Method of Integration. 83 



or assuming 



z.^'^^f^, . . . (41) 



wc obtain 



(^)-(4)i ■ ■ ■'-> 



Therefore an equation of the order 2"+^ has been reduced 

 to one of the order 2". Then by addition to the last 

 equation of the auxihary equation 



we obtain an equation which may be written in the form 



or assumnig 



_d^''-\ ^ d^'-'z, 

 '''dx^""'^ dy'^'-'' 



we may write 



^/2""., 



(45) 



(46) 



Therefore by the apph'cation of the method we have 

 reduced an equation of the order 2""'"^ to one of the order 

 2"-', It is evident that by the aid of the successive 

 auxih'ary equations we shall finally obtain an equation of 

 the first order of the form 



dx dy 

 of which the integral is 



-2»+i = «/'i('*^ + y) • • • • (48) 



By substitution we should get another equation of the first 



order of the form, 



dz.n dz 



^4-^=./n(a; + 2/), . . .(49) 

 dx dy 



