308 REPORT— 1873. 



CO 



-\' 



-GO 



e,(v, 5) = 2„(-l)»?""^^e^'"+'V=^l(e''-e-^)-5t(e"'-e-"')+ • • • • . 



(2«+l)2 



-GO 

 ^ (2»t + l) 



0,(1^, 2) = S„2 * e^'''^''v=qi{€^+e-'^) + qt(e^<' + €--^'') + q-T{e^'' + e~^'')+..., 



-Xi 

 CO 



flgC^j 2) = 2„5»V'"' = 1+2(62" + 6-2") + 24(e4«' + e-4")+ . . 



-co 



these functions are singly periodic, and their ratios doubly periodic. We 

 have ab-eady seen that this periodicity has been fuUy discussed by ScheUbach. 

 Now let us assume four new variables connected with the original varia- 

 bles by means of the equations 



2i', =v+v' + v" + v"', or 2v =v^ + v^' + v^' + i\"', 



2Vj' ~v-\-v' — v" — v'", or 2v' =v^-\-i\' —v^" — v^", 



2v^" =v-v' + v"-v"', or 2v" =v^—v; + v^' — v^", 



v^ =v — V — V -\-v , or 2v =?'j — ?.'j — Vj -ri\ > 



whence 





Hence dXv)d,{v')dXv")ds(v"') = ^ ^°^'^ S logoff, 



where 



'P=(v + n log^ q)-^(i>' + n' log^,^)' + (?'" + ?i" log^qy + (v"' -\-n"' log^ qf 



will remain unchanged, if v^, v\, v'\, v'"^ be substituted for v, v , v", v'", 

 provided that 



2rtj =n + n' + n" + n"', or 2n =)i^ + n/ + n/' + n/", 



2»j' ^n + 'ii' — ?i"— n'", or 2n' =)),j+n/ — n^" — '>\"', 



2n^" =zn—n' + n"—n"', or 2)i" =Wj — n/ + 9?,/'— n/", 



2n/" = « — ii'— n" + n"', or 2>i"'=Wj — w^' — ji/' + n/". 



Now n, 7i', »", n'" are all whole numbers from +co to — co ; but as these 

 equations are written we should have n\, n'\, n'\, n"\ including forms +r 

 and +(r-{-i) when r is any integer. 



This inconvenience is removed by assuming that 2n, 2n', 2n", 2n"' must 

 be subject to the same condition to which 2n^, 2n\, 2n'\, 2n"\ are subject, 

 namely of being all at once odd, or all at once even. Rosenhain shows 

 (p. 373) that this necessitates the introduction of functions 0^, and that we 

 have 



