520 
MB, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
will represent a circle whose coordinates are 
a + b + c + cV 
r) = &:c. ; 
unless a-\-l>-\-c-\-d= 0 , in which case the equation represents a straight line. 
Inversion .—§§ 07, 68 . 
67. Let xyzw be the power-coordinates of any point P with respect to the system 
of circles (l, 2 , 3, 4); let P' he the inverse of P with respect to any point 0 ; then 
— L,F - - is unaltered 
O.l^O.P 
ordinates of P' referred to the system which is the inverse of ( 1 , 2 , 3, 4), we must 
have 
X=aX, y=/3Y, z = yZ, W=SW, 
where a, J3, y, S are some constants. 
Thus if the equation in power-coordinates of any curve be f (xyzw)—0, the equation 
to the inverse curve will be f(aX., /3Y, yZ, SW) = 0 . 
68 . The system consisting of two rectangular axes, the point of intersection, and 
the line at infinity, is clearly the inverse of a system of two orthogonal circles, and 
their two points of intersection, the centre of inversion being one of these points. 
For instance, the equation of a parabola expressed in power-coordinates is clearly 
by inversion, it follows that, if XYZW be the co- 
smce by § 3, — 7 = 
1 v 1 
X 3 =2aYZ. 
Hence the equation to the inverse of a parabola is of the lorm 
x~—2 ctyz, 
x, y having reference to the orthogonal circles, 2 , w to their two points of inter¬ 
section. 
Similarly the equation to the inverse of a central conic must be of the form 
aar+/fy 2 =yz 2 , 
a, /3 having the fame or different signs according as the conic is an ellipse or 
hyperbola. 
