(jOO PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



rf-3 1 



L=(_l)-3/_2(^-fl)2=_-^Gr-a)2 





dx"^ x — a 



hence the coefficient of h-^ is x^—2x{x—a)-\-{x—af. 



The other coefficients may be found in a similar manner. It is remarkable 

 that our formula gives us not only the correct results, but, further, it gives the 



(x + hY 

 order in which the different parts occur. If we expand -^^ — r-^^ in negative 



X — it "T €t 



powers of h, we get not merely c^ as the coefficient of h~^, but the very terms 

 .T^-2x{x — a)-{-{x—af. 



Let us proceed to find the value of the coefficient of h" . 

 1 



d 



x — a 



d x" x — a 



d"^— „ d-'-^ . . 2d'' 



x — a _ or ^ Q 2 J. ^~" 0(0-1) x — a 



dx" x — a dx~^ 2 d x"' 



^ +2*. 0/(0)-0(a7-«)/(0) 



x — a 



x — a 



^ + 2«/(l)-(a;-n)/(l) 



..i 



xf 



and /(I) ^-j^o — 1 



^ 



therefore the coefficient of If is +2x -x-a. 



x — a 



(x + hf 



Now we have two expansions of — . - involving a term not containing 



h : we have of course obtained the sum of them by our process of expanding the 

 term ; consequently that coefficient which we seek is the diiference between this 

 quantity and the other coefficient, or u. 



Therefore, the coefficient of h' in the expansion which involves negative 

 powers of // is, 



\daf ) f(l) ^ ^ 



To find the coefficient of h\ 



d^ „ ^ d-^ 



x — a X^ r. 1 n ^~" 



+ 2x- — -_ +0 ., ,^_i 



dx'^ {x — aY x — a dx' 



