;»• 



PROFESSOK KELLAND ON GENERAL DIFFERENTIATION. 571 



For the second proposition, let v=2Be"^ 

 and 



dxT dz'' 



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

 in the form 2 A e""^ : 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 by reversing M. Liouville's process, are Mr Great- 

 heed's. 



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

 lowing equations are satisfied by it : 



i '- — + - 



rf'' d' di'^'u 



dx^ dx' dxi"" 



Now let t/=f"" , 



dy 

 dx 



... (1) 

 ... (2) 



and 



a"' di/ df^e"" 

 ■ T- - "* 



dz''^^ dx^ 



or -r ■ — - =m —^ ido.l 



''^ rfr^ di" 



hence, by integraticm, 



dxT 



V P 



Now if „-^ <^' d'" ...toyterma (.,,„ 



dxf dxf ... to q terms 



p_ 

 or . mf =C^ , C=»8». 



— nfi r'"' 



ds^ 

 VOL. XIV. PART II. 5 t 



