Of SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 555 
and either 
eM OM 
B= kX OV Anan oo 8 Angi ay, 12 = 0 
Also (8) becomes 
oM M 2M M OM 
(u — 2) {array es + seep, 2 ca} +2 {5 ay Oe + aan Cet + 5a de =O. 
§ 43. First, let » =a. Then 
M = ¢(, y, Y2) = 0 




OM _ 96), oH Ob 
au ae Tay, + Gy, = 
oM o . oF een ee ee Ob 
One — ax? + 2C) Ox Oy, + Cy 0x OY + Cy Oy? ar 2C)€5 Oy), OY» + Cy Oya” rie 0, 
OM de, ti oM de, Ue Op de, , 0 de, ne 
Ob, dx | 0b, dz ~\ Oy, dz " Oy, dz 

The last equation and the equation (8) are satisfied in virtue of the first three and 
the equations p, = ¢), Py = Cp. 
The integral of these equations represents a series of curves on the surface ¢ = 0, 
each tangent to each curve being an inflexional tangent to the surface. 
Second Singular Solutions. (SS 44, 45.) 
§ 44. If there is a singular solution of these equations, it is given by supposing two 
consecutive curves of the series to intersect. At their point of intersection the 
inflexional tangents will then be in the same direction, and the singular solution 
therefore appears to be the locus of parabolic points on the surface. We shall, 
however, find that this curve is not a solution at all in general. The consideration of 
the reciprocal surface suggests that a cuspidal curve may supply a solution. We 
begin with the parabolic curve. 
We take, as a trial solution, 


Od , o> Op _ ob 
Ox? eG Ox OY, we Ox Oy, Oe? 
Op op Op _ op 



Ox OY; mi Oy TiS 2 ein | Gyn 
Od ae ep os 
gues 1 Oy, Oya ate Oy? “Ge 
4B2 




