316 REPORT — 1873. 



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0,^ , {V, w) = %S-l)V\^'^'dri^o + 2mK q), 



-00 



0,^ (^. iu) = %^{-iyY'e^''"ei,v + 2nk, p). 



(2;«+l)a 



,^,^ ^ {y, w)=\p 4 e^^"*+^^V(«' + (2m + l)A, 5), 



-00 



<p,.^iv,w)=%^q 4 e^""+^)\(^ + (2n + l)A,p), 



-00 



(2m+l)- 

 ^,^ ,. («, iv)=X-iy"P * e^^+^)\(tt; + (2m+ 1)A, 3), 



(2w+ir- 



where r denotes one of the indices 0, 1, 2, 3. It is manifest from this that 

 there are sixteen of these functions, which may all be expressed under the 

 form 



where a„ „, 6„ ., c, , are linear functions of v and w. 



The periodicity of these functions is given by Eosenhain, pages 409, 410 ; 

 and he then proceeds to develop the following theorem : — If 



2v^ =v-^rv' -^-v" -\-v"', 2ii\ =w-\-w' + w" -\-%v"', 



2v; =v-{-v'—v"-v"', 2w; =w + tv'—w"—iv"', 



2vj" =v—v' + v"-v"', 2iv^" =w—iv' + iv"—iv"', 



2v^"=v—v'—v" + v"', 2w^"'=w—w'—w"+iv"', 



also if 



M =03. 3 (^. ^) t>S, 3 W' ^') 03, 3 i^", ^") 03, 3 (^"', ^0'") 



+ 03, 2 (^' ^) 03, J (^'. «'') 03, 2 (^". '"") 03, 2 (^"'> «'"')> I 



M' =02, 3 (^', ^o) 0,^ 3 (i;', It;') 0,_ 3 (^", t«;") ^^^ 3 (^"', t<;'") ^ 



+ ?.,, , (v, w) 0,^ , (v', w') 0,_ ,, (t;", lu") 0^^ ^ (i;'", ly'"), 

 M" =0,^ 3 (v, w) f^^ 3 (v, w') 0j^ 3 (v", w") 0,^ 3 (v", iv'") 



+ 01, 2 (^» ^) 01, 2 (^'» w'') 01, 2 (^"' "'") 01, 2 (^"'' «'"'). 

 M" -00, 3 (^. w) 00, 3 K'^') 00, 3 (^". «'") 00, 3 (^"'' "'"') 

 + 00, 5 (*'. w) 0,, , (i^', to') 0,^ , (?;", w") 0„^ J (v'", w"'), 



