" Further Thoughts on Mr. Herapath's Demonstration." 355 



all connected with the value of r, or in any ways dependent 

 on it ; which is all that Mr. Herapath appears to mean by 

 saying, the two quantities are " independent in point of value." 

 Probably, had Mr. H. designated these quantities " indepen- 

 dent variables," as he does just below, it would have been 

 more to the point; though he has committed no error by a 

 different denomination, in the sense which he has evidently 

 employed. " Mathematical magic" need not therefore be called 

 in to establish Mr. Herapath's position : but what except " ma- 

 gic " of some kind could induce Mr. Davies to draw a com- 

 parison between the relation of r and v, whose sum is an in- 

 determinate integer, and that of an angle and its complement, 

 whose sum is an invariable quantity, 1 do not know. The va- 

 riations of r and i\ I have already proved, may, by giving to 

 ji an indeterminate value, have no relation : but it would re- 

 quire "mathematical magic" indeed, to prove the .same of the 

 variations of " an angle and its complement, or of a number 

 and its reciprocal : " the variations of which quantities have 

 fixed, or at least definable relations. 



Having demonstrated the absolute independence of the va- 

 riations of /• and V, I shall proceed to prove the remaining 

 part of Mr. Herapath's arguments on much more general prin- 

 ciples than he has done. This will give me an opportunity of 

 introducing a theorem in functions of no contemptible impor- 

 tance, which I have never met with elsewhere ; but which there 

 <;an be no doubt Mr. H. had his eye full upon when he com- 

 posed his demonstration. 



Theorem. 



Let pr, qu be any two functions of r and v respectively; arid 

 let f{r,v) = ¥{p„q,) (1) 



Then, if the changes ofr, v be indepetident the one of the other; 

 and if fr be the function in the left-hand member of {I) con- 

 taining r only, fv the function contaiiiing v only, and f h; v) 

 the function coiitaining those terms only iichich have r and v 

 mutually combiried; arid if F, p^ ^i qn F3 {Pri qr) are like 

 functions in the right-hand member; I say that, besides equation 

 ( 1 ), the three follcfwing isoill subsist simtdtaneously, 



fr=F,pr (2) 



fv=F,q, (3) 



/a (r, v) = F3 (;;,,, q,) (4) 



For since the changes ot r and v are independent, we may 

 suppose one of tliem, r for example, constant; while the other 

 varies. Regaiding therefore (1) as functions of the variable 

 V only, and in)agining these functions developed according to 

 the powers of v, each side nnist contain identically the same 



V y 2 terms. 



