MR. R. LACHLAN ON SYSTEMS OP CIRCLES AND SPHERES. 
545 
And it may be easily verified that 
e+r+ c 2 +o>*= (aV 3 + 3 +cV 3 + dh v y ; 
so that 
(cdx ' 2 + b 2 y ' 2 + c 2 z ' 2 + d 2 v/ 2 )i 
(a — b)(a — c)(a—d) /a , (b — a)(b — c)(b—d ) /R ( (c — a)(c — b)(c — d) /?i 
■ — a’ ’-f~ y \ 2; 
? 1 ^’2 ^3 
(d — a)(d—b)(d — c ) ,, 
w 
• (135) 
PART II.—SYSTEMS OF CIRCLES ON THE SURFACE OF A SPHERE. 
Chapter I. —General Systems of Circles. 
The Equation of a Small Circle on a Sphere .—§§ 125, 126. 
125. Let ABC be a spherical triangle, having all its angles right angles : then, if we 
denote the sines of the perpendiculars from any point P on the sides of the triangle 
by x, y , z, we have at once 
^ 2 +r+- 3: = 1 ; 
and again, if ( xyz ), ( x'y'z') be any two points, <fi the angular distance between them, 
- xx' + yy' -\-zz' = cos <f>. 
So that the equation of a small circle is of the form 
ax + &.V+ cz= 1, 
the coordinates of its centre, and its radius, being given by 
x y z cos r . 
a b c 1 ’ 
and if (xyz) be any point, whose angular distance from the centre of the circle is <£, we 
have 
ax +iy+ cz— cos (f) sec r. 
It follows from this that, if the angle of intersection of the circles 
ax -\-hy -j-cz =1, 
a , x-\-b'y-\-c'z= 1, 
4 A 
MDCCCLXXXVI. 
