574 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
powers of a point with respect to the three circles of reference ; then by § 136, 
equation (147), the equation of any circle orthogonal to w will be given by 
ax J r(3i/- i r y~ = 0, .(207) 
where a, (3, y are proportional to the triangular coordinates of the pole of the circle 
referred to the triangle formed by the poles of the circles (x, y, z). 
Suppose, now, the circle given by (206) to be a bitangent circle of the spheri- 
quadric (205), then we must have 
ci, k, g, a |=0 ..., .(208) 
h, b, J] (3 
9, f c, y 
a, (3 , y, 0 
Hence it follows that the poles of all bitangent circles belonging to the same system 
lie on a sphero-conic; or again, the spheri-quadric (205) is the envelope of circles 
whose poles lie on (208), and which cut a given circle w— 0 orthogonally. 
195. If the circles ( x , y, z ) are the other three principal circles of the curve (205), we 
know that the equation of the curve is of the form 
a.T 2 -j- by~ -f - cz 3 = 0, 
hence the equation of the sphero-conic is 
o >02 2 
-+f+^= 0. 
Thus the sphero-conic corresponding to one principal circle is self-conjugate to the 
triangle formed by the poles of the other three principal circles. 
196. Again, in the case of a nodal-spheri-quadric the equation of the curve referred 
to its other principal circle, and the two points in which its two principal circles 
intersect, is of the form 
ax*-\-by*+2fyz=0, 
so that the equation of the sphero-conic must be 
f~ct~ — aby~ + 2 af(3y— 0. 
Thus the sphero-conic must pass through the node. 
197. Let (y, z) be any two bitangent circles, (x) the circle which passes through their 
four points of contact; the equation of the curve takes the form 
* 3 =2 fyz, 
