510 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
we have, giving x the successive values 4, 1, 2, 3 :— 
where 
A 1 cos 4 
A L COS CO, 4 = Ao — A 3 cos y — A 4 cos f3 | 
A 1 COS co, o= — A 3 cos y + A g — A 4 cos a 
A 7 cos w„- )3 = — A 3 cos /3 — A 3 cos a+A 4 J 
Aj= COS w 4)5 — COS w 4|6 — COS C0 4i? — COS "1 
Ao= — COS C0 4 5 + COS C0 4 g — COS 0) 4h7 — COS CO^g 
> . 
Ag= — COS W 4l5 — COS C0 4i6 + COS COS CO^g 
A 4 — — cos co 46 — cos co 4i6 — cos co 4i7 + cos co 4)8 
But we also have 
Therefore 
n 
'z, 4, 1, 2, 3 
4, 1, 2, 3 
= 0 . 
— A T +A 3 cos co 1]C +A 3 cos &> 2 ,z+ A 4 cos co 3> _-+4 cos w 4)S =0. 
( 66 # ) 
Hence we must have 
16—A 1 3 +A 2 2 +A 3 2 +A 4f 2 —2A 3 A 4 cos cl— 2A 4 A 3 cos (3— 2A 3 A 3 cos y— 0, 
which may be written 
16 + 2 cos 2 w 4i5 (l — cos cl— cos (3— cos y) 
+ 2 COS 3 tu 4i6 (l— COS a+ COS / 3 + COS y) 
+ 2 cos 3 co 4) 7 (1 + COS CL — cos / 3 + cos y) 
+ 2 cos 3 0 ) 4i8 (l + cos a+ cos (i — cos y) 
+ 2(1- COS a) COS 0)4,5. cos W |:(;— 2(1 + COS a) COS 0)4,5. COS 0)4,5 
+ 2(1- cos (3) cos co 4i5 . cos co 4i7 — 2(1 + cos (3) cos co 4 g. cos co 4i8 
+ 2(1— cos y) cos co 45 . cos co 4i8 —2(1 + cos y) cos co 4i6 . cos <y 47 =0. 
Referring to (65) we see that this equation is satisfied, provided we choose the 
groups of circles so that 
cos co 46 . cos co^g. cos co 4i7 . cos 4i8 is positive. 
Thus if we denote the tangent circles of the system (1, 2, 3) by the symbols 
(t, t), (t 1} t'i), (t 3 , t' 3 ), (t 3 , t' 3 ), where r, r l5 r 3 , r 3 correspond to the positive values 
of cos co, cos co 7 , &c., as given by (65), then we see that the groups 
