356 Defence of Mr. Herapath's Demonstrati07i. 



terms. If not, by transposition we should liave an algebraic 

 equation of v and constants, which would of course limit this 

 quantity to an invariable value, and thus make it a constant, 

 against' the hypothesis. But in these developments of (1), 

 yi ;•, Y^pr are the whole of the functions which contain r only; 

 and consequently by the property of identity just proved 



J\r = F,pr 

 By similar reasoning it follows that 



fzV= F.,<7,„ and/3 {r, v) = ¥,{pr. q,). 

 It should here be observed that the function F (;;,., q^) is 

 not confined to powers and products of p^ and y„ ; for it may 

 have the more general signification of 



F[p (;•), p, (>•), p, (r), ,q {v), q, {v\ q, {v), . . .] 



in which p, ;?,, jhi • • • ^"^^ (Zj ?i> ^^j • • • ^^'^ ^^ different func- 

 tions of?- and V respectively, though for brevity p (r) and q [v) 

 only are expressed. 



Hence the truth of Mr. Herapath's proof is apparent. I 

 could have easily established the above theorem by putting r 

 and V separately = 0, and not in the least affect the legitimacy 

 of Mr. H.'s arguments. But apprehending Mr. Davies might 

 think he had reason for playing against such a procedure the 

 surprising inferences (Phil. Mag. for October, p. 276) he has 

 drawn on the supposition of r and v being put separately = 0, 

 I was willing to show that our resources are not so meagre as 

 to depend exclusively on a solitary process. 



It is now manifest that Mr. Herapath's demonstration is 

 correct and complete, whether the exponents be real or ima- 

 ginary. Of this I hope Mr. Davies will be convinced ; and 

 if he be, I am persuaded, from the honourable and manly 

 sentiments he has displayed, that he will not be backward in 

 acknowledging it. Should he however not be satisfied, it will 

 be incumbent on him to show by actual examples the fallacy 

 of the preceding theorem, which comprehends the principles 

 of Mr. Herapath's process. 



Mr. Davies complains of my having wrongfully charged him 

 with a wish to overturn most of Mr. Herapath's mathematical 

 labours. From the general tenourofthat gentleman's first paper, 

 and his particular allusion (Phil. Mag. for Aug. pp. 117, 118) 

 to Mr. Herapath's " reasoning on periodical functions," which 

 does certainly not at all depend on the binomial demonstra- 

 tion, I did imagine Mr. Davies intended a sweeping objection 

 to the whole or the greater part of Mr. Herapath's mathema- 

 tical writings; and I conceive almost any other individual 

 would think so too on reading the paper in question. How*. 



ever. 



