594 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
1 ^ 
, COS CO 
l 
1 
1 
1 
n 
P 
l_ r ’ 
r \ 
2 
V 
r i 
1, 
-B 
cos co 2 1 95 
COS CO 2 . > 3 ? 
COS 
1, 
cos o).f } ] 5 
-B 
COS 
COS <W 2i4 , 
B 
COS C 0 3 , J, 
COS Ct)^ 0} 
-B 
COS W 3i4 
B 
COS (0 
COS &J40, 
cos o)^ 
— 1 
= 0 
( 251 ) 
223. There is no analogous theorem, in general, to Feuerbach’s theorem. In 
the case, however, when the given spheres (1, 2, 3, 4) cut at angles such that 
cos aj li3 =cos <y li3 =cos <u L4 =a ; cos = cos cos w 33 =/3 ; it maybe shown that 
the two spheres touching the given spheres, all externally, and the eight spheres 
touching three externally, or three internally, may be divided into two groups of five, 
each sphere of either group touching a certain other sphere. 
Thus, let the tangent spheres be denoted by [a, b, c, cl, e), let 2 denote the sphere 
which touches them. Suppose z touches (a) internally, and (6, c, cl, e) externally. 
From the equation 
(z, 1, 2 , 3, 4, o\ 
VL + b, c, cl, e) ’ 
we deduce 
A COS co-, x + B COS (Ox J +C cos co x , 2 ~b I) cos co x , 3 ~b E cos co x ,^~\~1 cos (Ox'j — 0 ; 
where 
A = 
2 cos 
OJ„, 
5 
cos 
M b,5~ 
cos 
cos 
°V, 5 ~ 
cos 
w *,5> 
B = — 
■ cos 
OJ„, 
5 + 2 
cos 
co b,5~ 
cos 
<+■,5 — 
cos 
cos 
**5» 
C=- 
■ cos 
co„, 
5 
cos 
o + 2 
cos 
<+,5— 
cos 
"A 5 ~ 
cos 
5j 
D= — 
• cos 
(o„, 
5 
cos 
cos 
w c,5+2 
cos 
cos 
°+, o’ 
E= — 
■ cos 
(O ff , 
5 
cos 
5 — 
cos 
*+, 5 
cos 
**5 + 2 
cos 
**5» 
F= — 
■ 6 . 
By taking for x, 1 , 2, 3, 4, 5 in succession, we can find cos w- 15 cos w- i2 , cos co. t3 , 
cos co. A , cos co. 5 , and then substituting in the equation obtained by putting z for x, we 
find the required condition. 
By equation (250) we find at once, 
and 
COS' (o ir 
3(« + l) 3 
3a 2 + 2/8—1 
0 3(a — l) 2 
C0S ~ "^= 3 . 42 / 8=1 ; 
COS' a» ( . 5 = COS~ COd,?,-— COS' W C)5 : 
(a 
l) 3 (/8 + l)9 -4(a 8 -l)Q8 3 -1) 
(/S + lj 2 (3 » 2 + 2/3-1) 
