(JOQ PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



&c. - &c. = + ^(-i-«)7(0) 



hence the coefficient of /s-^ is a-2-2a'(a'-a) + (a;-a7. 



The other coefficients may be found in a similar manner. It is remarl^able 

 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 in negative 



X ft "J" a 



powers of //, we get not merely a^ as the coefficient of Ir^, but the very terms 



.T^ — 2x{x — a) + {x — af. 



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



x — a I 

 d xf ~ x — a 



r- 1 1 



d x° x—a dar^ 2 d x-' 



aP 



x — a ^ , n o ^~" 0(0—1) x—a 



1 — 7. hU . Z X j 1 , 



/ r" X — a d x~ 2 d X—' 



+ 2x. 0/(0)-0(a^-«)/(0) 



+ 2^/(l)-(.r-a)/(l) 



x — a 



x"- 



x — a 



d'l- 



x° 



and /(I) --j^-^ 



therefore the coefficient of h° is ^3^ + 2x -x-a. 



(x + hY 



Now we have two expansions of ^^_ - involving a term not containing 



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

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

 quantity and the other coefficient, or u. 



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

 powers of // is, 



/d° u \ 1 o / \ 



I « I ^^7^ = 2x — (x—a). 



Kdx" ) f{\) "• 



To find the coefficient of /t'. 



r2 1 



x-a ofi o 1 , n -g-" 



— 7-^i — = —7 \i + 2x h , _■ — 



