ON THE CYCLIDE. 
67 
[569 
569] 
L l K 1 = L 1 K 2 = L,K 1 = L. 2 K. 2 
nti-points. Observe that 
real; but the distances 
:d that one is real and 
¡m pure imaginaries. To 
K 1 K 2 , RS as horizontal, 
the same plane) the two 
tre of symmetry, and let 
the two circles in A, B 
points, (therefore A, B' 
is of the circles in the 
S, and not only so, but 
at a distance SL l = SL 2 , 
fints L ly L 2 and K lt K 2 
the two circles have a 
cles meets the line K 1 K 2 . 
uate in the perpendicular 
centre of symmetry R; 
the centre of symmetry, 
r 2 for their radical axis, 
sir plane with L X L 2 , the 
particular, if their plane 
>ned circles C, O', having 
fies C, C' or D, D', we 
itruction one or other of 
nfinity for a nodal line: 
t has besides the points 
K 1} K 2 , Lj, L 2 , that is, a system of skew anti-points, for nodal points; these determine 
the cyclide save as to a single parameter. In fact, starting with the four points 
L 1} L 2 , K x , K 2> which give S, and therefore the plane of the circles C, G'; the circle 
C is then any one of the circles through K x , K 2 \ and then drawing from S the two 
tangents to C, there is one other circle G' passing through K 1} K 2 and touching these 
tangents; C' is thus uniquely determined, and the construction is effected as above. 
Hence, with a given system of skew anti-points we have a single series of cyclides, 
say a series of conodal cyclides. 
If in general we consider a quartic surface having a nodal conic and four nodes 
A, B, G, JD, then it is to be observed that, taking the nodes in a proper order, we 
have a skew quadrilateral ABGD, the sides whereof AB, BC, GD, DA, lie wholly on 
the surface. In fact, considering the section by the plane ABG, this will be a quartic 
curve having the nodes A, B, G and two other nodes, the intersections of the plane 
with the nodal conic; the section is thus made up of a pair of lines and a conic; 
it follows that two of the sides of the triangle ABG, say the sides AB, BG, each 
meet the nodal conic, and that the section in question is made up of the lines 
AB, BC, and of a conic through the points A, G and the intersections of AB, BG 
with the nodal conic. Considering next the section by the plane through ACD, here 
(since AG is not a line on the surface) the lines GD, DA each meet the nodal conic, 
and the section is made up of the lines GD, DA and of a conic passing through 
the points A, G and the intersections of the lines CD, DA with the nodal conic. 
Thus the lines AB, BC, CD, DA each meet the nodal conic, and lie wholly on the 
surface; the lines AC, BD do not meet the conic or lie wholly on the surface. 
A quartic surface depends upon 34 constants; it is easy to see that, if the surface 
has a given nodal conic, this implies 21 conditions, or say the postulation of a given 
nodal conic is = 21, whence also the postulation of a nodal conic (not a given conic) 
is =13. Suppose that the surface has the given nodes A, B, C, D; the postulation 
hereof is = 16; the nodal conic is then a conic meeting each of the lines AB, BC, 
GD, DA, viz. if the plane of the conic is assumed at pleasure, then the conic passes 
through 4 given points, and thus it still contains 1 arbitrary parameter; that is, in 
order that the nodal conic may be a given conic (satisfying the prescribed conditions) 
the postulation is =4. The whole postulation is thus 16 + 13 + 4, =33, or the quartic 
surface which satisfies the condition in question (viz. which has for nodes the given 
points A, B, G, D, and for nodal conic a given conic meeting each of the lines 
AB, BG, CD, DA) contains still 1 arbitrary parameter: which agrees with the foregoing 
result in regard to the existence of a series of conodal cyclides. 
It is to be added that, if a quartic surface has for a nodal line the circle at 
infinity and has four nodes, then the nodes form a system of skew anti-points and 
the surface is a cyclide. In fact, taking the nodes to be A, B, C, D, then each of 
the lines AB, BC, GD, DA meets the circle at infinity; but if the line AB meets 
the circle at infinity, then the distance AB is = 0, and similarly the distances BC, 
CD, DA are each = 0; that is, the nodes (A, G) and (B, D) are a system of skew 
anti-points. 
9—2