280 PROFESSOR KELLAND OX GENERAL DIFFERENTIATION. 



I ,-D /-D /-D J 



i' 



which beinff reduced gives 



2(D + l)(D + 2)L^^ + 2«.-^D(D + ])^™^.y 



or ~ (Tij-fswDj-A^ (Tij. t;^•3' + ''* 



D 



(D + 3)(D + 4)(D + 5)-^'^"'' (D + 2)(D+3)(D + 4)-^ 



'(I)T2f(DT3)^ = r'-^W«--' 



Now3/=/(a + ?), but since a is itself a function of -, we cannot proceed fur- 

 ther with the reduction of this equation by division, but must proceed to obtain a 

 relation between n and R or a and z. 



To do this we shall expand / (a + --j by Taylor's Theorem. • 

 The result is 



d" f(OL) '" 



j^=2 ~^-r 7==- which being substituted in the reduced equation, gives by (A) 

 rfa" /« + ! ° ^ ° ■' ^ ' 



(d"/(a) 



I 



/ 'f"/(a) e"' \ r 

 V da'' ;« + !/ I 



{n + 3) (w + 4) {n + 5) 



+ a 



(w + 2) (n + 3) (« + 4) ^ "^ (« + 2) (n + 3) ] ~ 3 -^ " " 



But a = a-R = a-^ : hence 



— -^ ^ ^ = ^ — i — ^ 4. &c 



rfa" (fa" rfa^ + i 2 



which being substituted for j *-^ ' , the sum being taken for n and ;i>, we get 



(-!} >■ d"+P fa 1 f _^«+p 



"'^ 2/' rfa"+?' /wTlF+T I (» + 37(w + 4) (>r+5) 



+ («+2)(w + 3j(» + 4) "*" (w + 2)(« + 3)J ~"3"? 

 every integer value of n and p being taken from to 00 . 

 When «— 0, p=0, the left-hand side gives 



—fa a^/a 



