540 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
§ 25. If ¢ = uw and v is not constant, then TU = 0 and v =#, or else (1 + ¢) 
(a + At) (b+ Bt) (e+ Ct) = 0. 
The latter condition leads to no solution. 
For the second singular solution we must take ¢ = «=v, whence we find that 
each determinant of the matrix 
ax’, Byi', yy’, 1 
| 
Cb, b, C, i |=, 
if a, B, y are the cube roots of 
Aa(aB — Ab) (aC — Ac) 
(6 — B) ¢—C) 

, &e, 
§ 26. In the geometrical interpretation it will fix the ideas if we suppose the two 
conicoids projected into confocals. 
The complete primitive represents their common tangents. The first singular 
solution gives the geodesics on each that touch the other, and the second includes the 
curve of intersection, which is the envelope of these geodesics. 
The common tangent planes envelope a torse, whose generators will be common 
tangent lines. Of the four common tangents to the surfaces from any point of the 
torse, two will coincide with the generator of the torse. The equation to this torse is 
¢ = u, and every generator of it is a generator of one confocal of the system. Thus 
the equations ¢ = uv, v = const. represent the different generators of this torse, and 
their occurrence as an apparent singular solution is accounted for. 
The cuspidal edge of this torse is a second singular solution, and is represented by 
the equations 
ax’, By’, ye, L | 
a, b, @ i) 0; 
The General Case. (§§ 27-38.) 
§ 27. The straight lines 
= G01 5, 2d 2 ee ee 
x= Coe by OS Teal 
where the quantities b,, b, c,, cy are connected by two relations, but are otherwise 
arbitrary, form a congruency, and, if the relations are properly chosen, may be made 
to represent any congruency. 
