512 
MR. R. LACHLAN ON SYSTEMS OE CIRCLES AND SPHERES. 
Three circles produce, as may be seen by drawing a figure, four pairs of triangles, 
each pair consisting of a triangle, and its inverse with respect to the orthogonal circle. 
Thus, supposing a, /3, y are the angles of the triangle P, Q, Pi, the angles of the 
triangle P', Q', R' are either a, J3, y or else it —a, n — /3, n — y. Again, the angles of 
P', Q, R are a, tt — /3, n — y. Hence, having obtained the formulae for the radius of 
any circle connected with a particular triangle, we can easily obtain the formulae for 
the other circles. It is also evident that there must be eight circles corresponding to 
the circum-circle, and eight circles corresponding to the nine-points circle of a plane 
triangle. 
Chapter V.—Power-Coordinates. 
Definition .— §§ 47-50. 
47. We have already seen that any circle (straight line or point) is completely 
determined when its powers are known with respect to any four circles which have 
not a common orthogonal circle. Hence, given four such circles, which may be called 
the system of reference, any multiples, the same or different, of the powers of a circle 
(straight line or point) with respect to them, may be defined as its power-coordinates. 
We shall find it convenient to denote the coordinates of any circle by ; the 
coordinates of any point by xyzw; and the coordinates of any straight line by \gvp. 
If a, (3 be the Cartesian coordinates of the centre of any circle whose power- 
coordinate with respect to a circle be £; if a, h be the Cartesian coordinates of the 
centre of the latter ; and Pi, r be the radii of the two circles : we shall have 
£oc (a — a) 3 -f-(/3— 6) 3 — R 3 —r 3 ; 
so that the power-coordinates of any circle are quadric functions of a particular form 
of the Cartesian coordinates of the centre of the circle. 
Similarly, if x be the power-coordinate of a point whose Cartesian coordinates are 
x cc (a—a) 3 +(y8—6) 3 — r 3 ; 
or the power-coordinates of a point are quadric functions of a particular form of the 
Cartesian coordinates of the point. 
In the case of a straight line, whose Cartesian equation is 
we shall have 
x cos aff-y sin a=p, 
X cr-p — ci cos a — b sin a. 
Thus the power-coordinates of a straight line are linear functions of a particular 
form of what may be called the Cartesian coordinates of the straight line. 
