570 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



rf2 . e'^' 





_^2gmi 



T— = fe"' ^dx = m~ ^ e 

 dx~^ 



&c. &c. 



■ ^ onf f"^ ' 



mr e 



dx^ 



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



d/'e'^' w X ™- ., .„ , n .. P , d" di - 



Or e V 



that z=f(m)e'"'; then, if ^ be a fi*action — , we must have 



rfa/' ? 



^ p_ 



dx' dx" 



the symbol of differentiation being repeated q times, equal to /{mJJ^e"". 

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



hence f(m)\^=mF 



f{m)\ = m^ 

 p_ 



= ^?^ . e 



p_ 

 dx'' 



d^ e'" -^ 



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



dx'^ 



Hence, if any function of a; 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 : 



d'' r d 



•fi.+ 1 



. u — — . U 



dxf- dxf d^^' 



(T , , df" u df- I, 



[u + v) = 1- 



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 



and 





« = 2Ae»'--^ 



"• 



dx' 





dx' 





</^+'. u 



dx 



« + V 



