116 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



It is in this case, as X C sin {V(&) x + a }> 



)a; + gn 

 J ' 



a sm {V (K) x + a} _ . x , ... , ^ sm { v (K) x - 



dk $ k $ 



therefore 



y = i^^i [&k* cos (V(&) x + a ) ~ sm (V(&) # + }]> 

 or integrating the first term by parts, 



d' 1 

 = JcC sin 



But 



sn > x + a = cos 



dk~ l *' ' ' ' x 



2 . . .... , 



-f - sm |v (A?) a; + a], 



/y* I * \ / J ' 



therefore, if kC^G^ 



y = C l [sin (V(&) ^ + a} 



3 3 



+ r cos y(k) x + a} - y-j sin {VW fl? + all, 

 ^ 2 A:a? 



which is the required solution. 



Equation (13) corresponds to (11) : but there is another form 

 of the solution in the case of p 1 = gin, which we shall just 

 mention, and which is the counterpart of (12). It is 



It would be a needless repetition to go through the steps 

 which lead to this result. 



All the operations indicated in these symbolical solutions are 

 practicable. This will appear by considering the nature of the 

 function JT, which, in its most general form, consists of the sum 

 of terms, of which the type is 



