MetJiod of Integration. 

 and if we assume 



dv dv ,^^^ 



then (20) may be written in the form 



^ = ^. (09) 



dx d,f' ■ ■ ■ ■ V--; 



therefore, by this method we have succeeded in reducing 

 an equation of the fourth order to one of the first order ; 

 V will be a function of x and y ; therefore, following the 

 usual method, the integral of the last equation will be 



v = ^i{x + y) (23) 



or by substitution from (21) 



du dv , . „ , 



;^^7^ = 0i(^ + S'). • • • (24) 



Integrating a second time we obtain, 



V = <p,X^j + x) + (l>s(y - x) . . . (25) 



Substituting for v its value given by (17), the last equation 

 may be put in the form 



dh dh 



^2 = - ^^^2 + Hy + ■^) + h{y - *) • • (26) 



In order to continue the application of the method at this 

 stage a change of one of the variables must be made ; 

 assume the equation 



^ = v^^4 (27) 



By means of this equation we obtain 



dh_ d^ 



dx^- ~di^ • • • ■ ^'^^^ 



Substituting in (26) we obtain 



%=-%- h(y + ^' ~i) - <ps{y -^~U) . (29) 



The auxiliary equation will therefore be 



— -^ 



didy~ dydV ' ' ' ■ ^^^ 



