552 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
If the roots are equal we have the cuspidal curve traced by the point 
(Be, Bae, 20%), 
It is easily verified that the tangents to either of these curves are included in the 
third congruency of bitangents to the surface, and that accordingly the curves will 
satisfy the differential equations to that congruency. 
The cuspidal curve on this surface has the property discussed above (§ 36), and it 
is for this reason that its tangents are included among the bitangents to the surface. 
/ 
Example IV.—Inflexional Congruencies. (S$ 39 to 50.) 
§ 39. When the two values of dc, : de, given by the equation db,dc, — de,db, = 0 
(§ 28) coincide identically, the lines of the congruency are inflexional tangents to a 
surface. 
For if the direction-cosines of the normal to a surface at the point (x, y), y2) are 
proportional to /, m, n, the directions of the inflexional tangents are given by the 
equation 
dadl + dy,dm + dy,dn = 0. 
Hence, from the last equation of § 29, we find that the directions of the inflexional 
tangents in that case are given by 
dddy, = dx (de, + dde, + cad). 
But the equation 


ob 0b, |= 0 
| OO; 2 
a 0c, ’ Oe, | 
ob ob 
| ee 2 
| agg ae inane 
has equal roots, so that 
ob Ob, 
Be trae Se 
Yes? (5, i Zl 
Oa ( 0b, 2) 
0c, 2 \ de, 0c, |’ 
= (de, + Adcz). 
1 
Thus 
Ob, 
dy, — eda = \ (de, + ddc,), 
1 
