128 Mr. 0. Heaviside. [June 15, 



i./M = /frzi) (93} 



/(*) /(-!)' 

 and from this we may derive, when r is a positive integer, 



'-$$-+*+-.+&%$ 



and also 



Therefore, when r is positively integral, we have 

 /('+*)_ /(-'-i*) 



which makes the coemcient of x in (92) become 



where, for brevity, F stands for /'( i)//( J). It is readily seen 

 that the complete coemcient of F vanishes, and the remainder reduces 



- (97) 



Therefore the coemcient of x in (92) contributes one-half of the total, 

 and the other half must be given by (or rather, be equivalent to) the 

 sum of the terms involving x. Although I have not thoroughly 

 investigated this, there did not appear to be any inconsistency. 



Remarks on Equivalences in Factorial Formulce. Verifications. 

 53. If it is given that 



F(s) = 2^0(0, (98) 



it does not, as already remarked in effect, follow that 0(r) is a 

 definitely unique function of r. But it is sometimes true, and then 

 the equation 



(99) 



may require the vanishing of every coemcient. For example, using 

 (88) above, if we differentiate it to x we obtain 



(100; 



