PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



But further. 



599 



rfS 



■s/x — i 



d^-Vx-a 3 d^Vx- 



dx'i 



dx 



d X'- 



= . (-1)- l\ l\ 



IT 



|/(l) 



Hence, dividing this by/(f ) we get as the coefficient of A^ , ^ y\ 



/(I) 



= 1. 



In both cases our results are obviously correct. 



Lastly, to obtain the coefficient of -7^ 



V h 



d~^ x'^x—i 



xd-^ 



dx 



— i 



dx 



d i\/x — a 

 dx~- 



=<-l) 



and 



(l)M-i. 



1 1 , /^ 1 



siinr ^ ^ 2 ^ ^ /g sm2'7r 



— i '2 



(^-a>' 



1 sin TT 



rf^'^w 



/r 



x(x—a) 1 



--(^-a>'. 



.^^•^ /(-I) 2 



In the same way we might find the values of the other coefficients. 



A very natural question to be asked now is this, What is the coefficient of h 

 or of k^ in this expansion ? Should we proceed to the determination of that coeffi- 

 cient, we might expect to find zei-o as the result ; but a little consideration wUl 

 convince us that such a conclusion would be ill founded. In fact, we here de- 

 termine each coefficient independently of its connection with the others, or of its 

 connection with the actual expansion. Now (x + h) \/ x—a + h may be expanded in 



terms of positive integral powers of h ; consequently the value of -r— is the coeffi- 

 cient of h in this expansion, and not in the expansion above, which does not con- 

 tain such powers of h. 



We confess the subject labours under a slight difficulty in one or two points, 

 to which we shall call attention presently. 



Ex. 2. To expand 



by the theorem. 



rfar-^ x — a 

 VOL. XIV. PART II. 



r.;-:;=/(0)=(-l)-^ -/o 



= (-l)-V^ {x-n) = - /(T. (.r-fl!)= +(j7-«i/(0) 



d ar~^ • x — a 

 rf-- 1 



5q 



