84 The Rev. Robert Harley on 



up, and the transformation carried forward to a satisfactory 

 conclusion, by observing that, in conformity with the laws 

 of the factorial notation, [ — a]' b may be replaced by 



[a -If 

 as may be readily shown, thus : — 



[a]" = (a)(a-l)(a-2) . . . (a- 6 + 1), 



[a-b] c =(a-b)(a-b-l){a-b-2) . . . (a-b-c + 1), 



and therefore 



[af[a-b\c = [a]^ (1) 



an equation of identity which, we will assume, holds uni- 

 versally, that is to say, for all values of a, b, and c. 

 First, in (i), make b=o ; then 



[a]°[«] c = [a] c , or[a]°=l. 

 Next, in (i), write — b for c ; then 



[af[a^]- 6 = [«]°, =1; 

 changing the signs of a and b, 



[-a]'\b -«]*=1: 

 or, since 



[6-a] 6 =(6-a)(6-a-l)(6-a-2) . . . (-a + 1), 



= (-lf[a-lf, 

 therefore 



r_ a -i-& = _lzJJL . ... (2). 



L J [a- If V ; 



Applying the formula (2) to the differential equation (jo), 

 we have 



[-(„< + i)D' + i]-«= [( ^c; rw 



and, therefore, 



/ _ I w+i 

 r _ (n'jy + i)Y n ' +1 = v -' 



(»' + !)"' 



[{n'+l)D'-2f +lV ~[>iD']"+S l/ ~ 



