OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 535 
From these and the former equations we find 
—l, —17?2 — 2 — 1,2 
[Pa eH oy == oes av; Cy a ee 
ae SS GD oO 
Th = tae ae 
Ch, ores 
The equations (7) are the second singular solution. 
We have here an example of what was pointed out in § 13. If we differentiate the 
equations () and eliminate y, and py, the resulting differential equation of the second 
order is 
d*y,/dx? = 0. 
This is satisfied by the complete primitive (8), but not by either of the singular 
solutions. 
§ 18. To find the singular solutions from the original equations, we have first to try 
if the equation 
x+2 Do, L 
ay a 22 ame 8G) 
Pe a 2, - Py 
can be used as a first integral. 
The elimination of p, and p, gives 
4(y, + Yo)? + 2? (yy + Yo)? + 4aey, + 18ay, (yi + Y2) — 27y°=0. . (y). 
It is easily verified that the derivative of this equation is an algebraical consequence 
of it in virtue of the original differential equations. 
§ 19. If we take «, y,, y, as Cartesian co-ordinates, the equations (8) represent a 
line common to two osculating planes of the twisted cubic (7), (¢) represent the conic 
enveloped by this line when one of the osculating planes is fixed and the other 
variable, and (u) is the equation to the torse enveloped by the oscuilating planes. 
The General Case. (§§ 20, 21.) 
§ 20. For any other twisted curve there is a similar theory. 
Let 
z+ Ay,+ By,.4 C=0 
be the equation to an osculating plane, A, B, C being functions of a parameter yp. 
Let A,, B,, C, denote the same functions of y,, and let A’, be written for dA,/du,. 
