528 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
If, however, x=0 be real, we may take as system of reference this circle, a circle 
cutting it orthogonally, and their two points of intersection ; then, since the absolute 
is of the form 
the discriminant becomes 
(/v-f a) 
x 2 -\-y' 2 = 4zw, 
I'd- b, J, m 
f, c, n — 2k 
m, n — 2k, cl 
which can only have three equal roots when 2 b——7i, m=d= 0 ; in which case the 
equation of the curve takes the form 
aar-b by 2, — Abziv- f- 2fyz J r cz z = 0, 
and by taking instead of y= 0, the circle y-\-\z, which clearly cuts x orthogonally, we 
can get rid of the term cz 1 . 
And since 
x~ J r y~= 1 izw, 
this equation can be further reduced to the form 
ax^-\-2fyz= 0. 
84. Thus we see that the equation of a bi-circular quartic can be reduced to one of 
three forms :— 
(A.) cox 2 -\-by 2 -\~ cz2j rdw 2 ^= 0, 
in which case there are four principal circles, the equation of the absolute being 
x 2 d-.r+z 2 d-w 2 = 0 . 
(B.) ax 9, -\-by 2 -\-cz 2 =0, 
the equation of the absolute being 
in which case there are two principal circles, which must be rea], and a node which is 
one of the points of intersection of these circles. 
(C.) ax % -\- 2fyz= 0, 
the equation of the absolute being 
x~ j -\-y-=±ziv. 
In this case there is only one principal circle (a?=0); the curve passes through the 
two common points of ( x, y), and the point (cc=0, y= 0, 2 = 0 ) is a cusp on the curve. 
It is also clear that circular cubics can be reduced to one of these three forms: 
since we have seen that the equation of the second degree represents a cubic when it 
is satisfied by the coordinates of the line at infinity. 
