222 Mr Glaisher, On the zeta-fanction in ellijjtic functions. [Feb. 2, 



and Z(x) has the meaning assigned to it by Jacobi. Denoting 

 by accented letters the same functions of k' that the unaccented 

 letters are of k, it can be shewn that 



EK' + I'K=\tt, 



and 



ez (x + 2mK -f 2m'iK') = ezx + 2mE — 2m W, 

 iz (x + 2mK + 2m iK') = \zx + 2ml — 2m'iE', 

 gz (x + 2mK + 2m iK') = gz x + 2mG — 2m iG'. 



The function eza; is the same as Jacobi's E(x), and \zx is the 



Al'x 

 same as Weierstrass's function -r^— . The functions ezx and \zx 



Alx 



form a pair of corresponding functions in which E and I, and E' 



and I', are interchanged, but gzx stands by itself and is such that 



corresponding to an increase of argument 2K the increase of the 



function is 2G, and corresponding to an increase of 2iK' the 



increase of the function is — 2iG'. 



Three other functions uzx, vzx, wzx, denned by the equations 



U 



uz X = j^X + Z(x), 



V 



vz X — -^. x + Z(x), 



wz# = ~ x + Z(x), 



where 



U = E-i(l + k' 2 ) K= I + i k l K, 



V = E-±k' 2 K = I + ±(l+k 2 )K, 



W=E-\K = I + \K, 



were also considered. These functions correspond exactly to ezx, 

 \zx, gzx, the quantities E, I, G, E', T ', G' being replaced by IT, 

 V, W, V, V, W respectively. There is thus a reciprocity between 

 uzx and vzx; but wz x, like gzx, is complete in itself. The 

 general form of Zeta-function, and the systems of Theta-functions 

 derived from these six Zeta-functions (corresponding to Weier- 

 strass's A\x, derived from iz#), were also referred to. Those derived 

 from gzx and v?zx are the most complete and symmetrical. 



