MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
541 
Chapter VIII. —Miscellaneous Theorems. 
Equation of an Anallagmatic Curve referred to Three Circles Orthogonal to the same 
Principal Circle .—§§ 115-121. 
115. If the system of circles (1, 2, 3, 4) be such that (4) is orthogonal to (1, 2, 3), 
the equation of the absolute must be of the form 
w 3 + f(xyz ) = 0 ; 
also if (4) be a principal circle of an anallagmatic curve, its equation must be of the 
same form ; by subtraction we see that the equation, 
ax 2j r by 2 + cz 2 -f- Tfgz + 2gzx -f- 2hxy = 0,.(128) 
may be considered as the general equation of such a curve referred to any three circles 
cutting one of its principal circles orthogonally. 
Thus, for any theorem proved in the case of conics we can easily derive an analogous 
theorem for bicircular quartics or circular cubics. 
116. Any bitangent circle of the system which cuts the given principal circle ortho¬ 
gonally, must have for its equation 
a:r+/3y+yz=0,.. (129) 
and since it touches (128) , we shall have 
I a, h, g, a 
K b, f, 
f c, y 
a, ft, y, 0 
Referring to § 24, equation (22), we see that a, (3, y are proportional to the areal 
coordinates of the centre of the circle (129) referred to the triangle formed by joining 
the centres of the circles (1, 2, 3), provided that x, y, z are proportional to the powers 
of a point with respect to these circles. 
We see, then, by equation (130), that the locus of the centres of all bitangent circles 
of the same system is a conic; which is called by Dr. Casey the focal conic of the 
system. 
117. Suppose now the circles (1, 2, 3) to be the other principal circles, then the 
equation to the curve must be of the form 
= 0 . 
(130) 
ax' 2 A by~ -j-cz~— 0, 
