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



+ oo */ — 1 and + — . And as relates to the second form 



A 



of ca, it is to be rejected. There is certainly room for discussion 

 as to whether the quantities p and p' are to be determined or 

 assumed. I assumed them, and took the least values, because 

 it did not affect my conclusions. Were I to discuss the va- 

 rious points to which this difference between me and Mr. Cay- 

 ley gives rise, I should extend this paper to too great a length. 

 And as I think I can place the subject in a clearer light by a 

 much shorter process, I prefer doing so. 



But first I must beg to call Mr. Cayley's attention to a 

 real transformation at page 54 of Jacobi's work. It is derived 



K' V^T 

 by the aid of imaginary quantities, and from an co as , 



and is therefore of the third form. Will Mr. Cayley be pre- 

 vailed upon to make trial of it in its simplest case, or when 

 n = 3, and see if he find it to be a transformation ? It is but 

 right to say that I have done so, and did not make it to be 

 one. And if I am correct, this must be fatal to the third form 

 of oo. 



And I must observe, that though Jacobi has shown the 

 possibility of such a transformation as he has given, by show- 

 ing that there are sufficient equations to determine the con- 

 stants, he has not shown that any and every value of od will 

 give one. Suffice it that there is one value, or a series of 

 values, namely those included in the first form of this quantity. 

 Nor has he assigned any reasons for the different forms of it 

 which he has suggested. Moreover, he has set out from an 

 assumed equation 1 —< g =y(a?), page 39, from which all the 

 rest of the formulas are derived. In this assumed equation 

 he has not actually determined the constants, but only assumed 

 them. If they were actually determined, it might appear 

 that they are not susceptible of that generality which their 

 author and Mr. Cayley suppose. 



M. Jacobi's formulae, as Mr. Cayley has reduced them, 

 are 



sausa (u + 2a>) sa (u + 2 (n — 1) w) . 



sav=s sfl(K-2«)s«(K-4eo)..sfl(K-2(«-l)«) "' I 5 



cauc a(u + 2co) c a (u + 2 (n — 1 ) w) ,„ . 



c a v = 5 -r 1 7T-, tt ' — - ... (2.) 



The numerator of (1.) when developed is 



sau (s* a l 2 a> — s* a it) 



and that of (2.) is 



ca«(s 2 fl(K-2w) -s 2 am) 



