MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
567 
If then (£r)la>) be so chosen that 
dp dp dp dp 
dg _ dr] __ dco _ 
d±~d±~d±~d^~~ p* say ’ 
dg dy d£ dco 
then the circle cuts p(ccyzw ) orthogonally in each of the four points in which it 
meets it; we see at once that p. must satisfy the equation 
H (p+yp) = 0, 
which is the same equation as in § 168 ; the coordinates of the four orthogonal circles 
corresponding to the four values of /x, being proportional to the minors of the deter¬ 
minant Tl(p-\-jJcp). 
The Principal Circles. —§§ 175-179. 
175. The four orthogonal circles found in the last article are usually called the 
principal circles of the curve. By § 168, we see that a spheri-quadric is the envelope 
of a series of circles which cut one of the principal circles orthogonally, and it is 
evident by inversion that the curve must be anallagmatic, i.e., its own inverse with 
respect to each of its four principal circles ; also each point in which a principal circle 
cuts a spheri-quadric must be a cyclic point on the curve ; there are in general sixteen 
such points. Again, the double great circle tangents are the tangents which can be 
drawn from the poles of the principal circles. 
176. It is easily proved, as in § 81, that any two circles corresponding to different 
values of k given by H(p-\-kp) = Q, cut orthogonally; hence, if the four roots of the 
discriminating quartic be different there are four principal circles which are mutually 
orthotomic, and the poles of these circles must be such that the arc joining any two is 
perpendicular to the arc joining the remaining two. 
177. If the roots of JT(p-\-kp) = 0 are all different, then we can reduce the equation 
to the form 
ax 1 + by 3 -f cz 2 + dw z = 0, 
the system of reference being the four principal circles,, and a, h, c, cl being the roots 
of the discriminating quartic. 
178. If two roots of the quartic H (ppkp) are equal, then taking the two principal 
circles corresponding to the two other values of k, and any other circles forming with 
them an orthogonal system, as circles of reference, we can reduce the equation to the 
form 
ax 2, + by 2 + cz 2 -f- dip -\-2nziv=0, 
and, exactly as in § 83, we see that if one of the two circles (x, y) be imaginary, 
then the discriminating quartic can have two equal roots only when 
c—d, ii = Q ; 
