PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 571 



For the second proposition, let v=iBe"* 



u + v = '2.Ae""' + lBe"' 

 and Jl^J^±lL=Im'' Ae-^' +ln'' Be"^ 



d'^ u of V 



= + . 



It must be observed that all functions are supposed to be susceptible of expansion 

 in the form 2 Ae'"'^ : with the correctness of this assumption we have no concern, 

 provided we limit our results if the assumption is not correct. 



3. The demonstrations above exhibited are due to M. Liouville. The fol- 

 lowing, which are deduced hj reversing M. Liouville's process, are Mr Great- 

 heed's. 



A general differential coeflficient is defined to be such a function that the fol- 

 lowing equations are satisfied by it : 



d'^{u-{-v) _ d''ii d'^v /-j^^ 

 rfa;" dx^ rfx" 



d'' rf' df' + 'u 



u = 



dz^ dx' dx''"' 



Now let .'/=^ 



(2) 



W X 



dy 

 dx 



df" dy di^e 



and . -^ = w 



m X 



dx^ ^^ dxT 



df' + '^y di'y ,, ^, 



or ■ , = m — - ( by 2) 



dx^^' d^ ' • 



d di'y d^^y ,, , 



or V • — ^ —in — — i^do.i 



dx da^ dx^ 



hence, by integration, 



dxf' 



p p 



Now if ^=4 , ^^^^'^- toy terms ^^^^,^„ 



9 



dxfl dx^ ...to q terms 



p 



or . mP —G" , C = »j'. 





dxf" 

 VOL. XIV. PART II. 5 t 



