MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Gil 
(C.) axx + by* 4' 2 hwz = 0, 
the absolute being 
0 i o | 0 i 
xx -f y~ -\-z~= 4 wv. 
This surface has two principal spheres and. a binode (iv— 0). 
two other nodes. 
(D.) 
the absolute being 
ax 3 + 2 hyz -f- dvr = 0, 
ox + y~ -j-z 2 = 4 iov . 
If a=b there will be 
This surface has only one principal sphere and a cmc-node. Also, since the general 
equation represents a cubic cyclide, when it is satisfied, by the coordinates of the plane 
at infinity, it follows that the equation of a cubic cyclide can always be reduced to one 
of the four forms given above. 
Chapter VI. —Classification of Cyclides. 
Cyelides have been classified by Darboux and Casey according to the nature of 
the focal quadrics. We shall find it more convenient here to discuss the different 
forms of cyclides in order, according to the number of different roots which the 
discriminating quintic has. We have seen that there are four distinct forms to which 
the equation of a cyclide may be reduced; and it is proposed to discuss briefly the 
different species represented by similar equations, and a few of their properties. 
A. The General Cyclide .—§§ 25G-264. 
256. The equation of the surface is of the form 
ax 3 -f by 3 + cz 2 -j- dvT + ev z = 0, 
the equation of the absolute being 
^ a +2/ 3 +z 2 +w 3 +'y 3 =0 . 
and the coordinates of the plane at 
radii of the five principal spheres. 
257. The coordinates (^y^&jzrr) of any tangent sphere at the point (x'y'z'w'v') must 
satisfy 
£ 7j £ CO _ ® 
(i a + lc)x ' (b-\-k)y' (c + k)z' (d+k)w' (e + k)v' 
4 I 2 
infinity -, —, —, —, i.e., the reciprocals of the 
r s r 4 r s 
