OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 56] 
Hence the point is a parabolic point on the surface, and the line of the congruency 
is the intersection of tangent planes at two consecutive parabolic points. All such 
lines will generate a torse, and they would belong to the first singular solution were 
they not already in the complete primitive. 
§ 49. The intersection of two consecutive generators of this torse is given by the 
equations 
y, = 0,0 + by, 
Yo == Cyt ++ dg, 
« = — db,/de, or — db,/dey, 
where by, ¢, by, c, are connected by the equations 
M0 0M op O02 Mion —) Oe N10) 
These equations are satisfied if P = 0, and therefore the cuspidal edge of this 
torse is the second singular solution given by taking N = 0 = P. 
A Particular Kxample. (§ 50.) 
§ 50. As an example of an inflexional congruency, we may take the system 
of. lines 
y = 30°*b'x + 40° (1 — 6a’), 
z= b? (1 + 2a3) w — 3 a'd°. 
These are half the system of inflexional tangents of the surface of § 38. 
It is easily veritied that the cuspidal curve is a second singular solution, and the 
nodal curve not. 
The first singular solution is given by 
a=ab>, b= constant. 
lt is 
a“ 
— 1 j9 
y= tah, 

2=4 5B + ba. 
The system 
y= (4 ey 2 — 90, 
z= 2b°x + 3 atb® — ab , 
MDCCCXCV.—A, 4 © 
