PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 293 



Ex.5. A" ^— =(-!)" f-i + &C.] 



= (-1)"a/— ; \Saf = a + x 

 Ex. 6. To find a^^. 



By formula (2), ^*| = ^^{i - ^ " H^-&c.} 

 Let ''=?-|£^-&- 



then f^ =^^-^(1-1^-2-71-^^ «=•=•) 



and A*- = ^A^l/' y^-i(l-^)*dy 



Ex. 7. To/wc? A" -• 



A»U(-l)"fi-^ + ^^) 1 -&c.| 

 iT ^ ^ la? z + 1 1.2 ir + 2 J 



Let «=r_^^i;! + &c." 



then g=^-i(l_^)" 



and A''U(_i)"/'V-^a-yr«fy 



^ ' ' ^ / V. / jz + n + l 



Cor. 1. If n be a whole number, jz + n+i =x {x + V) ... (« + «) /J 



A-i^(_ir ^•^-'' - 



X ^ ' x{x-^\) ... (a + «) 



CoE.2. Ifn4^*U^/3l £I=i^AIT^/^-4 

 Ex. 8. To find, a" ^, m fteiri^ awy integer. 

 a" - =(-1)" f - - « r^ + ^fe^ rA^ -^c. ) by formula (2). 



^.m V. -^; (_^m (i; + l)™ 1.2 (a; + 2)™ J "^ ^ ' 



VOL. XVI. PART III. 4 E 



