1893.] On Operators in Physical Mathematics. 517 



myself. I cannot say that my results are quite the same, though there 

 must, I think, be a general likeness. I can, however, say that it is a very 

 interesting subject, and deserves to be treated in works on the Integral 

 Calculus, not merely as a matter concerning differentiation, but 

 because it casts light upon mathematical theory generally, even upon 

 the elements thereof. And as regards the following brief sketch, 

 however imperfect it may be, it has at least the recommendation of 

 having been worked out in a mind uncontaminated by the prejudices 

 engendered by prior knowledge acquired at second hand. I do not 

 say it is the better for that, however. 



Differentiation Generalized. 



16. The question is, what is the meaning of y"j if V signify d/dx, 

 when n has any value ? This is, no doubt, partly a matter of con 

 vention ; but apart from all conventions, there must be fundamental 

 laws involved. Now observe that the effect of a whole differentiation 

 V upon the function x n is to lower the degree by unity. This applies 

 universally when n is not integral. When it is integral, there seem 

 to be exceptions. But we can scarcely suppose that there is a real 

 breach of continuity in the property. We also observe that a whole 

 differentiation y multiplies by the index, making v# M = nx n ~ l ; and 

 again there are apparent exceptions. Now the first thing to do is to 

 get rid of the exceptions. Next, the obvious conclusion from one v 

 lowering the index by unity, v 2 by two, and so on, is that v w lowers 

 the degree n times, whether n be integral or fractional. Further, since 



when n is positively integral, and \n is the factorial function 

 1 . 2 . 3 . . . . n ; and, similarly, 



V"j = I, (33) 



whatever positive integer n may be, it is in agreement with the 

 previous to define generalized differentiation by the last equation, for 

 all values of n, provided we simultaneously define \n_ to be given by 



M, = n|n 1, (34) 



for all values of n from -co to -f oo, and to agree with the factorial 

 function when n is integral, that is, [1_ = 1> |2 = 1 . 2, |3 = 1 . 2 . 3, 

 &c. We shall still call \n the factorial function, and (|^) -1 the inverse 

 factorial. 



2 M 2 



