554 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
whence as a particular case, 
(cot rj+cot r 2 -fcot r 3 +cot r 4 )(cot r 5 -j-cot r 6 -f-cot iy-|-cot r 8 ) 
= 4+ 2(cot r 1 cot r 5 -bcot r 2 cot r 6 +cot r 3 cot r 7 -fcot r 4 cot r 8 ). . . (158) 
An example of this would be, when (1, 2, 3, 4) are the inscribed and escribed circles 
of a spherical triangle, and (5, 6, 7, 8) are the corresponding nine-points circles. 
Chapter IIL—Circles connected with a Spherical Triangle. 
Regarding a spherical triangle as formed by the arcs of three small circles, most of 
the theorems concerning the three species of circles, connected with a triangle formed 
by great circle-arcs, may be readily extended. It will be seen that there is a much 
greater resemblance than there is between the corresponding formulae for plane 
triangles formed by arcs of circles and straight lines. We shall suppose that the 
circles intersect in the points P, Q, R, P', Q', R', the former points lying within the 
triangle formed by arcs joining the poles of the circles ; and we will call the angles 
of the triangle P, Q, R— a, /3, y ; then the formulae for any other of the eight 
triangles which make up the whole figure may be at once written down by changing 
two of the angles into their supplements. We shall use r z , r 3 to denote angular 
radii of the circles, and r to denote the angular radius of their orthogonal circle. 
The Circum-circle of a Triangle .—§§ 145, 146. 
145. If x denote the circle which passes through the points P, Q, R, the points of 
intersection of the circles (1, 2, 3); and if (4) denote the orthogonal circle of the 
system (1, 2, 3), we shall have, since 
n 
exactly as in § 39, 
/s.l, 2,3. 
\S, 1, 2, 3, 4/ ’ 
But 
^4 . + 
■ul 1 ' 2 ’ 3 VD 
4,® 
1 
! Vh 2 ,3/ ! 
I -^4,4 J 
+ 
+ 
n 
7 r 4t4 { V(1, 2, 3)} 3 sec 3 r x sec 3 r 3 sec 3 r s , 
gsec»r j8 ec*r,{V(P,2,3)}» 
where R is the radius of the sphere, and Y(l, 2, 3) denotes the volume of the tetra¬ 
hedron formed by the poles of the circles (1, 2, 3) and the centre of the sphere. 
