546 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
be oj, and the distance between the centres <f>, then 
tan r tan r cos w = 1 — cos <f> sec r sec r 
= 1 —aa—bb' — cc . 
(136) 
126 The equation of a great circle will be of the form 
ax-\-by-\-cz—0 ; 
and the angle of intersection of this with the circle 
a'x-\-b'y-\-c z= 1, 
will be given by 
tan r cos oj= — cos <£ sec r' 
= —(aa'-\-bb'-\-cc); 
(1ST) 
and for two great circles we shall have 
cos (0= -(aa'-j-bb'-j-cc'). 
The Power of Two Circles on a Sphere. — §§ 127, 128. 
127. If r, r' be the angular radii of two small circles on a sphere, </> the angular 
distance between their poles, and cj their angle of intersection, then either of the 
expressions 
tan r tan r cos a>, 1 — cos (f> sec r sec r , 
may be defined as the power of the circles. Denoting the power by tt (or if S, S' are 
any circles by 7r Si s'), we see that 7r SiS .=0 if the circles cut at right angles, and if they 
touch 7r S] S /~= 7r Sts . 7r S / j s #. In the case of a pair of points P, P': 7r Pil v— 1— cos <£=2 sin ~ h<f>, 
where <^> is the distance between them. 
It will be convenient to define the power of a great circle, with respect to a small 
circle, as the product of the tangent of the radius of the small circle and the cosine of 
the angle of intersection : thus 
tt = tan r cos w = — cos <£ sec r ; 
and the power of two great circles, as the cosine of the angle between them. 
128. If O be the pole of a small circle, radius r, and P any point on the sphere, then 
if Q be taken in the arc OP so that 
tan tjOP . tan iOQ = tan 2 pr, 
