536 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
Then by eliminating p, and p, from the four equations 
1+ Ap, + B, p, = 9, a+ Ay, + By + C; = 0, 
1+ A,p, + By p, = 0, a+ Ay, + By, + Cs = 0, 
we have two relations which may be put in the Clairaut form 
Y, = Pye + $y (Pu Po); 
Ya, = Pot + hy (Pr; P2)- 
§ 21. The complete primitive of these equations will represent the trace of any one 
osculating plane on any other. The singular solutions are given by 
A py 
dz UO: 

/ , d / 7 / 
(Ay, + Biy. + Ch) =e = 0, (A’oy, + Bioys + C's) 
For the first singular solution, either , or py is to be equated to a constant, while 
the other satisfies the equation 
A'y, + By, + C = 0. 
The first singular solution is, therefore, the trace on any osculating plane of the 
torse which they all envelope. 
For the second we must suppose both p, and p, to satisfy the equation 
A’'y, + By, + C= 0. 
This gives the original curve, which is the cuspidal edge of the torse, together with 
the nodal curve of the torse. The latter is not a solution of the differential equations. 
Example II. A Bitangential Congruency. (S§ 22-26.) 
§ 22. Take for another example the equations 
(A+Bp/+Cp,*) {B(y—pir) +C (v.—pox)?—1} = {Bp, (y,— p12) + Cys (yo— por) }*, 
(a+bp,+ep.*) {0 (y:— pix) +e (y2—poe)?—1} = {bp, (y,—p,#) + eps (Yy— pox) }?. 
These define y, — pv, and y, — px as functions of p, and pg, and are, therefore, of 
the Clairaut form, and may be integrated by substituting ¢, for p, and ¢, for p,. 
The integral in this form will represent four lines of a congruency, which consists 
in fact of all the common tangents to the two quadric surfaces 
