MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
485 
Again, the power of (4) and 
— 2x cos /3—2 y sin /3-f-2^= 0, 
is given by 
7 r = —2 cos (3 (a—/3) = —2 cos a cos /3 — 2 sin a sin /3 ; 
which ma}^ also be considered as a particular case of (3). 
The equation to the line at infinity may be written 
0cc+0y-|-1 = 0. 
Hence, denoting this line by the symbol 6, we shall have 
TT ex =l, if x denote any circle, or point, 
tt 6 x —0, if x denote any straight line, 
7t m = 0, of course. 
and 
and 
5. It will be convenient to observe here, that if n denote the power of the two 
circles which are respectively the inverse curves of (1) and (2) with respect to any 
circle, whose centre is the origin 0 and radius E ; then 
, R 4 
7r = -77 ; 
C]Co 
i.e., denoting the circles by S 1; S 3 , and the circles inverse to them by S^, S' 2 , since 
C l~ 7r 0,h’ C 2, = jr 0 
hs'l, s '2 
\/ IT 0, s\ TT 0, s' 2 \/^ 0, 0, s 2 
and the formula is still true if either or both circles degenerate into straight lines. 
Thus if x, y denote any circles, straight lines, or points, the expresssion 
1 
\/ ' Tr Q,x- 7r Q,y 
is unaltered if the circles be inverted with respect to any circle whose centre is O. 
General Theorems. §§ 6-8. 
6. If we have given a system of five circles (1, 2, 3, 4, 5), their powers with respect 
to any five other circles (6, 7, 8, 9, 10) are connected by identical relation, which may 
be expressed in the umbral notation by 
'1, 2, 3, 4, 5 
n U 7, 8, 9, 10/ °- 
The word “ circle ” being intended to include a point, a straight line, or the line at 
infinity. 
