5.96 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Chapter IV. - Power-Coordinates. 
Definition .—§§ 224-227. 
224. Since a sphere, plane, or point is completely determinate when its powers are 
known with respect to any five spheres not having a common orthogonal sphere, we 
may define the coordinates of a point (sphere or plane), referred to such a system of 
spheres, as any multiples, the same or different, of the powers of the point with 
respect to them. 
Thus if ( xyzwv ) he the coordinates of any point, whose Cartesian coordinates are 
(a/3y), we shall have 
x proportional to (a—a) 2 +(/3—&) 3 +(y—c) 3 —r 3 , 
where (abc) is the centre of the sphere of reference, r the radius. Thus [xyzwv) 
are quadric functions of a particular form of the Cartesian coordinates of the centres 
of the spheres of reference. 
We shall find it convenient to restrict the use of (x, y, z, iv, v) to denote the 
coordinates of a point; the coordinates of a sphere will be denoted by (£, y, £, co, ct) ; 
and the coordinates of a plane by (X, p, v, p, <x). 
225. Let us denote the system of reference by (1, 2, 3, 4, 5); then if ( xyzwv) be 
the coordinates of any point P, we see that, 9 denoting as usual the plane at infinity, 
so that 7r 9iP =l, then, since 7r PiP =0, the coordinates of P must satisfy a homogeneous 
quadric relation 
/p 1234 5' 
n( ’ ’ ’ ’ )—0 
P, 1, 2, 3, 4, 5 ' ’ 
and a non-homog'eneous linear relation 
O 
n 
/F, 1, 2, 
\0, 1, 2. 
Let us suppose the coordinates of P defined by the equations 
x=k 1 .7Tp tl , y = Jc,.7T l , i o, z=k s .7T P ^, iv=fi. tt va , r=V v 77- Pi5 . 
Then the quadric relation which {xyzwv) must satisfy is 
