SOLUTIONS OF DIFFERENTIAL EQUATIONS. 191 



15. To find, without the aid of the complete primitive, 

 whether a solution w = o of a differential equation of the 

 first order and of the nth. degree is singular or particular 

 is a problem the answer to which is not by any means 

 obvious, and, if it be solved, which is doubtful, by the 

 process given by Boole (see supp. vol. p. 30), the solution 

 is accomplished by a troublesome transformation, accom- 

 panied by a definite integral, which, by the bye, is often 

 impracticable. 



The following answer to this difficult problem may be 

 interesting. 



Reduce, by algebraic methods, the given differential 

 equation into its quartic factors, of which the following is 

 one, viz. 



where r, s are given functions of x, y. 



If the solution w=o be of the envelope species the 

 results of the substitution of (w) in (r) and (s) will be 

 equal. 



It will now be necessary to compare (58) with the first 

 factor in equation (n), paragraph 4, omitting the affixes ; 

 then 



/— dv dw ( /— dv dw\ , x 



W »di-di =r \ Ww ij-3j)> ■ ■ ■ (59) 



,— dv dw ( ,— dv dw\ ., . 



2VW-j T-^SllVW-y -7-);. • . (60) 



da? da? \ dy dy/ v ' 



from these equations 



, x dv . x d^w d^w ., x 



(•-') a = (r+#) TS—m ~jp ■ ■ (61) 



, x dv dVw , , x d*/w ,, x 



