On the Eelatiyely Abelian Corpora. W 



-f- 



we get 



u 1 /•" d- 



This shews that, in tlie parallelogram OABC, the locus of the 

 point u, for whicli z(;ii) is real, consists of two straight lines, on 



whicli 



i( ill. 



or — 



is always real. Taking into account tlie values (."')) and the relation 



f>-(fn() =z f'-'ifu) = r(u), (4) 



we arrive at the following conclusion : 



z(^u) is real along the straight lines OB, AC, 

 pz^7() „ „ „ OC, AB. OE, BG, 



frT(7() „ ., „ OA, BC, OF, BII. 



The direction of OB, along whicli t{v) is real and positive, makes 

 witli the real axis of the (Jaussian plane an angle, which is equal 

 to the phase of the numl)er V/' — k i.e. the angle of 75^. 



In our specialised ^-function, since ^.j = 0. we liave j[co) = 0. 

 Hence 



which gives 



;,' = i: //> or -^^ io'-. 



Between our ^-function and the function sn with the singular 

 modulus 7c = io, til ere exists tlie relation 



Hn) =^.^— r^- (5) 



Bll- It 1 —o 



and 2K = (0, '2iK'= oj + o/. 



If we comhine (4) with (5). the followiug formulae can easily 

 he deduced : 



