G20 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
D. Nodal Cyclides, with One Principal Sphere. —§§ 277-280. 
277. If the cyelide have only one principal sphere, we have seen, in § 255, that its 
equation may be reduced to the form 
ax 1 + 2 hyz -f- d iv l =0, 
where x is the principal sphere, w the node, and( y, z ) two other spheres cutting x and 
each other orthogonally, and cutting x in the points (iv, v). 
The equation of the absolute being 
o i o i o a 
r+ y "+— 4 w%\ 
and the coordinates of the plane at infinity —, —, we have seen that the node 
r l r 3 r 3 6 £ 
is in general a cmc-node ; but it is a binode if a= 0, and a unocle if h — 0. 
278. The coordinates of any tangent sphere at the point (; xyzwv ) must 
satisfy 
£ r) £ —2d — 2co 
(«. + k)x' hz' + ky r hy' + kz' dw'—2kv' —2 Jew’ 
The equation of the tangent plane at ( xy’z'w'v ) will be 
ctx hz . hy dw \ // ( /./ o' o / \ 
+ — b~+ _ )(x x~ry y-\-z z — 2 wv —2 vw) 
r l r 3 r S e i 
= (— + “+ — 2^—^— j [axx + hy'z + hz'y + dww'). . . 
The equations of the normal at (x y'z'w'v') are 
X, 
y , b 
— 2v, 
— 2w 
x', 
y ’ ~ 
— 2v', 
— 2w' 
ax', 
hz', hy', 
dw', 
0 
1 
1 1 
2 
2 
5 
r i 
»a r z 
e 
6 
279. The system of bitangent spheres is given by 
c_n + t*) + 2hyZ d(0~ 0)d 
- ^5 0 7 0 ” 1 o 4 
op — lir a ~ a 
(301) 
(302) 
(303) 
and the corresponding focal curve is given by, 
cc=0, 
4 hrwv . ah , d 0 
i +^^ y z +~ w ~- 0 
a~—h~ «. 2 — ld“ J ‘ a 
y~-\-Z : = ±WV 
