570 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



rf2 . e"^ 



dx^ 



—m'e"' 



— =r fe""(lx — m--^ e^' 



&c. &c. 



whenever /z is a positive or negative integer. Let us retain the general form, viz. 



that — — = /('«) «'"''; then, if fi he a h-action — , we must have — — — j e"", 



the symbol of differentiation being repeated q times, equal to /(»«)/' e"*. 



But the result is also 711" e""" , since, by repeating —j q times, we get j^^ ■■ 



dx" 



hence f(m)\''=m>' 



f{m)\ = m^ 

 p_ 



cf . e"" _Ji 

 p 

 dx^ 



,m z 



m' . i 

 p 



d'^ e"" 



and by the usual extension, we obtain — '- =»»''. e"" whatever /^ may be. 



dx^ 



Hence, if any function o/Lx can be expanded in terms of e""' &c., we can find 

 its general differential coefficient. 



2. From the above proposition, we deduce the two following : 



jT d^ _Jlll_ 



(t , W u d>' „ 



. (11 + v) = + • 



dx^ dx^ dx" 



The former of these propositions was requisite for the demonstration above : we 



may, however, assume the result of the last article as the definition. In this case 



we can prove the formula, before us thus : 



Let «=2A«""- 



and 



dx 



