Verification of an Elliptic Transcendent Identity. 437 



and from these I find that to so small a depth does the super- 

 heating extend, that the surface of the ocean at the equator 

 requires to stand only four and a half feet above that at the poles 

 in order to the ocean being in perfect equilibrium. In this case, 

 if we suppose, in order to constant circulation, that the polar 

 column is kept in excess of the equatorial by the weight of, say, 

 two feet of water, there would then remain only a slope of two 

 and a half feet between the equator and poles. 



James Croll. 



LVI. Verification of an Elliptic Transcendent Identity. 

 By J. W. L. Glaisher, M.A* 



ryiHE identity in question is 



2v/7rfl J"l 2ttx - 4 J?1 4nrx \ 



= -< - + e " 2 cos f- e « 2 cos \- . . . >, . (1) 



a L2 a a J 



on which see Cauchy, Exercices de Mathematiques, 1827, p. 148; 

 Abel, (Euvres, vol. i. p. 308; Schlomilch, Analytische Studien, 

 part ii. p. 31 ; and Sir W. Thomson and Cayley, ' Quarterly 

 Journal of Mathematics/ vol. i. p. 316. 

 When x = 0, (i) becomes 



= {i-\-e * + e a* +e «?+...}, . (2) 



which, expressed symmetrically, takes the form 



V iog^(K? + ? 4 +? 9 +.-0=Vios^(i+^+^+^ 9 +.-.). 



where 



logq . logr = 7r 2 . 



The identity (2), as Abel remarked, is deducible at once from 

 Jacobi's formula, 



2K\ 

 ir)> 



and (1) itself, as was shown by Cayley, is easily obtained by 

 elliptic functions; but the most simple proof of the latter for- 

 mula is that given by SirW. Thomson, and which depends on 



* Communicated by the Author. 



l + g + g 4 + q s +... = i + i X /(- 



