120 Mr. T. S. Davies on the Demonstration of certain Fonmdce. 



ment of a" as demonstrated; nor, on the other hand, for 



Mr. Herschel's denominating the asserted equahty of the 

 function and its development " a definition*." That the 

 theorem is true, provided the expression can be considered in- 

 telligible, there is a high degree of probabihty ; but that pro- 

 bability can only arise from the identity of its general form of 

 expansion with the general form of expansion of the binomial 

 theorem. Indeed, I cannot doubt, on the faith of the general 

 principle, the truth of r ^ _ ji « ^ ^ »« 



in their developments, whatever be the form assigned to n : 

 against the affected rigour and consequent inconclusiveness of 

 its demonstrations rest the only objections I have to make. 



To conclude, and in reference to notation, I shall just re- 

 mark that Mr. Babbage's viewsf on this subject do not ap- 

 pear to be quite correct, though the phraseology which he 

 generally uses indicates a near approximation to such correct- 

 ness ; perhaps a nearer appi'oximation than can be found in 

 the writings of any other mathematician whatever. However, 

 concerning new notations, " not to agree with, but to include 

 the former," we ought to remark that this including notation 

 is merely a statement of facts (or in truth, a more general 

 theorem), so contrived as to embrace in one expression the 

 facts of two or more of the old notations ; just in the same 

 manner that n is used to include 2, 3, 4, 8cc. as indices in any 

 power of <p X. It is true, that fi'om this new statement, or no- 

 tation, new relations might be observed, or new theorems sug- 

 gested, or the relations so expressed might excite some col- 

 lateral inquiries arising out of mei'e analogy, as in the diffe- 

 rential theorems to which I have so often referred, and which 

 were suggested by the binomial. The formulae were not in- 

 tended originally to " include" negative, fractional, irrational, 

 or imaginary values of n ; the signification of the formulte was 

 so interpreted afta- experiments had shown that the formulae 

 held good in every tried case of all those classes of values. 

 Improvements in notation are in truth identical with the dis- 

 covery of new theorems, or with a more extended application 

 of the old. — But I have far exceeded the limits which I had 

 prescribed to myself (notwithstanding several important consi- 

 derations on these subjects remain untouched, and not even 

 hinted at): I shall, therefore, defer the further inquiries to 

 Avhich the considerations already stated so obviously lead, till 

 a future and more convenient period. 



Bristol, July 19, 1825. T. S. D, 



• I infer that it is to Mr. Herschel that Mr. Herapath alludes, Phil. 

 Mag. May 1825, p. 326. 



t Cambridge Phil. Trans, vol, L pt. i. ; and Edinb. Encycl. art. "Notation." 



XVI. Che- 



