566 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
170. If the coordinates of a bitangent circle satisfy the equation of the absolute, the 
circle reduces to a point and corresponds to a focus of a plane bicircular quartic— 
there are clearly sixteen such foci, four on each of the circles which cut the bitangent 
systems orthogonally. Dr. Casey (“ Cyclides and Sphero-Quartics”) calls these 
single-foci. 
171. It is clear that, if by any linear transformation of coordinates the equations 
0 , xfj=0 become respectively <X>=0, ' V P=0, then the same value of k which satisfies 
H.(p+kp) = 0 must also satisfy H(G>+/t’'d , ') = 0. 
Hence the coefficients of powers of k in the equation (186) are invariants. 
Equation of the Normal at any Point .—§§ 172-174. 
172. Let (£ylco) be the coordinates of any circle which cuts the curve p(xyzui)= 0 
orthogonally at the point ( x'y'z'w ), then we must have 
(187) 
for all values of k. 
173. It follows that the coordinates of the normal (i.e., great circle) at ( x'y'z'w ') 
must satisfy 
_ n 
dw' 
dp 
die. 
ydp _ 1 _ 
f a7 +,) 
dp 
¥ 
—co 
dp 
dh' 2 
< 
equation of the 
normal 
is 
dp 
dp 
dp 
dp 
dx ’ 
¥ 
aT* 
dw 
def) 
C(f) 
d<fr 
dej) 
dx° 
¥’ 
¥ 
dp 
dp 
dp 
dp 
dp 
a? 
¥ 
df’ 
dw' 
dp 
dp 
dp 
dp 
d/f 
dh’ 
¥ 
¥ 
equation (187) 
we can 
deduce 
= 0 . 
(188) 
