276 Mr. T. S. Davies on Mr. Herapath's Demonstration. 



It is for the reason above given that we are to admit the 

 identity of tliese two equations: (Phil. Mag. vol. Ixv. p. 324.) 



ir+vf~^+ o(>- + v)'~^+ ", (>• -f ")'"!+ :i^_ +2(0-1) +....q 

 and 



r.-2 (?-l) ir^ ^v 



And further, for the mdependence of\ and r, we are to admit, 

 that " this equation evidently gives 



„ _ ,. ('•-l)(>-2)(»--3)- I (.-l)(»-2) ... 



9r = ' • iT2...(y-0 ^^ ~ 1.2...(<,-1) 



■ro^fc/i completes the proof ^ — (Ibid.) 



As this latter conclusion is derived by taking v and r sepa- 

 rately equal to zero, a momentary recurrence to the original 

 equation of condition (r + u = integer) would have convinced 

 Mr. Herapalh that if every preceding step had been perfectly 

 legitimate, still these resulting equations are not for fractional 

 values of V and \, but for integer onli/: and I am confident 

 tliat that gentleman will see the force of this suggestion, and 

 withdraw his claims to having given the " most complete and 

 o'eneral demonstration which has yet been published of this 

 celebrated theorem." His demonstration I confess is as good, 

 but 7iot better, than any other that has been given : and more, 

 it is as good as any that ever 'will be, or that ever ca7i be 

 given. 



It is now sufficiently clear, that in either view of INIr. Hera- 

 path's demonstration the same fallacy is involved, — the same 

 gratuitous assumption employed; viz. an extension of the 

 values of r and v to forms not consistent with the conditions 

 originally stated as the basis of the investigation. 



I hope I shall not be charged, even by implication, with 

 any wish to do injustice to the mathematical labours of Mr. 

 Herapath. No man can entertain for that gentleman's inde- 

 fatigable spirit and splendid powers of intellect a more sin- 

 cere respect than 1 have ever done. The mistake of Mr. He^ 

 rapathis a very general one, — it originates in a principle which 

 has obtained universal credit, — and is so far from involving 

 anything derogatory from his mathematical character, that it 

 is almost invariably found amongst the writings of men of the 

 very first order of scientific merit. — This remark would have 

 been unnecessary, had not P. Q. insinuated that my wish was 

 " to overwhelm in the ruins of the binomial demonstration" 



the 



