MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
557 
and we also infer, from § 46, that this circle cuts the sides of the triangle PQK at the 
angles fl—y, y—u, a—/3. 
Comparing equation (164) with equation (161) we see that if the circle PQR cut 
the orthogonal circle at the angle oj, then cos m=2 cos oj. Whence we infer that in a 
spherical triangle formed by great circle arcs, the radius p of the nine-points circle, 
and the radius P of the circum-circle, are connected by the formula 
cot p=2 cot R. 
In the case of a general spherical triangle, this is replaced by the formula 
cot p — cot p — 2(cot R — cot Pd), 
where p, R are the radii of the analogous circles connected with the triangle, and 
p', R' the radii of the corresponding circles connected with the inverse triangle, with 
respect to the orthogonal circle of the triangle. 
Chapter IV. —Pow t er-Cooedinates. 
Definition .—§§ 149-151. 
149. Given any system of circles, say (1, 2, 3, 4), on the surface of a sphere, then 
any circle (great or small), or any point, is completely determinate when its powers 
with respect to the system (1, 2, 3, 4) are known, provided that this system be not a 
system having a common orthogonal circle. 
If then P be any point, we may define the coordinates of P referred to the system 
(1, 2, 3, 4) as any multiples, the same or different, of the powers of P with respect to 
these circles ; thus denoting the coordinates of P by ( xyzw ), then k 1} Jc 2 , k ?j , /q being 
any constant multiples, we may take 
X — /*|.77p_ -]_, 1J — 477" Z —■ ko. 7T P) o, J.V — lc^.7T-p^. 
Since 77p )P =0, and 7r Pi@ =l, we see at once that the coordinates of any point must 
satisfy a homogeneous quadric relation, viz., 
P, 1, 2, 3, 4 
n \P, 1, 2, 3, 4 
and a non-homogeneous linear relation, 
= 0 , 
n 
/P, 1, 2, 3, 4\ 
\e, 1, 2, 3, 4 1 
= 0 . 
The former is called the equation of the Absolute, and will be usually denoted by 
xp(x, y, z, w), and then the latter may be written 
