MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
551 
Let (1, 2, 3, 4) be such a system, then if ( x , y) be any other circles we must have 
from the equation 
(x, 1, 2, 3, 4\ 
U L2,3, U ’ 
_ 7T Al' 7r y,l 1 7r J-'.2' 7r y,2 1 7r ^,3- 7r '/,3 1 7r -t',4- 7r '/.i 
«*.»=-+--h-- 
7T 
1,1 
7T, 
2,2 
7T. 
. . (148) 
3,3 
4,4 
whence we have as particular cases 
(i.) TTx t 1 cot 3 1 \+ 7T.r t a COt 3 7^+ 77^,3 COt 3 7^ + 77 *, 4 COt 3 += — 1, 
(ii. ) n Xt £ cot 3 + + Tr.f t £ cot 3 r 2 + n Xt 3 3 cot 3 r 3 + 77*, 4 3 cot 3 r 4 =tan 3 r 0 
where a: denotes any circle, radius r z ; 
(iii.) 7 r, a cot 3 71 + 77-3 cot 3 73 + 77-3 cot 3 r 3 + 77^ 4 cot 3 r 4 = 0 , 
(iv.) 77 - ! 3 cot 3 ++77- 2 2 cot 3 7’ 3 +7r_- ]3 3 cot 3 r 3 +77« )4 a cot 3 r 4 = 1, 
where 2 denotes any great circle ; 
(v.) cot 3 7*!+ cot 3 r 2 + cot 3 r 3 + cot 3 ?q= --1 ;.(119) 
so that one of the circles must be imaginary. 
138. If the circles (1, 2, 3, 4) form a system not having a common orthogonal circle, 
we may find four other circles, (5, 6, 7, 8) say, such that each of the latter is orthogonal 
to three of the former. One such system may be called the “orthogonal system " of 
the other. 
Let, x, y denote any two circles, then since 
(150) 
whence we obtain as particular cases 
1 + + ++ 0 , 
\y, 5, 6, 7, 8/ 
we shall have 
TL.r y - 
7Ti, 
L +‘ 
77.- 
J + -^-^ + 
1 8 -" y, 4 
2,6 
'3,7 
'4,8 
+ H-b — 77ff, 9 — 1 , 
^ 1,5 7r 2,6 * 4,8 
and, x denoting any small circle, 
7r 4', 6 | 77 t -,6 
7r l,5 7r 2,6 
+ 
ZLi7_l_+L§_] 
^S, 7 7r 4,8 
(151) 
