526 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES 
where k is a root of the equation 
a + k, 
h, 
9> 
l 
h, 
b+k, 
f 
m 
9’ 
f 
c+k, 
n 
h 
m, 
n, 
d+k 
Let now (g'yf'aj') be the coordinates of the principal circle corresponding to another 
root k' of this equation; then multiplying equations (102) respectively by (ff £V) 
and adding, we obtain at once 
(k~k')(^' +777/ + ££'+6)<u , )=0. 
Hence, if k, k! be unequal, the two circles must cut orthogonally. 
82. If the curve have four principal circles—he., if the roots of the discriminating 
quartic H(<£+L//) = 0 are all different, the curve cannot have a third double point—for, 
inverting with respect to any principal circle, the inverse must also be a double point, 
unless the point lies on the principal circle; since then a quartic curve can have but 
three double points, in the case of a bicircular quartic, the third double point must lie 
on each principal circle. Hence two of the roots of the discriminating quartic must 
be equal, and there are only two principal circles. 
Similarly, if this third double point be a cusp, it is easy to see, by inverting with 
respect to a principal circle, that any circle touching the tangent to the cusp at the 
cusp must touch the principal circle ; and hence there is only one principal circle, and 
the discriminating quartic must have three equal roots. 
Reduction of General Equation. —§§ 83, 84. 
83. If one of the principal circles be a circle of reference (say #=0), then it is 
clear that the terms involving xy, xz, xw, must be absent from the equation. 
Supposing, then, that the equation H(<£-j-fo/>) = 0 has all its roots unequal, then there 
are four principal circles, and taking these for circles of reference the equation must 
reduce to the form 
ax s + by 2 -j- cz 3 -f did 1 — 0. 
Suppose, now, that two of the roots of the discriminant are equal; then taking the 
circles corresponding to the unequal roots, and tw r o circles cutting them orthogonally 
as circles of reference, the equation will be of the form 
ax 1 + by 2 -|- cz 2 + did 1 -f-2 nzw — 0 ; 
and the system of reference being orthogonal we have for the absolute 
& 2 +?/ a +z 3 +w 3 = 0; 
