548 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
8 
Ss 
S 
= 
a 
2 
> 
I 
co) 
ad 
| ao Yoo 2.9 Cy 
This proves the theorem. 
It is clear that the cuspidal edges of developable surfaces belong to this category, 
and thus the second singular solution in Example I. is accounted for. 
Example III. (§ 38.) 
§ 38. As an example, we will take the system of lines represented by the differen- 
tial equations 
Y, = Prt + Fpo’, 
eas 1 2 
Yo = Pot + gPy’- 
It will be more convenient to use y, z, p, q instead of ¥,, Yo, Pi, Po 
The complete primitive is clearly 
y = pu + 30%, 
2 vet op. 
The singular solutions are given by the system 
cdu+vdv = 0, 
vdv+ pd = 0. 
We find 
ee 
pi dp +vdv= 0, 
3 + $v? = c, a constant. 
Thus, for the first singular solution, 
go? == 378 (26 — v7), 
y = pau + gr” 
= zhv (18¢ — dv’), 
By eliminating v from these equations we have the two equations to a curve 
belonging to the first singular solution, that is to an envelope of the bitangeuts. 
