82 Dr. James Bottomley on a « 



and proceeding as before we get the equation 



^2 dz 



■ ^, + (i^= 04(y + ^0 + 05(y + V - li) + fr,{!/ - J-n) . (31 ) 



Integrating this equation and substituting for ^ we obtain 



finally 



z = 07(y + x) + (/)«((/ -x) + (poii/ + sf - l.x) + 0io(y - J -l.x) (32) 



The above method may now be applied to the equation 



''^ -''■ . .,33) 



• (34) 



, . (35) 



. . (3G) 



the integration of which has just been effected ; hence the 

 integral of {t,t,) will be obtained by writing ax for x in (32). 

 Next, consider the equation of the order 2""''^ 



Qfidr ■ • ■<-) 



The auxiliary equation will be 



\Mfyh^\dydxP ' ■ • • (38) 

 by addition we obtain 



\dx'h + KdMlyJh = \dy')h + \dydx)'^- ■ (3^) 



but each side may now be written as a complete differential, 

 for it may be put in the form 



