
7, _L 7. _1_ /. _7. —I 
"a— j > 'H— K ±— • 
/1 / ^ 
We shall have 
xJj(x, y, z, w)=x*+y 2 -4:zw 
K= —4 
T(cq 5, c, d) = a 3 +6 3 —ccZ 
(92) 
65. Thus the angle <£ at which the loci 
intersect, is given by 
or 
cos </> = 
COS (f 7 
ax -\-by +C 2 fi- dtv =0, 
dx -\-b'y-\-dz-\- d'w =0, 
««/ + &&' + cd + del' 
' + v* + c' 2 +Wy 
_ aa' + bb'-^cd' + r'd) 
(a 3 + lr—cd){ci'~ + Z / 2 — c'd') ’ 
according as the system of reference is the “ orthogonal ’ or the “ semi-orthogonal ” 
system. 
66. Occasionally it may not be convenient to take for system of reference either of 
the systems just considered. In some cases, however, the equations may be simplified 
by referring the coordinates of a point to one system of circles, and the coordinates of 
any line or circle to the system cutting the former system orthogonally. Thus, if the 
system of reference be (1, 2, 3, 4), and (5, 6, 7, 8) denote the system orthogonal to 
this, then taking /q, 1c 2 , /q, Zq equal to unity, the equation of the circle (or line), whose 
coordinates referred to (5, 6, 7, 8) are y, £, to, referred to the system (1, 2, 3, 4) is 
j_^L + 
7r l,5 ^2,6 lr 'Z, r l 
WOO 
=0. 
1 4,8 
The equation 
ax + by + cz-j- di v=0 
