208 Mr. Cayley on Jacobi's Elliptic Fimclions, 



lines are a resjilt of the motion, and that they correspond with 

 the veins of glaciers; the lines incline most when the surface 

 is steepest, as at h, fig. 3, and are very faint and nearly hori- 

 zontal at /, where the surface of the stream is nearly so too. 

 I left Gateshead without having an opportunity of getting a 

 sectional view of this stream. I can get no real Stockholm 

 pitch in Glasgow, else I should have made the experiment you 

 have incited me to attempt here. 



" I am, &c., 

 " Glasgow, January 31, 1845." " Lewis GoRDON." 



XXVIII. On Jacobi's Elliptic Functions, in reply to the 

 Rev. Brice Bronwin ; aiid on Quaternions. Bj/ Arthur 

 Cayley, Esq., B.A., F.C.P.S., Fellow of Trinity College, 

 Cambj'idge. 



To the Editors of the Philosophical Magazine and Jour?ial. 



Gentlemen, 

 \ S my last paper on Elliptic Integi'als does not appear to 

 -^^ have met with Mr. Bronwin's approbation, I will, with 

 your permission, say a few more words on the question, and 

 these will be the last I shall trouble you with on the subject. 

 As Mr. Bronwin's last paper hardly professes to bring for- 

 ward any new arguments, and complains of my not having 

 condescended to reply to his previous ones, I shall endeavour 

 at present to repair that omission. Mr. Bronwin says (Phil. 

 Mag. S. 3. vol. xxiii. p. 90), " Moreover, he [Jacobi] has set out 

 from an assumed equation 1 —y=sf(^x), p. 39, from which all 

 the rest of the formulae are derived [and correctly derived, for 

 Mr. Bronwin says afterwards, that " he supposes each parti- 

 cular step to be quite correct"]. In this assumed equation he 

 has not actually determined the constants, but only assumed 

 them. If he had actually determined them, it might appear 

 that they are not susceptible of that generality which the 

 author and Mr. Cayley suppose." 



Jacobi is, of course, entitled to assume y any function of .r 

 that he pleases, and he might, if he had thought proper, have 



K' < 



made co perfectly determinate in his assumption, e. g. co = — . 



He then proves that this assumed value of 7/ gives 



dy d X 



^T^H — jfy'^ ~ M v'l-.r^ \^T^k^ x^' 

 If then 



a; = s a (w, 1c), y = sa {v, A), 



