The Rev. B. Bronwin's Reply to Mr. Cayley's Remarks. 91 



These formulae, therefore, by suitable values of u, are con- 

 structed to fulfil the conditions s a » = 0, c av = _+ 1, and 

 also sa»= + l, c a w = 0. And it must be possible to 

 satisfy them both. For at page 40, in deriving the value of 

 y — s a v from that of 1 — y, Jacobi finds y m when u = 0, 

 2 co, &c. And at page 41, in finding the value of M, he 

 makes x ~ s a u = 1, y = 1. Also the values of A, and of 



1 ± ^y depend on those of M and of y. Both these condi- 

 tions therefore are at the very foundation of Jacobi's theory. 



He also makes u, u + 2 w, &c, and even 0, 2 co, 4 co, 

 &c, successive values of u. This decides the form of u. If 



the general form of co be , that of u is 



1 - ; and Q' being real elliptic functions, having 



the common amplitude >j, and the moduli h and k' respectively. 



When tj = 0, y, p, &c, = 0, K, 2K, &c; fl'=0, K', 2K',&c; 



and m=0, w, 2 eo, &c. For the three forms of co which we have to 

 consider these values of u fulfil the condition sav=0,cav=: + l. 

 For the first form of w, when the denominator of (1.) reduces 



tosacosa3co , the values u = 0, 9 co, &c. satisfy 



the condition saw = jfl, cai; = also. But for the third 

 and fourth forms this denominator cannot be so reduced, nor 

 can u be made to take any of the forms K — 2 co, K — 4 co, 

 &c, or 2 co — K, 4 co — K, &c. For it could only assume 



them when »j has some of the values — , tt, &c, and conse- 



quently 6 and 9' some of the corresponding values K, K', 2 K, 



2 K', &c. But for none of these values will u become any one 

 of the quantities 2 r co — K. The third and fourth forms of 

 co therefore will not fulfil the conditions to which the formulae 

 have been subjected, and consequently they must be rejected. 



We might take a shorter course. It is sufficient to observe 

 that the first form of co only will satisfy the conditions sau= 

 and sau = 1 required by Jacobi's theory, pages 40 and 41. 

 Mr. Cayley says, I have brought no objection against any 

 particular step of Jacobi's reasoning. I suppose them to be 

 all quite correct. But any assumed form of co will not neces- 

 sarily fulfil all the required conditions. It must be remem- 

 bered that these forms are assumed, not determined. 



B. Bronwin. 

 Gunthwaite Hall, June 15, 1843. 



