550 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
so that (145) may be written, 
—G: i (i 1{" (1:1; 5) 1 G: J; 1)} '■ 
Bat if r be the radius of the common orthogonal circle, we have, by § 133, 
u/ 1 ’ 2,3 ) nf 2,3,4 
+ 2 2 2 2 2 \1, 2, 3/ 3 t> 3 \2, 3, 4/ 
Re tarn r 3 =cos 3 r x cos- r 3 cos 3 r 3 {V( \’ 2> V,)p = cos r 2 cos- r 3 cos- ^ {V ( 2 ; 3 , 4) } 2 ~ 
36 
:&C. 
Hence we have 
7r, i4 ,cosr 4 .Y(l, 2, 3)=77-.,. 4 .cos n 1 .V(2, 3, 4)+7r* )3 .cosr 3 .Y(l, 4, 3) 
4 7r., 3 .cosr 3 .Y(l, 2, 4) ; . (146) 
which result may also be written 
7r^ )4 .cos r 4 =a.cos rj.TT^^+yS.cos r 3 .7r^ 2 +y.cos r 3 .7r^ 3 ; . . . (147) 
where a, /3, y may be defined as the areal coordinates of the pole of the circle (4) with 
respect to the triangle formed by the poles of the circles (1, 2, 3). 
Thus, if A, B, C be the triangle, P the pole of (4), then 
Y(P, B, C)_sin (perp. from P on BC) 
a V(A, B, C) sin (perp. from A on BC)’ 
As a particular case of (147), let x be a point, O say, then A, B, C being the centres 
of (1, 2, 3), and P being a point on the circle which cuts them orthogonally, we shall 
have 
1 — cos OP=a( cos r 1 — cos OA)-]-/3(cos r 3 — cos OB)+y(cos r 3 — cos OC), 
or more generally, P being the pole of a circle which, with (1, 2, 3), has a common 
orthogonal circle, 
cos r 4 — cos OP=a(cos i\— cos OA)+/3(cos r 3 — cos OB)fi-y(cos r 3 — cos OC). 
Orthogonal Systems .—§§ 137-139. 
137. Four circles may be said to form an orthogonal system if each one cuts the 
other three orthogonally. It is clear that the pole of any one of four such circles must 
be the orthocentre of triangle formed by the poles of the other three. 
