g02 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



1.2 ' a'^ 



in + m 



= (-!)». 



f/" 



: n" . 



\ ax ) 



In + m a" 



1 



1 \ "+>» 



which is the proposition to be proved. 



28. The example given in the last article wUl furnish us with a ready means 

 of exemplifying a theorem analogous to that of Maclaurin ; for the coefficient 



of X-" in the expansion of -„ in tei-ms of negative powers of x, is, accord- 



ing to that theorem, supposing it extended to the case before us : {x being = 0) 



(1 + « xY \ which is equal to 



/-«+ 



/ — » + »» a~" i 



when a' = 



l—n + m or" In ^ 



-^ '-> [^ (l + «a-)-»+'»(-l)-" /O ' 



l—n + m a"" In 



For instance, if n-^m; since /O == co , the coefficient is zero. 

 If n=m, the coefficient is 



/ O" or-" I'm 

 Im /O" 



If n = m + r, it is 



/^ /o" ' ' ' ' /o 



Now, [^ — -1 /-T = (-1/ /r^ /^r if r be a whole niunber : in this case 



the coefficient is 



C_1V "«(» + !) ••• ('» + »'-l) 1 

 ^ r72 T^ r „»' + '•■ 



But if r be a fraction, /^- is finite and /o~ infinite, therefore the coefficient 



is zero ; results which are all obviously correct. With one more example we 



shall conclude the present memoir. 



