OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 543 
through each point of the surface for every double tangent that can be drawn to touch 
the surface there. 
§ 33. As to the second singular solution, it is apparently given by the equation 
\ = p (these are the two values given for — de,/de,), or 
Gens ee, 
Ge WE Gla, Geamccy 
If there is such a solution, which is not generally to be expected, it will represent 
a curve on the surface every tangent to which meets the surface in four consecutive 
points. Such tangents to the surface do not, however, generally envelope a curve 
on it. 
Case of a Nodal Curve. (§ 34.) 
§ 34. The case of a surface having a nodal curve deserves consideration. 
The tangent to such a curve meets the surface in four consecutive points, and is 
therefore to be counted as a bitangent. The curve satisfies the same differential 
equations as the bitangents, and the two values of x given by the elimination of the 
differentials from (3) and (4) are equal, so that the curve is to be reckoned as derived 
from the singular solution of (5), or as a second singular solution of (1) and (2). 
But here a paradox presents itself. There are two tangent planes, and therefore 
two values of \, whereas the equations (6) and (7) only yield one unless they are 
identities. 
We have then 

ae 0c, 0c, 
Cli Oi 
a an =O 
Generally, these equations are not consistent, and therefore, generally, there will 
be no nodal curve. 
But on the other hand it will generally be possible to find a singly infinite series of 
values of #, ¥;, Y,, such that the equations (1) and (2), solved for b,, bg, ¢, ¢a, shall 
haye two pairs of coincident solutions, and the corresponding curve will be a nodal 
curve on the surface. 
The explanation of the paradox is that the surface will generally have other 
bitangents as well as those belonging to the given congruency. These will form 
the surface by a variable tangent plane and the tangent to this section touches the surface again at a point 
on the generator along which the plane touches the surface. The generator is therefore the locus of the 
second point of contact, and as it is the intersection of consecutive tangent planes it is included in the 
complete primitive. 
