330 REPORi'— 1873. 



also 



and r„ s='"fl „ ' ^^^^ function 6 is defined by the following equations : — 

 whence 



It will be observed that these assumptions coincide with those of Weierstrass 

 (Crelle, xlvii. p. 303), and which we have given in the Report (Brighton) for 

 1872, p. 345, by putting in the formulae of Weierstrass 27rVj, 27r(',. . . .27rt'p 

 for v^, v^. . . . Vp, and 2Gj ^ . . . for G^ ^ ; then it will be found that Jc and a 

 are equivalent. 



We easily obtain from (2), 



and 



d(v^ + n^r,, j + »i,, T,^ „ ''o + '^ir,, , + H^r,.„ . . . . ) 



and writing 



0K + ^., ^. + r, t'p + r^)=e-^V-'V+^)'^'a(V',....„^). 



This assumes, of course, that n^, n^, . . . .n are integers ; when they are not, 

 Konigsberger assumes another transcendent, as follows : — 



2» (2v +T )7r , 

 and calls it 6(t'v„. . . .v n,n„. . . .n ). 



^1- p,l- p-' 



Then we shall have, if r'^=n>„ ^+n\r^^^+ "V^.p)' 



(remembering that Sn'^r^=2«^7'^) 



— ?z'(2y +r')Tri , 

 = e " " 0(^, '^a v^, n^ + n,, n^ + 71^ ). 



Konigsberger furthermore assumes transcendents with the notation : — • 



0(",^.- • • •^p)A=e(^+>i- • • •^+>p ; inl . . .i»^^); 



also 



where?n^=m^ + TO;;- (mod. 2), »,^=«|; + »;^ (mod. 2), and »^ and iif &c, 

 arc given by the Tabic, p. 20, 



