SOLUTIONS OF DIFFERENTIAL EQUATIONS. 181 



which is not a solution of (24), therefore the principle of 

 equal roots applied to the complete primitive fails also to 

 give the singular solution. 



The elimination of (c) from (27) by the condition 



-j- = o and -7- = o leads to the equation (28). The usual 



methods, therefore, of finding the singular solution from 

 the complete primitive fail in this example. 

 The primitive (27) admits of the form 



(c+ Vx + y—x) (c + y — x) = o, 

 from which we have the two equations 



c+ \/x + y = x, 

 c + y = x. 



The first represents a series of parabolic curves, and the 

 second a series of straight lines to which the singular 

 solution x-\-y = o is perpendicular. 

 Let 



\AJiAj vis • iAj 



x sin - 



y 



Required the singular solution if there is one. 



This example is given by Sir James Cockle, F.R.S. &c. 

 (See ' Quarterly Journal of Mathematics/ No. 54, 1876.) 



Its complete primitive is 



x 

 c + x — cos-=o (30) 



Compare this equation with (22), then v = x, z=—i, and 

 f(w) =cos-. Therefore /' (w) = — sin -, which does not 



y y 



become infinite for any value of x or y. 



The equation x = o satisfies (29), but it is a particular 



