MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
595 
Taking cos w c>5 , 
COS CO, 1 r, 
w 
itli the same sign, we have 
Hence 
and 
A = 2 cos co „ 5 — cos co^ 5 — 3 cos co,. 5 , 
B = 2 cos o)f Ji5 — cos q )„ i5 — 3 cos co C:5 , 
C=D = E= — cos ct) a 5 —cos gj 4 i5 . 
A cos oj 2 > 1 =B — 3 aC, 
A cos co. 0= A cos co- 3 =A cos cd- i4 = — ccB-)-( 1 ~ 2 / 3 ) 0 , 
A cos co- 5 = — 6 , 
-A + B cos co^.+SC cos a>. : ,.-+F cos 6i .= 0 . 
Substituting, we must have 
- A'+ B 3 - 6aBC+ 3C~( 1 - 2/3)+36=0, 
or 
I 2 + (cos co-p —cos co 5>a )(cos 0J 5,« + COS CO- /, — 6 COS cO- e ) 
+ 2a(cos cog ) n -j-cos oj- j i)(2 cos co 5) /,— cos co 5i „ —3 cos co 5 f ) 
— (2/3— l)(cos co 5i j+cos co 5i „) 3 =0. 
The coefficient of cos co- if =(a +1) cos (a — 1) cos (o 5 „, and choosing the signs of 
cos (o 5ta , cos co 5 4 , so as to make this vanish, we can easily show that tin’s condition is 
satisfied. 
( a + l)\ 
/o 
Thus if r l5 t\ denote the spheres for which cos co=ffi and if r 2 , rh 
1 V oar + ip— 1 
denote the spheres for which cos co= : f y 0 3 and if (r 3 , r' 3 ) (r 4 , t' 4 ) (t-, t'-) 
< v fJCi ~j~ -1- 
denote the other spheres, we see that the groups 
( T n r -2> r 3’ r 4> r 5) ( r n T T 3> T n T 5 ) 
( r n t 2 > T 35 T n r 5 ) ( T n T 2 > r 3> T n r 5) 
have each a common tangent sphere, which touches t 1 or t\ in the opposite sense to 
t 2 , t 3 , &c. 
4 G 2 
