OF SIMULTANHOUS ORDINARY DIFFERENTIAL EQUATIONS. 565 
The result of eliminating « is 
ce da® + 3(cda + ade) { (1 + 2c)eda + (3 + 4c) adc} = 
The complete primitive of this is 
a — bacra — bac? = 0, 
and for the singular solution of it we must take 
a= 3ac? = — 2ac, 
The solutions a = 0, c= 0 arise by giving « the particular value zero, and the 
true singular solution is 
2+ 8¢ = 0. 
Hence for the first singular solution 
a? — 6acta — 6a7%c? = 0, 
acy — 307% + 80y — 1 = 0. 
For the second 
ye 
This is the equation to a parabola. 
The curves of the first singular system are hyperbolas, having their asymptotes 
parallel to the axes of coordinates and having contact of the second order with this 
parabola. 
The complete primitive represents a series of parabolas with axes parallel to 
the axis of y, and each having contact of the second order with some one of the 
hyperbolas, and, in fact, with two of them, since « is given in terms of a and ¢ by a 
quadratic equation. 
§ 57. In order to make up further examples we only need to take :— 
(1.) A curve A. 
(2.) A series of curves A,, depending on one parameter, each having contact 
of order 7 with the curve A. 
(3.) A series of curves A,, involving two parameters, each having contact of 
order 7 with some one of the curves A,, and so on, till we have a series of curves A,, 
involving 1 parameters, each having contact of order n with some one of the 
curves A,_). 
Then A, is the complete primitive of a differential equation of order n, A,,_, will 
be a first singular solution, A,_, an 7" singular solution. 
