558 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
for the first factor clearly vanishes, since ), p, @, are proportional to the direction 
cosines of the tangent plane. 
Also 
Op C2 Op op 
Dae + (m + p,) D ee + (mz + p,) D ay, + mp, D ay 
o op 
+ (1, Po = Mep;) De. = ar M2P e OY,” 
= (A my + pygmy) (DA + p; Diy + P2 Dery) 
+ (A + papi + MP2) (DA + mm, Duy + 172 Dury) 
= (A+ papi + pops) (DA + m, Dy + my Dpy), 

and 
ae p?¢ oe on Oe 
De® + 2p, Dae + 2p,D a Pa a ee no De 
= (A+ mpi + MP2) (DA + p, Dy + py Dyts). 

Thus the point (7, m,) lies on the cubic, its polar line is \ + pp; + pap, = 0, and its 
polar conic consists of this line and another. Hence the three solutions of the two 
equations coincide, and we have p,; = m,, py = Mx. 
Thus the cuspidal edge is enveloped by the inflexional tangents, and is a solution 
of the differential equation of the congruency. 
If the surface has a nodal curve the equations M = 0, 0M/du = 0, 0?M/op? = 0 are 
apparently satisfied along it, but these equations, as they stand, are not enough to 
determine ¢, and ¢,, and when ¢, and ¢, are evaluated by means of another differen- 
tiation they are not generally equal to p, and p, taken along the nodal curve. In 
fact there are two inflexional tangents in each sheet at every point, and the tangent 
to the nodal curve is not generally the same as any of the four. Hence the nodal 
curve, as such, is not a solution.* 
Another Second Singular Solution. (§ 46-49.) 
§ 46. Let us now take the alternative of § 42 and suppose that 

=e, ss 5, es = (0) 
* The lines that touch the surface at points on the nodal curve form such a degenerate inflexional 
congruency as was discussed above (§ 40), and they will satisfy the differential equations to the inflex- 
ional congruency of the surface in the unreduced form in which we have used them. The tangents to the 
nodal curve form a first singular solution included in the complete primitive, and the nodal curve belongs 
to the second singular solution which also includes the cuspidal edge of the torse enveloped by the 
tangent planes. 
