542 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
§ 30. Hence if the equation (5) gives unequal values for de, : dc, the lines of the 
congruency are bitangents to the surface whose equation is found by eliminating 
C1, Cg, X from (1), (2), (6), (7). 
The singular solutions of the differential equations are the solutions of (5), that 
is, of 
ab, 
‘de? + 
2 
ab, ob, 
pce 
ob, 
ped ee A Ae eu tete Qe 
ae a de, dey a dc—s0. 
an ordinary differential equation connecting ¢, and c,, from the solution of which two 
equations are to be found free from ¢, and ¢, by help of (1), (2), (6), and (7). 
§ 31. The equation connecting X, ¢,, ¢ is 

52s, (8 By) Bh 
oe ae 
Now 
du, = c, du + adc, + db, 
| ab. a 
Gr dx — Me dc, | as de., by (6), 
abe ay 
dy, = c, dx + aE de, — ot de,, by (7). 
1 2, 
But if the differential equation is to be satisfied we must have 
Qi — CN OY — CO a 
and therefore 
de, + pde, = 0, 
where 
so that p is the second root of the quadratic for A. This can only hold at one point 
of contact. 
§ 32. The integral of the equation (5) thus represents a double system of curves on 
the surface, one curve being traced by each point of contact of the double tangent. 
One curve is such that every tangent to it touches the surface again. This is the 
first singular solution. The other curve is the locus of the other points of contact 
of the tangents to the first and is not a solution.* One curve of each system goes 
* Tn the case of § 17, this curve is found_to be the straight line 
yy = x (we — Qn) + Bn? — 25, 
Yo = — wp? + Ww, 
and it is included in the complete primitive. 
This will always happen if the surface is developable, for the first singular solution is then the section of 
