598 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



connexion with any of the others, it is evident that, when two or more expan- 

 sions involve the same power of h, we shall obtain all (if any) of them by the 

 above process. 



Consequently, should an expansion involve a power of h, which is also in- 

 volved in the ordinary expansion of Taylor's theorem, and in no other, we can 

 at once determine the value of the coefficient, by subtracting from the general 

 value of the corresponding differential coefficient, that particular value which is 

 o))tained by ordinary differentiation. Should it be demanded to explain how it 

 happens that the general differential coefficient is different from the ordinary one, 

 we should find some difficulty in answering the question. We say there would 

 be considerable difficulty in explaining the reason for this difference ; but to shew 

 how it arises is easy enough, as the examples Avhich foUow will evince. In fact, 

 the particiilar cases which form the basis of induction are Umited as to the num- 

 ber of their terms, whilst the general form (even when the general symbol has 

 been replaced by one of the particular numbers on which its existence depends) 

 is unlimited ; and although a series of the terms may be each zero, it may happen 

 that, under certain circumstances, results may be deducible from them. Zero 

 may, in fact, be a divisor of the form in the final state, and thus its appearance 

 may be chased away. 



2f). Ex. ]. Let u—x'\/x—a : to find the coefficients of A^, }fi, &c. in the ex- 

 pansion of u'. 



^ , „ , d" uv d" V du d"~^ V 



By the formula ^^ = " "^ + » ^ J^^ + ^«- 



weget ^^W^^^^V^U^l^^ 



= =»/( 9-) +-^(-1 ' J- . (x-a) 



but /(|)=(-l)i/L^(v iili;:±^ = -cos0=-lwhen0=o" 



\ 2 / I smdTT \ sin 



Hence the coefficient of h^ in the expansion of m' is x + -^(x-a). 

 But by actual expansion, we obtain 



(r.f/,^\/x-(i + /i = (x + h)J/i f 1 + 1.^^-1 (x-ay ^_ 1 



= ,= + {.,.4(.-a) } V7, + (-l:IZi^ 4.(.-.) ) -L. 

 which completely verifies our operations so far as they go. 



