THE 



PHILOSOPHICAL MAGAZINE 

 AND JOURNAL. 



SV MAY 1825. 



LIII. Ofi the Binomial Theorem^and the Application of somePro- 

 loerties of A'". o" to General Differentiation and Integration. 

 By John Herapath, Esq. 



FROM the time I first became acquainted with the fluxional 

 discoveries of Newton, it forcibly struck me that his cal- 

 culus was only a particular branch of a much more general 

 one. I felt, therefore, surprised that mathematicians had suf- 

 fered upwards of a century to pass without any attempt, I 

 might almost say without even a hint, towards its generaliza- 

 tion. They have given us methods by which we can generally, 

 by successive operations, obtain the value of any order de- 

 noted by an integer of the differentials or integrals of a func- 

 tion ; but to extract, if one may so call it, any differential root, 

 or even to assign the value of dl" . x^ when r is only a fraction, 

 to say nothing of its being an irrational or imaginary number, 

 seems to have been looked on as an impossibility. Lacroix, 

 I have heard, has said something of fractional differentials ia 

 liis great work on this calculus; but he considers them, it seems, 

 as cjuantities of a different kind fi'om integer differentials. 

 M. Brisson too in the Journal Poly technique, 14 cahier, page 



199, has given a series for — !^ whatever be the value of »; 



dx 



but the series is of a most troublesome description, and by 

 proceeding according to the integer differentials of ~~^ is, in 



my opinion, nearly if not quite useless. The development of 

 d^u may, if that only be sought, be obtained by a much sim- 

 pler and neater method. 



Mr. Babbage has likewise in one of his functional papers 

 incidentally alluded to fractional differentials in mentioning 

 M. Lacroix's opinion ; but he has not that I remember said 

 any tiling which might lead to their calculus. 



1 should also mention that Mr. Herschcl, in a letter dated 

 Vol. 65. No.325. Ai«// IS'irj. Ss October 



