- ls Mr. 0. Heaviside. [Feb. 2, 



No\v, by the above 



V ,f = ^ = 1 , ( 35 > 



if x be positive, as we shall suppose throughout. We conclude that 



|0=1. 



Further differentiations give 



We, therefore, conclude that ([-l)- 1 = 0, (h-2)- 1 = 0, &c., or that 

 the inverse factorial function vanishes for all integral negative values 

 of n. We therefore know the value of the inverse factorial for all 

 integral values of the variable, and a rough curve can be readily 

 drawn. Say 



y being the ordinate, and n the abscissa. It has evidently a hump 

 between n and n = 1,-is positive for all + values of n, asymptotic- 

 ally tending to the n axis as n is increased, and is oscillatory on the 

 other side of the origin. This is not demonstrative, but only highly 

 probable so far. 



The Inverse Factorial Function. 



17. Now seek an algebraical function with equidistantly spaced 

 roots on one side (either side) only of the origin. The function 



vanishes at n = 1, 2, 3, &c., up to r. It has no other roots, and is 

 positive when n is negative. Also, its value at n = is 1. Similarly 

 the function 



(39) 



vanishes at n= 1, 2, &c., up to r ; is unity at n = 0, and is 

 positive when n is positive. These functions are identically the 

 same as 



|2 |3 



and 



( Q} 



, .... - 



