268 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



n + i 



4_7r-w(4-2'7r)-2\/'7r'^^ = 



which is satisfied by n=i and «=1. 



Hence 



A B 



CoK. 2. If n be a whole number ?■; /n=l . 2 . . . (/-l) 

 and rn + i—^ . f . . . (»•— J) x/TT 

 1.2... (r-l) + a a/^T Vtt 1 ■ 3 • • ^(2 /"- 1)_^^ . 1 . 2 . . . r = 0; 



will determine the integral values of n. 



If «=>-+i, jVi :^ i-3-- -(2^-1) ^-^ ;;m=i . 2 . . . ;■ 



and 



1.3...(2r-l) /- /— r^ T ., , 1 . 3. .. (2»- + l) /- . 

 V J ^TT + aV -1 . 1 . 2 . . . r~l> i ^ VTT =0, 



Or Or+l 



2'' 2*' 



which determines the fractional values of n which have 2 as their denominator. 



Now it is evident that these are the only forms which n can assume ; there- 

 fore the determination of the values of n is reduced to the solution of these two 

 equations. 



Ex.4. y + aVcc i-+bx-r' = X. 



Let X=2 6, e~'\ then 



,_./_D+J- ^/-Dja 



-D 



D + l \ -1 



= 2 a „ e ' 



6, X' 



l + a-vZ-l' 



=r r=^ by (A) 



'-- h 



the values of n being determined as in Example 3. 



CoE. \i r=p, n=p, we obtain, as in other instances, 



6, X-' 



1 — aV— 1^ — 7= b — =— 



Is 



IS 



where 



Ex.5. 



C=- 



a^zn^lP^-, 



dx" 



- + ax 



dp jp 

 dx'"-i 



d'"y ,_»-irf!:iijL + &e. = X. 



