544 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
another congruency, satisfying another pair of differential equations, which the nodal 
curve also satisfies. 
For instance, the normals to an ellipsoid are bitangents to the surface of centres, 
but there are other bitangents, which form three more congruencies (see SALMON, 
‘Geometry of Three Dimensions,’ § 511a). The double curve satisfies the differential 
equations to the congruency of “ synnormals.” 
If we reciprocate we find the same paradox in relation to the double tangent planes. 
The line joining the points of contact will generally belong to the second congruency, 
and the cuspidal edge of the torse which it generates will satisfy the differential 
equations to this congruency, and belong to its second singular solution, the corre- 
sponding first being included in the complete primitive. 
Case of a Cuspidal or Parabolic Curve. (§ 35.) 
§35. At first sight it would appear as if a cuspidal curve ought to satisfy the 
differential equations to the bitangents, since any tangent to it meets the surface in 
four consecutive points. But the consideration of the reciprocal surface shows that 
the tangent planes drawn through such a tangent include three coincident ones, not 
two distinct coincident pairs. The tangents to a cuspidal curve and the inflexional 
tangents at parabolic points are therefore not to be counted as bitangents. 
When a surface is varied continuously so that a nodal curve changes into a cuspidal, 
some of the bitangents become chords of the cuspidal curve, and among them are to 
be reckoned the tangents to that curve. In the same way the inflexional tangents at 
parabolic points are included in the congruency formed by the intersections of tangent 
planes at pairs of parabolic points. Thus in a sense the congruency of chords of the 
cuspidal curve, and that of double tangents to the torse enveloped by the tangent 
planes at parabolic points, are limiting forms of bitangential congruencies belonging 
to the surface, though they cannot be considered as true bitangential congruencies 
belonging to it. 
Digression on a Certain Singularity of Surfaces. (S$ 36, 37.) 
§ 36. If, however, the curve is both cuspidal and inflexional, in a sense which we 
shall shortly explain, the tangents to it are true bitangents to the surface. 
If we suppose the inflexional tangent at each parabolic point to coincide with the 
tangent to the parabolic curve, then that curve must be plane or else a double curve 
on the surface. For the tangent plane at each parabolic point coincides with that at 
the consecutive point along the inflexional tangent, and hence the tangent plane is the 
same all along the parabolic curve. A double curve will, however, satisfy the con- 
ditions, if we consider the tangent plane at each point as indeterminate. Buta nodal — 
