OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 557 
and 
19, 
0 ra) 0 
Ge tag, +%a,) 4 
is a perfect square, so that the first equation for c, and c, is nugatory, and the second 
may be written in any of the forms 
as Od Og 
ae oF aP 





Ox OY, 2 Oy OY me 
Ob Od op 
Ox OY), aca oy? T Oy, OY, = 
og Ox) Op 



Bet, © layer, 2 ae 
The equation (9) will therefore not contain the differential coefficients of ¢, and co, 
but will be available to determine c, and c¢, themselves. 
If there is a cuspidal edge these things hold at every point of it. Let m,, m, be 
the values of p,, p, taken along the edge, and let us write D for the operator 
6) 0 6 
an +m, an + Tivos . 
Also we may put 
BS Bea db ae) she Coes, 
fe? Beg ED Gee OED Gy LP aaey, Ee Bs 

The equations giving the inflexional tangents are then 
LF py P+ BoP. = 0. 
= 


mt BPA =o + 3py = gis Big -eariaiate OD e anamians te OL 2. anny 
Op 3 Op 
ee 
og od 
ae Die Oy, 33 3p? ay + 3p py" 
We shall show that these two equations will, if p,, p, are considered as coordinates, 
represent a plane cubic and one of its inflexional tangents. 
We have 

ee EDN eee 
a2 
Thus, 

Caray Op 5 Op apie 
fe Hee a 2D Ee 4+- m,? Des 5+ 2mm, Tae. + m,° D 
= 2(A + ym, + sts) (Dd + m7, Dp, + mz Dy) 
As 
