Demonstration of certain Formulce. \ \ 9 



ties, I should not have appeared before the public on this oc- 



casion. 



Mr. Herschel was aware that to deduce the differential 

 formuljE in the usual way had very much the appearance of 

 " an inductive process:" and to evade the difficulty he resorts 

 to a method equally inductive^" the calculus of generating 

 functions *." In the proof by this method, were no other in- 

 ductive principle involved, there is yet the binomial theorem. 

 But there are other inductions mingled up in the investioa- 

 tion : for instance, the assumption of a series to represent the 

 development of ip^; which series, especially as following a re- 

 gular and successively dependent law, we do not know to be 

 always possible. It is true Mr. Herschel attempts to parry 

 this objection, by assuming the series as the " dejiiition" of 

 the function p / : but even in this view, except considered as • 

 induction, it is inadmissible; for how can we define any func- 

 tion, except by stating some constantly observed quality or 

 circumstance belonging to it, and which may serve as a dis- 

 tinguishing character of that function ? In any other case 

 than this it is not a definition, but an assumed property ; and 

 our assumption is therefore merely a fiction, to be subjected to 

 investigation and analysis,— not. less so than any theorem in 

 the ancient geometry ; and by comparing its results with pre- 

 viously established principles, we learn its consistency with 

 those principles. If correspondent, we admit the fiction ; if 

 not, we reject it. Here, however, since this series is consi- 

 dered a general representation of every numerical case of 

 every possible series, we can only compare it with each sepa- 

 rately, and admit its partial truth by the successive probabi- 

 lities which arise out of those comparisons. Upon what 

 then, but induction, does the principle of investigation itself 

 depend ? We have found in all the cases we have yet tried, 

 that cf) t may be developed in the form of the fundamental 

 equation of the calculus of generating functions: and we can- 

 not question the probability, though we must ever make a re- 

 serve of the term " absolute certainty," that every possible 

 form of ip is such as would admit of such an evolution. In- 

 deed, Mr. Herschel's view of 9^ = ... series ... being a defi- 

 nition, is at once resolvable into the principle which LaGrano-e 

 calls the " M('o;7/ of series :" and this principle is merely an 

 observation that certain properties belonged to every sejiarate 

 function which had been examined, and an induction of the 

 universality of a law, in consequence of no deviation from it 

 having yet been noticed. 



I'akiiig this view of the subject, there appears no reason for 

 Mr. Ilerajjath's considering the expression for the develop- 

 • App. TransL Lacroix Diff. el Ivl. Calc. p. A^'2. 



ment 



