600 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
dp dp dp dp dp 
_ dr] _ d£ _ dco _ day __^ ^ 
dp dp' ' dp dp dp ^ ‘ ^ 
d£ dr] d£ dco dm 
then the sphere (£rj ’QmtP) will cut the cyclide orthogonally at every point of its 
curve of intersection. 
We see at once that p, must satisfy the equation 
H(p+ixp) = 0, 
and thus there are five such spheres; and the coordinates of the sphere which 
corresponds to a particular value of p are proportional to the minors of the con¬ 
stituents of any row in the determinant II (p -f- jip). 
These spheres are evidently the same as those which were mentioned in § 244, as 
being orthogonal to the five systems of bitangent spheres. We can easily prove by a 
similar process to that in § 81, that any two of these spheres cut orthogonally. They 
may be called the principal spheres of the surface. 
The Principal Spheres. —§§ 251, 252. 
251. By § 244, we see that at every point on the surface of a cyclide can be drawn a 
tangent sphere cutting one of the principal spheres orthogonally, and touching the surface 
elsewhere; and hence it follows that the surface must be its own inverse with respect 
to each principal sphere. Hence these species of surfaces have been called by 
Moutard anallagmatic surfaces, and the principal spheres hare been called by Casey 
the spheres of inversion. 
We have seen that the principal spheres cut the cyclide orthogonally, and it is 
evident that at points along the curve of section the corresponding bitangent sphere 
will not touch the cyclide elsewhere, but the curve of section will be a line of 
curvature on the cyclide. 
252. If a cyclide have a node, then, by the principle of inversion, this node must 
lie on each principal sphere; and thus in this case there can be but three principal 
spheres, and the node will be one of their points of intersection. 
If a cyclide have two nodes, they must be the two points of intersection of the three 
principal spheres, and any other two spheres forming with these an orthogonal system 
may be regarded as principal spheres ; this case corresponding to that of a quadric of 
revolution. Similarly if the cyclide have four nodes they occur in pairs, and lie on the 
only principal sphere ; but if we denote the nodes by P, P', Q, Q'; and the principal 
sphere by S ; then any pair of spheres orthogonal to S and passing through P, I v , 
which with any pair orthogonal to S and passing through Q, Q', make up an orthogonal 
system, may be considered as principal spheres. 
But if a cyclide have three nodes, then there are only three principal spheres. 
There are also cyclides with only two, and with only one principal sphere. 
