612 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Hence the equation to the tangent plane at the point ( xy'ziv'v) is 
//,/,/> / , / \( ax ’ , W , cz> ■ dvj ev 
(x x-\-y y-{-z z-\-w iv-\-v v) —fi- -+—+ + 
V ' 1 r 2 r s r i r 5 
= (ax x + by'y + cz'z + dw'iv + ev’'v) (—+-+—+~+~), 
\ r l r 2 T S r 4 r 5 / 
( 278 ) 
and the equations to the normal at the point {xy'ziv'v) are 
= 0 . 
X, 
y> 
IV, 
V 
r 
r 
, 
/ 
r 
X , 
y> 
b 
w , 
V 
ax', 
by, 
cz, 
dw', 
ev' 
1 
1 
1 
1 
1 
+ 
r 2 
r ’ 
'3 
V 
of bitangent 
spheres are 
given 
t— A 
0 
T 4- 
^ + 
0 
or 
0 
w 
s — 
b-a^~ 
c—a 1 
d-cr 
c — a 
*3 
II 
o 
? 4- 
^ 4- 
0 
(D~ 
0 
1- ^ 
a-b “ 
c — b' 
d—b 
*“e—6 
+ + =0 t ; 
a —c 5 — c — c c — c 
<-0 Q 03 0 
_a r , r , , w- _ A 
0=0, -7+7-7+ ~,+ 7—0 
a—a o — a c — d c—d 
03 O OO O 
n r , r ,6“ . n 
tjt= 0, -+ 7 - +-+ y—=0 
a — e b — e c — e d — e 
(279) 
(280) 
and the five systems of focal spheri-quadrics by 
x=0, -— 4 
vr . v 
7 ,7 f- -=0 
b — a c — a d—a e—a 
y 3 4-z 2 4-w 2 ++=0 
. . . (281) 
and similar equations. These curves have evidently, for their principal circles on the 
sphere, the circles in which the sphere cuts the other principal spheres of the surface. 
259. From the form of the equations of the focal curves, it is clear that any surface 
represented by the equation 
x- 
r 
vr . v~ 
« 2 + /e /3 2 + /c 7 2 + tc 5 2 + /c 1 e 2 + /c 
=o,. 
(282) 
