1885.] Results in Elliptic Functions. 317 



dnu=V] c 'r k e 2K ' \, , > 



ttK 



where r=e~ k' 



<I>(w) is, in fact, connected with Jacobi's Q(u) by the equation 

 #(«)-f-$(0)=e-4KK?OW-7-e(0). 



3. Expansion of $>(u) in a Hyper-harmonic Series containing odd 



Multiples of ^r t . 



2K 



From the above materials it is found that — 



*(«)=8 { ^rcosh^+^cosh jg+ cosh 

 + . . . ocZ iw/Zn.}, 



ttK 



where r=e~ k' and cosh a5=J(e* + e *). 



4. Some Consequences of the above Theorems. 



Among the numerous results which flow from the above I notice 

 the following, viz. : — 



T , Vr sinh-^--j-3v // r 9 sinh ... 



, n „ J' *■ 2K' 2K' 



K' 2K' v - . ™ 4 /-5 . 3ttu 



v r cosh 2j^+ " cosn 2K 7 + ■ • • 



If this be combined with Jacobi's 



Z(„) = «- r =_j_J_ sl n_ + ... ) 

 we have the curious relation — 



4/— 7TU 4 y 37TU 4/— hlTU 



u i v r sinh ^jFT + 3 V r 9 sinh 2£~/ + 5 v ? ' 25 sinn 2K' ' ' * 



*KK' K' 4/ _ 7m 4 /— - 37rifc 4/ — bmi 



vr cosh 2g7"r" vr 9 cosh + vr 2 ° cosh ^7 "T • • • 



* Other relations follow from the z function z(u)=a- .? + '- l ~,u, which deserves 



to be studied. As regards the transformation of the function the results are 

 very similar to those obtained in the case of Jacobi's 0. — April 29, 1885. 



