610 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
and since the absolute is of the form 
x 2 -\-y 2 -\-z 2 = 4 wv, 
the discriminating quin tic becomes 
(Z’+a) 
Jc-\-b, h, 
g> 
li, Z' + c, 
f 
9’ f 
d, 
l, m» 
n — 2k, 
This can only have four equal roots if 
e=l=m= 0, 
2b=2c= 
Thus the equation can be reduced to the form 
ax 3 + by 2 -{- bz 2 -\- dw 2 -\~ 2fivz-\- 2gwy -f- 2 liyz— ibwv= 0. 
This may also be written 
ax z -\- div~-\- 2 fwz + 2 givy + 2 hyz = 0 ; 
and by suitably choosing for spheres of reference z-j-yw) instead of (y, z) this 
equation can evidently be reduced to the form 
ax 2j r 2hyz-\- div 3 = 0. 
The point w= 0 is a node on the surface : it is in general a cmc-node; but if «=0 
it is a binode, and if h= 0 it is a unode. 
Thus there are four distinct forms of cyclides : their equations can be reduced to 
one of the four forms 
(A.) ax 2j r by 3 + cz~ -f dw z -\- ev z = 0; 
the absolute being 
cr 3 +y 3 + z 2, ++ v* 2= 0- 
There are five principal spheres : if d—e there are two nodes, and if b=c, d=e, 
there are four nodes. 
(B.) 
the absolute being 
<xc 3 + b y' : -f- czr -\-dw~= 0, 
x 2 + y 2j r z 2 = 4 wv. 
This is the general inverse of a quadric surface : there are three principal spheres 
and one cmc-node. If b—c there are three cmc-nodes, this is the inverse of a 
quadric of revolution. 
