MR. K. LACHLAN ON SYSTEMS OF CLRCLES AND SPHERES. 
525 
If then (be chosen so that 
00 00 00 00 
0f _0/; _0f _0w __ 7 
00 0-0 00 00 ' ^ 
0£ dr) 0£ 0« 
the circle (£rj(oj) will cut the curve 0 orthogonally in four points, and we see at 
once, that k must satisfy the equation 
H(0+&0) = O ; 
and then (^yCco) are proportional to the minors of the constituents of any row in the 
determinant H(0+&0). 
Thus it appears there are four circles which cut the curve 0 = 0 orthogonally : and 
these circles are identical with the four which are mentioned in § 73, as being 
orthogonal respectively to the four systems of bitangent circles. 
The Principal Circles .— §§ 80-82. 
80. The four circles considered in § 79 have been called by Moutard the principal 
circles of the curve. And the curve may be considered as the envelope of a system 
of circles, which cut one of these principal circles orthogonally : it follows then that 
the curve is its own inverse with respect to any one of the principal circles (hence the 
principal circles must cut orthogonally); also the four points in which any principal 
circle cuts the curve must be cyclic points ; so that there are in general sixteen cyclic 
points. 
81. We may prove independently that the principal circles cut orthogonally, thus; 
taking for our system of reference an orthogonal system, so that the equation of the 
absolute is 
0 = x 3 +y-+A+?y 3 = 0 ; 
then the coordinates of any principal circle being (cp^ai) we must have, if 
0 = ax z -\- 6 y 3 +C 2 2 -f dur-k-Zfyz-CHgzx-k-2hxy-\-2lxiv-\-2inyiv-\-2nziv=0 ; 
a£-\-hr)+g£,+l co=— kg "] 
g€+fy+cC+na)=— JcC j 
l a——Jcco j 
h£-\-b rj —j- f( > -{-mco = —krj 
(103) 
