86 Dr. James Bottomley on a • 



To integrate (57) by the proposed method assume the 

 equation 



4 = ^V - 1 ; 

 then changing the variable (57) becomes 



d'^v _ d^v 

 Wdf' 



of which the integral as obtained by the method is 



^ = <pi{y + + </'5(y - 1) ; 



or by substitution, 



^2-^2-</'4(j' + ^V -\)^<p,{y-x^/ -\). . (58) 

 Hence by the addition of (58) and (26) we obtain, 



2^ = f.(y + ■«) + h{y - ^ + Uy + ^- ^^ - 1) + <po{y - «v/ - i;, 



and by substraction of (58) from (26) we obtain 



Thus we have obtained two integrable equations ; if 



m these w 

 the equation 



from these we obtain -^ and -7^, then these values along with 



dz ^ dz . 

 dz = --T-dx + ~i-dy, 



will give on integration equation (32). 



In previous examples before applying the method, a 

 change of one of the variables has occasionally been made ; 

 the method, however, admits of some modification which 

 renders such change unnecessary. For example consider 

 the equation 



