with the Flow of Electricity in a Plane. 39 



of which "Wallis's expression is the particular case when 

 x = 0. 



The entire product Q also shows an evident resemblance to 

 Wallis's form, being 



_2/ l«r(l).3«r(3)... y 



Further than this I was unable to proceed ; so I showed the 

 products to Mr. J. W. L. Glaisher during the Bristol Meeting 

 of the British Association ; and he very kindly told me how to 

 evaluate the product denoted above by «r(#), and subsequently 

 worked out the compound product Q. Taking the well-known 

 trigonometrical identities 



. irx irx /' <£ 2 \/-, <£ 2 \/-, <^ 2 \ 



oos ^ = (i_f!)(i_|!)(i_f!)..., ^ 



Mr. Glaisher divided one by the other after putting x «/— 1 

 for x ; he thus obtained 



iirx iirx 2 2 + x 2 4? + x 2 l 2 . 3 2 . 5 2 . . . 



tan 



2 "" 2 l 2 + x 2 3 2 + ^ 2 "*2 2 .4 2 .6 2 



iirx , s 2 



whence 



a) = -t tan -~- = tann -^- . 

 This result gives us 



2 / ta n h|.temh^... y 



\tanh -^- . tanh — . . ./ 



The part inside the brackets may be written 



/ttQ _ l-g- ff < 1 — e-s* l + e~ 2 " t l + g" 4ff 

 V ~2"'~ 1 + «-*'l + 6-^"' 1 — e~ 27T ' 1— e~ 4 * " ; 



and this Mr. Glaisher saw at once was a special case of Jacobi's 

 products ; two of which (being those required in this paper) I 

 write down here — 



* Mr. Glaisher also obtained an expression for the more general product 



— . — — — . . ., which he has communicated to the Mathematical 



l n +x n 2> n +z n 



Society. 



