Mr. 0. Heaviside. [Feb. 2, 



cali-nlating the value of the function. It is, however, the series for 

 praciical use when at is big enough to make calculation by the con- 

 vergent ascending series very lengthy. Stop when the convergency 

 of (31) ceases. The result will be too big. Leave out the last 

 counted term, and the result is too small. Counting only half the 

 last convergent term, the result is nearly right, being a little too 

 big. There seems no possible way of hitting the exact value. But 

 still, when at is big, we can get quite close enough, to four or 

 more figures, or any other number we please when at is sufficiently 

 increased. 



Fractional Differentiation. 



15. Knowing in the above manner p$, the values of _pf, pi, &c., follow 

 by complete differentiations. But although, on the basis of the above, 

 a considerable amount of work may be done, and extensions made, 

 yet it is desirable to stop for a moment. For the whole question of 

 generalized differentiation is raised. The operator pk presents itself 

 in analogous problems, along with >T, &c. We want a general method 

 of treating p n , when n is not confined to be integral. Notice, however, 

 in passing a remarkable peculiarity of the above investigation. If 

 we had put L = in (17), as well as K = 0, we should have had the 

 form Y =pv to consider at the beginning, with no evident means of 

 treating it. By taking, on the other hand, a more general case, as we 

 did, we avoided the fractional differentiation altogether, and easily 

 obtained a convergent solution, viz., (21), through (18), (19), and 

 (20). It is not always that we simplify by generalizing. 



The sum total of the whole information contained in my mathe- 

 matical library on the subject of generalized differentiation is con- 

 tained in the remark made on p. 197 of the second part of Thomson 

 and Tait's 'Natural Philosophy,' paragraph (%), relating to the 

 process by which spherical harmonics of any degree may be derived 

 from the reciprocal of a distance : " The investigation of this gene- 

 ralized differentiation presents difficulties which are confined to the 

 evaluation of P,, and which have formed the subject of interesting 

 mathematical investigations by Liouville, Gregory, Kelland, and 

 others." 



I was somewhat struck with this remark when I first read it, in 

 trying to plough my way through the fertile though rather heavy field 

 of Thomson and Tait, but as the subject was no sooner mentioned than 

 it was dropped, it passed out of mind. Nor did/the absence of any 

 reference to the subject in other mathematical works, and in papers 

 concerning mathematical physics generally, tend to preserve my re- 

 collection of the remark. Only when the subject was forced upon my 

 attention in the above manner did I begins/investigate it, and not 

 having access to the authorities quoted, I was compelled to work it out 



