1893.] On Operators in Physical Mathematics. 510 



where (40) corresponds to (38) and (41) to (39). At first sight, 

 therefore, these functions might represent the inverse factorial, 

 positive or negative, on making r infinite, for the value at the origin 

 is correct, and the vanishing points are equidistantly spaced with 

 unit step all the way to infinity on one side only of the origin. But 

 something else happens when r is made infinite. The value of (38), 

 by (40), becomes (l l) w , meaning the binomial expansion in rising- 

 powers of the second 1. It is, therefore, zero for all positive and 

 infinity for all negative values of n. Similarly, (39) .becomes (l l)~ w , 

 which is zero for all negative and infinity for all positive values 

 of n. That is, from vanishing at detached points, the functions vanish 

 all the way between them as well. Besides, apart from this, we cannot 

 have the value of \n correct when n is integral. 



We may, however, readily set the matter right. To get rid of the 

 infinity on one side and vanishing all over on the other side of the 

 origin, multiply the functions (38), (40) by r re% and (39), (41) by r~ n : 

 Take 



. v^ */ ~~ v ' ~J \ - ' o / \ """ ' Q / * " * * ' \* u ) 



I. *,\ / M \ 



(43) 



We now satisfy all the requirements of the case, and when r is infinite 

 make the inverse factorial curve (37) be a continuous curve from oo 

 to 4-co, subject to (34), in agreement with the known values when n 

 is positively integral, and' harmonizing with the generalized dif- 

 ferentiation in (33). 



Multiplying (42) and (43) together, we obtain: 



sin mr>' 



The multiplication therefore brings all the equidistant roots into play, 

 on both sides of the origin. 



This gives us the value of \n in terms of \n._ Only the values 

 of \n from n = to n = 1 need be calculated, since (34) or (44) gives- 

 all the rest. But if we take n = J in (44), we obtain 



11 hi = *(hl)' = *. 



therefore | j = **, (45) 



a fundamental result. We now know the value of 4-J when n is 

 any integer, and this brings the matter down to the determination 

 of \n from n = to n = -J, for which a formula may be used. 



