4:2 Mr. 0. J. Lodge on some Problems connected 



this result being evident if numerator and denominator are 

 multiplied by 



3- + l r.9 2 + ^ 2 .15 2 + ^ 2 ... ; 

 and similarly 



. , trx 



*W \- 22 + * 2 4 2 + ^' 2 8 2 + ^ 2 1 2" 



W'~ tf , 6 2 + .^• 2 '6 2 + ^ 2,,, 4* . , 3 to 



b 



l + 3coth 2 ™ 



H. 



ence 



n - 1 + 3 tanh 2 —.,- l + 3tanh 2 -^ l + 3tanh 2 -^- 

 q = 3V3 2V3 2V3 ■ 2V3 



2tt 4 ' 4 ' 4 



l + 3coth 2 -^- l + 3coth 2 -^- 1 + 3 coth 2 - 6 



2 n/3 2 V3 2 V3 

 )\ . , 



This product I sent to Mr. Grlaisher ; and he returned it reduced 

 to a product of two theta-functions. He proved the two fol- 

 lowing identities : — 



_ 1 + m tanh 2 9 1 + m tanh 2 3d 1 + m tanh 2 50 

 1- ~ 1 + m 1 + m 1+m 



-V^m 



and 



p _ 1 + m coth 2 2fl 1 + m coth 2 40 1 +m coth 2 60 

 1 + m 1 + m 1 + m 



~ cos *V (2K) 3 M' \ 7T + *£ 



where 



~ m — 1 , aA K7 

 cos 2# = — -= and 20=zir ^« 

 m + 1 K 



In our present case 



m=3 and 0= 



2 n/3 7 

 so 



7T 



K= V3K' £ = sin 75°, and x- 



