PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 243 



i—l) ^^:^.l;then 



'^■S' / ^2^ + 1 



^Z' + l 



Now — —J 1:=0, except when ^ is a negative whole number; in which case 



d z 



.-. y = G / ; except when // is a negative whole number, in which case 



1/ = C e' , , — — &c. 



Now, m aZZ cases we omit the arbitrary functions in differentiation to any 



index ; they being readily supplied when required. But -, + &c., is evi- 



dently included in the arbitrary function, in the case in question ; we may there- 

 fore omit it, and write generally, 



y = C e-', or 



-^-i- =c^Ce- = c^Ce" . . . . (1) 



This result has been deduced from the definition without any assumption 

 whatever relative to the function T, except that it satisfies the condition /«+I=w/w. 

 We may, consequently, obtain the value of the constant C, by admitting, that 

 when n is positive, In coincides with Legendre's function j . In this case, 



tL=r e-'-'W-^da. 

 z" J ■' 



Therefore, differentiating, to the index //, 

 '^^=c/a''+"-^e"-"''rfa, by the definition and equation (.1). 



But if n + fjihe positive, jnTJi also coincides with Legendre's function, there- 

 fore, 



-^^^=/^-«''a''+f'-irfa,orC = l. 



