55 6 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
The Inscribed and Escribed' Circles of a Triangle .—§§ 147, 148. 
147. If the inscribed and escribed circles of the triangle PQR cut the orthogonal 
circle at angles co, cu 1} a> 2 , &) 3 , we have, as in § 43, 
wher< 
K.cos 3 
—2(1 + 
cos 
a)(i + 
cos /3)(l + 
cos 
y) 
K.cos 3 
"i — 2(1 + 
cos 
a)(l- 
cos /3)( 1 — 
cos 
y) 
K.cos 2 
&n=2(l — 
cos 
a)(l + 
cos /3)(1 — 
cos 
y) 
K.cos 3 
1 
l—J 
cP 
II 
CO 
3 
cos 
a )(l~ 
cos /3)(1 + 
cos 
y) ^ 
K= cos 3 
a+ cos 3 /3+ cos 2 y + 
2 cos a cos 
6 COS y- 
= 4 cos s. cos (.< s— a). cos (s — /3). cos (s—y). 
(162) 
In these formulae cos' 3 a> has been written for 
-t.* 
TT.l ±-TT. 
4 , 4 *" x.x 
, and so, if the given triangle 
be an ordinary spherical triangle, cos 3 oj must be replaced by —cot 3 r x ; thus the above 
formulae correspond, in the case of an ordinary spherical triangle, to the formulae, 
N cot r — 2 cos cos l?/3 cos ■g-y'l 
N cot r x — 2 cos \cl sin 4/3 sin \y . 
N cot r z = 2 sin cos \(3 sin \y j 
N cot r 3 = 2 sin \ct. sin 4/3 cos \y J 
where 
N 2 = — cos s.cos (s — a). cos (s — /3).cos (s — y). 
In our present case, the radii will be given by formulae similar to 
K(cot ?y+ cos (o cot r) + 
0, 
cot 
cot r 2 , 
COt ?’ 3 
= 0 
1, 
-1, 
cos y, 
cos /3 
1, 
COS y. 
-1, 
COS a 
1, 
cos /3, 
COS a, 
— 1 
(163) 
148. In exactly the same way as in § 45 we may show that, associated with every 
triangle, there are eight circles analogous to the nine-points circle of a plane triangle, 
each of them touching four of the circles, which touch the sides of the spherical 
triangle; that is, taking any one of the eight associated triangles formed by three 
circles, say PQR, the inscribed and escribed circles are touched by another circle. 
If this circle cut the orthogonal circle of the triangle at the angle we shall have, as 
in § 46, 
cos 3 73 = 4 sec s. cos (s — a). cos (.$—/3).cos (s — y), .... (164) 
