MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
495 
i.e., 
_ ir ->\ I• 7r //, I i 7r -‘V3- 7r 2/,2 i 7r J\3 ,7r V.3 i 7r -',4 -7r ;'/,4 
^•,y— -r--H..(23) 
’Al *8,8 'A,3 A A V ’ 
As particular cases, we have, since 7r ia = — 2rp, &c., 
o u o+ o o— — 2 .a, a——2 or 0 ; ..... (24) 
A A A A" 
according as x denotes a circle or a straight line. 
If x represents a circle, radius r x> we have 
A o 7 r «,l~ l I 7 r .u, a " 1 | A.g* I 7 AY /nc\ 
4 A- = “—y + —V + - A +~ 3 .(25) 
A 3 A A A 
And if a; represent a straight line, so that tt x%x — — 2, we have 
4=^+^+^+^... • ■ (26) 
A ’’s" A" V 
Again, since 7r 0i0 =O, we have 
0 —aH— 2 + “2“1—2 j.(21) 
A A A 2 A 
whence it appears that one at least of the three circles must be imaginary, and one at 
least real. 
26. By equation (16) we have 
_ aVa 2 
4 4{A(1, 2, 3)} 3 
Hence 
4{A(1, 2, 3)} s =-i* 3 V+W+«*iV; ...... (28) 
and we can easily find that, if p be the radius of the nine-points circle of the triangle 
formed by the centres of three of the* circles, 
( 29 ) 
27. If the circles (1, 2, 3, 4) be any system not having a common orthogonal circle, 
we may find four other circles, (5, 6, 7, 8) say, each of which is orthogonal to three 
of the former. Two such systems are connected by several interesting formulae, and 
one system may be called the “ orthogonal system ” of the other. 
Thus, x and y denoting any two circles, we have, since 
