MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
543 
Again, if ( yz ) denote the double tangents from the centre of the principal circle of 
the system, we see that the centre of the focal conic coincides with the centre of the 
polar circle of the centre of the principal circle, and the asymptotes of the focal 
conic are perpendicular to the double tangents. It follows, then, that the focal conic 
is an hyperbola or ellipse, according as these tangents are real or imaginary. 
It follows, also, that the focal conic of a circular cubic is a parabola, whose axis is 
perpendicular to the asymptote of the cubic. 
Circle of Curvature at any Point of an Anallagmatic Curve .—§§ 122-124. 
122. Let the equation to the circle of curvature at the point ( xy'zw ') be 
£x -f yy ++ (mw— 0. 
Then we must have £ y, £, &>, proportional to the minors of x, y, z, w, in the 
determinant 
X, 
y, 
25 
w 
X, 
y'> 
L 
IV 
Sx', 
%y', 
8z', 
Sw' 
SV, 
sy, 
8~z', 
3 V 
And as we are merely concerned with the ratios of the coordinates ( x , y, z, w), we 
may take iv constant; so we shall have 
Sy'SV-Sz'B y 8z'8 2 x'-8x'8V 8x'8hf-8i/8 2 x r 
123. If the equation of the curve referred to its principal circles be 
ax 2 + by 2 -j- cz 2 -f d w~ = 0, 
where the equation to the absolute is 
we shall have 
xr+f+C+iP= 0 ; 
ax 8x d~bySyfczSz=0, 
a'Srr+v/Sy-bzSz=0 ; 
x8x y8y z8z 
b—c c—a a—b 
say ; 
whence 
