The Rev. B. Bronwin ow Jacobi's Elliptic Fuiictions. *75 



room. May his mantle descend on our future presidents, and 

 his spirit long continue to preside over our councils, and ani- 

 mate our exertions in the cause he had so much at heart ! 



On the conclusion of the reading of the preceding Memoir, 

 the thanks of the Society were proposed by the Dean of Ely, 

 and unanimously voted to Sir John Herschel. 



It was then moved by Mr. De Morgan, seconded by Mr. 

 Donkin, and unanimously resolved, 



" That the Society feels it impossible to express in adequate 

 terms its obligations to its late President; and it desires to 

 impress on the minds of all the Fellows, that such imitation 

 of his example as their occupations will allow is the mode of 

 testifying their gratitude and respect for his memory with 

 which he would have been most pleased." 



VI. On Jacobi's Elliptic Functions. 

 By the Rev. Brice Bronwin. 



1 DID not expect to hear again from Mr. Cayley on this 

 •*- subject, as my paper printed in this Journal in August 

 1843 made the matter in dispute between us exceedingly plain. 

 Every step of it is clear. The most difficult is — But for the 

 third and fourth forms of w the denominator 



sa(K— 2«))sa(K — 4co) 



cannot be reduced to the form sflfws^.Sw And this 



is perfectly easy (see my paper in this Journal, April 1843, 

 p. 260). The next step — u cannot take any of the forms 

 2 /'CO — K — is evident; for the imaginary part of u must be 

 equal to the imaginary part of 2 r w, and then the real part 

 also will be equal to the real part of 2ric«, and not equal to 

 the real part minus K. This is all exceedingly easy, and 

 very unlike the slippery path Mr. Cayley has trod in his last 

 paper, a path in which a false step is easily made. 



Besides, I called Mr. Cayley's attention to a transforma- 

 tion depending on the third form of w, at p. 54 of Jacobi's 

 work, and requested him to make trial of it in its simplest 

 case. I pointed out to him the means of very easily testing 

 this form. The result would have been a stubborn fact on 

 one side or the other. I suppose the readers of this Journal, 

 whether they understand Jacobi's functions or not, will know 

 how to estimate this circumstance. 



I will now enlarge on the other mode of deciding this 

 matter, which I just hinted at in the conclusion of my paper 

 of August 1843. In order to determine the constant M, 

 Jacobi makes x = sa .u=l (see p. 41). No form of w there- 

 fore which does not satisfy this condition can consist with his 



