in reply to the Rev. Brice Bronwin 209 



it is proved that 



1 u 



dv = -^du, or v=^^. 



Admitting all this, how can a person deny it immediately, and 

 say that the proof only applies to particular forms of w? [Of 



.u f .u .. mK + m' K' I , „ 



course the proor assumes that co = , the reason or 



this being in order that the equation sa. [u + 4? w w) = sa ,u 

 may be satisfied ; and this is the case for any value of co in- 

 cluded under the above form.] 



But the real ground of Mr. Bronwin's objection is evidently 

 the different results to which his own reasonings lead him. In 

 the Mathematical Journal, p. 124, he says, as the foundation of 



his formulae, " Make cw = — . Moreover, when ?«=0, w, 2 w, ... 



-71 



suppose V — 0, H, 2 H, ..." (clearly meaning H to be analo- 

 gous to K). And he then assumes a certain expression for 

 sa .V) not to mention another one for c .a .v^ the identity of 

 which with the former is not satisfactorily proved (at least to 

 me, but I am not pressing upon this at present). I believe in 

 this case the two suppositions ol the correspondence of values 

 of M, V, and the assumed form of sa .v are correct; still one 

 ought to be deduced from the other, as Mr. Bronwin appears 

 to admit (Phil. Mag.), " there is certainly room for discussion 



whether the quantities p, p^ in the equation ^= pH +p'll' i 



are to be assumed or determined." This was my original ob- 

 jection, that they ought to be determined, and moreover, that 

 in the cases Mr. Bronwin objects to, his assumption was in- 

 correct. I had overlooked an equation in Jacobi (p. 59) which 

 tends to confirm this ; it is for the case of the impossible trans- 

 formation CO = , viz. the formula A! = — ^^jt, i. e. / A/=-=-;-, 



n ' ' ;^M/ ' M' 



u 

 so that when m = w, t; or ^ = A' i (instead of A as Mr. Bron- 

 win supposes [A is Jacobi's letter corresponding to H]). Of 

 course I am not quoting this as proving the point; it is only 

 that it enables me to retort Mr. Bronwin's challenge about 

 the above transformation. Let him begin with the assumptions 



<ti — , ?^ = 0, w, 2 CO, ... t; = 0, H' J, 2 H' *, ... and see what 



his theory will lead him to, I cannot undertake to do it my- 

 self, for I do not understand it ; I have 'worked out the parti- 



cular case co = — r- hy Jacobi's method, beginning, as I suppose 



