PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 599 



But further, 



d'^ x\/x — a d^\/x — a 3 d'^Vx — a 



: = X ; h 



dx'' dx- ^ dx'- 



TT x — a 



Hence, dividing this by/(|) we get as the coefficient of /i* , ^ \y =1 



/(I; 



In both cases our results are obviously correct. 

 Lastly, to obtain the coefficient of — ^=r • 



d'^'' xVx — a xd^'W x—a 1 d~iVx—a 



X i O • 2 



t^.x' - dx * ^ rfa? - 



/l 1 1 /I 1 



^^ '' /2 sinTT ^ ^ 2 ^ '^ /3 sin2'7r ^ '^ 



and / (y) = (-1)-*-^ ^^^7y^ 



d~^ w 1 irf-r — a) 1 . . 



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

 cient, we might expect to find zero as the result ; but a little consideration will 

 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 



(1 'ijf 



terms of positive integral powers of h ; consequently the value of -z- 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. 



X — a 

 d-^ 1 



d x~^ 'X — a 



=/(0)=(-i)-^ ./o 



= (_l)-2/3l(a7_«) = - /(F. (a7-a)= 4-(a;-a)/(0) 



dar^ x — a 

 VOL. XIV. PART II. 5 Q 



