SOLUTIONS OF DIFFERENTIAL EQUATIONS. 187 



Differentiate (42) with respect to (w), then 



J w w-h (w-hy 



substitute the values o£f(iv) andf"(w) in (41), then 



dp = ff(w) _^\ / dw _ dw\ 



dy \<l>(w)' r h-w)\ dy dx)' ' ' { * 6) 



When, however, w=h, which is the case for a singular 



solution,, then the equation (43) becomes infinite on the 



-, , .-. , dp . n ., .,. f dw dw\ 

 dexter side; hence -j-= infinity, providing (r-^ -=— I 



does not vanish. 



dt) 

 The condition, therefore, which satisfies -£-= infinity, 



and also satisfies the differential equation (39), may be a 

 singular solution, and will be if it is not contained in the 

 primitive by giving a constant value to (c). 



It is not difficult to show that the complete primitive of 

 (39), in terms of its integrating factor/'^), is 



c+§f\w)dy+^rf{w)dx-§ d ^dydx=o. . (43') 



13. The solution of the third question is more difficult 

 still than either the first or second. It will be seen by 

 what follows that Boole's solution to this question is not 

 regarded as altogether satisfactory (see supp. vol. pp. 28, 



29,30)- 



The assertion by Boole that F(a?, o) is a constant, or 



rather F(#, o) is not a function of x, for a particular solu- 

 tion is not universally correct. The theorem, therefore, 

 in problem (10), p. 28, supp. vol., is faulty in proof; it is 

 also difficult in its application and uncertain in its utter- 

 ance. 



Observe the following example : — 



o2 



