MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
496 
the relation 
which may be written 
77V 
iV ’ 
77 
y, 1j "l 
77V o, Oj 
7T ->\ 5> 7r - , A‘>' 77 *,7> 
0, 0 
0, 0 
= 0 
Wi.5, 0 
77. 
2, G> 
77 „ , 
y, 3> 
0 , 0 , 
77V 
77, 
y, n 
0, 0, 0, 
0 
7r 4,8 
. 7r ^',5- 7r y, 1 | 7 D,fi-' 77 V,2 I 7r -i-,T 7r ’J,3 | 7r -78-^,4 
1 1,5 
‘3,7 
'4,8 
(SO) 
whence as particular cases, we have, a; denoting any circle whose radius is r x — 
7r -i , ,a | 7r £ l fi 7r J.7 __j_ _ | 
77 "i,5 ’Lj.c vr 3,7 
TTv Z‘7Tx -\ 9r*, e .9r, f3 TT^^s^.TT, a*7Ta; 
— 2r, 3 , 
77 
1,5 
77 
2,6 
77 
3,7 
77 
■ • (31) 
• • (32) 
4,8 
If as rlenote any straight line, we have 
and 
, + LAfi + ^ + ^ = 0 , 
70,5 7r ° 
‘ 4,8 
(33) 
77.1-. 5 ,77j, i 77-j', 6-77.1', 3 | ‘77,f, 7-T7,r, 3 . 77 , w .77,. 
'1,5 
1 2 6 
77 
4,8 
1 1.1 1 
H - + + —^9,9 — 0. 
7r i, 5 77 ?,.1 "Yt.s 
(34) 
(35) 
28. This last result gives us an interesting theorem-—in the case when the given 
system (1, 2, 3, 4) consists of four points; then 7r L5 is equal to the square of the tangent 
from the point (l) to the circle passing through the points (2, 3, 4), and so on ; thus 
the sum of the reciprocals of the powers of each of four given points with respect to 
the circle passing through the remaining three is zero. One of these quantities must 
be negative, so that one of the four points must lie within the corresponding circle. 
Also by (31) the sum of the powers of these points with respect to any other circle, 
divided respectively by their powers with respect to the orthogonal system, is equal to 
unity. And by (33) the sum of the quotients of the perpendiculars from each point 
on any straight line, divided by the power of that point with respect to the circle 
which does not pass through it, is zero. 
29. There is another special system of circles which is closely allied to the orthogonal 
