92 Mr murphy, ON THE RESOLUTION OF 



in which x is to be integer or fractional, positive or negative, real or 

 imaginary : so anomalous have they appeared as to induce a belief in 

 some, that they did not admit of an algebraical expansion, and there- 

 fore might be supposed to affect some of the first principles of the 

 Diiferential Calculus. 



In fact, the application of Maclaurin's Theorem requires the know- 

 ledge of the differential coefficients, which can only be deduced a priori, 

 in forms which leave them still unknown, while the application of 

 Taylor's Theorem in Finite Differences introduces impracticable coefficients 

 of a nature more complicated to value than the proposed functions 

 themselves. 



As an illustration, suppose we denote by «^ the successive function, 



^j; times) 



6 



6 



€ being the base of Napier's Logarithms, then, to find its differential 

 coefficient, we have the equation 



CIH 



Ur+i — e"', and putting ~j-^ = w',, 



U 



therefore -j-^ = e"-' ■ 



M\r-l 



To solve which, put u. = e's 



b. — h-, — «,-,; 



therefore if x should be an integer, 



b. = const + Mo + Ml + «2 + «.-i ; 



or more generally, 



b, = ^.ii,; 



■ dx~^ ' 



