MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
619 
The equations of the normals will he 
X, 
y, 
Z 
— 2v, 
— 2 iv 
/ 
X , 
y\ 
Z 
* ? 
— 2y', 
— 2w' 
ax ', 
fy. 
hiv 
hz, 
0 
1 
i 
1 
2 
2 
j 
5 
— — 
7> i 
To 
r 3 
6 
6 
= 0 . 
(297) 
274. The two systems of bitangent spheres are given by 
£=0; 
i?=0 ; 
and focal curves are given by 
£C=0 ; 
V 2 . 4fem («£+Aco) 3 ") 
b—a ‘ a a? 
\ 
£ 3 4 wct (b£+7ia>y 
(298) 
a — b 
b 3 
= 0 
J 
b- 
b 2 2 A W 3 
r— zw— -iv j =o 
a a a- 
and 
O.o < 
y=o; 
a 0 2 A A 2 0 
- ~X~ — —zw — 75 IV~ = 0 
a — b b b~ 
>■ 
O . 
r+' 
: 4n’V 
275. Let us suppose the sphere y to be of infinite radius, then the corresponding 
focal curve will pass through the centre of the principal sphere x, when 
a 1 2 A 1 A 3 1 
-7iTF=0; 
a — b ry b er 3 lr e 
(300) 
and then the surface will represent a Cartesian having a binode. 
276. The surface 
ax 2 - J r hy z -\- 2 hzw— 0 
will represent a cubic cyclide, having a binode when 
a , b , 2A „ 
oT - oH— — o. 
7 Y r 2 r 3 e 
In this case the surface passes through the centres of the two principal spheres (x, y), 
and the tangents at these points are parallel to the plane 
ax , by , 7 2 , 7 w 
- + A +h-+h-=0, 
r i r -2 e r 3 
and so are the tangent planes drawn through the line at infinity on the surface. 
4 K 2 
