562 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
which includes the other inflexional tangents to the same surface, would serve 
equally well. 
This example shows that it is possible for the inflexional tangents to a surface to 
form two distinct congruencies. The parabolic and cuspidal curves, moreover, 
coincide. 
Example V.—System of Curves ii Space.  (S§ 51, 52.) 
§ 51, As another example, take the equations 
da? CRP 8 GkP 
e—1” f-17> #—1° 

The complete primitive is the result of eliminating ¢ trom 
1 1 1 
ZAP a 2y =at+—, 22 = bt +- — 
a and b being the constants of integration. 
The curves represented are conics touching the six planes 
e=+1, = se ils 2—-4-1. 
The first singular solution includes six forms— 
F 1 1 
ple 2y = u + —, Vif, == Oh, > === 3 
Uu u 
: 1 1 
YW = se ll DTA UTD aaa PH = (60) aS 
1 L 
; 1 it 
ole MEM 2) —" CW ae 
in each w is a variable parameter, and ¢ an arbitrary constant. The curves repre- 
sented are conics inscribed in the six faces of the cube contained by the planes 
Cs stale z2z=+1. 
The second singular solution consists of the twelve edges of this cube. 
§ 52. If we seek the singular solutions by means of the differential equations, we 
take 
pe — 1) SF 1) 0 
g(a —1) —(# — 1) = 0, 
and form the Jacobian with respect to p and gq. 
