m-r 

 n 



(xxx) 



(xxxi) 



190 071 Differential Equations 



28. Changing r into r — '^ we have, from (xxviii), 



m a — 1 • (1) 



_ L n J ^ ^ 



Hence, dividing (xxix) by (xxx), 



\ Tn + ar Y~^ 

 [rYUy [ n J 



But 



\ on + ar y-n-u^ 



\m + aT y-n-i l^^^n^ _J 



[^~n "~ J ~ [m+ar ^Y 



\ m + ccT Y'^ 



I n \ / -x 



= li— = (xxxi) 



\7n + a7' ^I'^rm + ar Y~" 



1 r + n — a 



L ^1 J L ^ J 



consequently 



u^.n in \ I n J 



or, making two obvious changes in the form of this result 



r -,^ \m + a7' J'^Tm — (n — a)T Y'"" a 



b'Y'^r—[-^ Ij [ ^— l-+n—a]^ %-n = 



(xxxii) 



This equation corresponds to Boole's equation (6) (ibid. p. 

 736) which requires correction in two places and should 

 stand thus : 



[r]" u„ + — ^ 1 I 1 u^_^ = (6) 



29. Multiply both sides of (xxxii) into x^ and it becomes 



\m + ar ^]«rm-(?^-a)r "|"-« „ 



[r]-iv.a3-— j^— Ij j^ ^---^_ + ^_aj . a;". u,.„. «;*-»= 



(xxxiii) 



