8o Dr. James Bottomley on a • 



the integral of which has been obtained ah'eady, so that it 

 is only necessary to make in that equation the substitution 

 indicated in (ii). 



As an example of the application of the method to an 

 equation of higher order than the second, consider the 

 equation, 



d^ z dh 



dx^ dy^' ' ' ' * V / 



If, then, we differentiate ^ four times, twice with regard to 

 X and twice with regard toj, since the result is independent 

 of the order of the operations, the subsidiary equation will be 



^^--^- . . -(14) 



hence, by addition, 



dh^ dh _ dh^ dh 



dx^ dx'^dy^ dy^ dyHx-' ' ' \ > 



But each side is now a complete differential, for we may 

 write the equation in the form 



d- r<^ ^\^d-rdh dh\ 

 ~dx^\j?^d^y~dy^\df'^d?J' • ■ ^ ^ 

 Now assume 



dh dh 



, = — •-{ ■ 



dx^ dy^ ■ 



(17) 



(18) 



then (i6) may be written in the form 



d^'v _ d^v 

 dx^ dy'^' 



Hence, by this method we have reduced an equation of the 

 fourth order to one of the second order, and by means of 

 the equation 



-^=— (19) 



dxdy dydx^ ^ ' 



equation (i8) may be presented in the form 



±fdv_j^dv\^£rd_v^'dv\ 



dx\dx dyj dy\dy dxP ' • \-' > 



