360 On the Values of the Theta and Zeta Functions. [Nov. 20, 



or ^(erfr— e~&— er¥/» + & c .) 



= ^( e -iv_ e -|v_ e -^ + e -¥v+ &c.) .... (7) ? 

 where /m, v are any quantities connected by the relation 



2 



Changing q into cp, (6) becomes 



g *_ 2 l_ 2 V + g f V+ &c. =B W* 



which also leads to (7). This identity is of the same kind as the well- 

 known formula 



^(l + 2e-^ + 2e-^ + 2e- 9 ^+ &c^=i> i (l + 2e^+2e^+&e- 9 *' + &c), 



2 



viz., in which a function of fx is equated to the same function of — . 



A* 



Identities of this class form the subject of a paper in the fifth volume 

 of the " Messenger of Mathematics,"* pp. 174 — 179 ; and the identity 

 which results from (3) and (5) , viz. : 



2ji i (er*p+ 6 -fr+er^+ &c.) = v i (l — 2e- v + 2e-± v —2e~ 9v + &c.) 



is given on page 177 of that paper. 



§ 9. From the third group of values in § 3 it might be shown that 



jA-jtt—grft+gW + grW- &c. =?^W*cos|0 . . (8), 

 jA-gtt-^W+gW + gW- & c . =^W*eiiiJtf . . (9), 

 where denotes the modular angle. We thus have 



gT6_gT6 — ^T6 + ^16 + £T6 — &C. 



It is a known theoremf that 



arc tan g* — arc tan + arc tan qy — &c, 

 so that arc tan q* — arc tan + arc tan q§ — &c. 



In the " Philosophical Magazine "{ for December, 1875, it is shown 

 that 



* "Notes on certain Formulae in Jacobi's c Fund amenta Nova'" (March and 

 April, 1876). 



f " Fundamenta Nova," p. 108. 



J " On some Identities derived from Elliptic-Function Formulae," ser. 4, vol. 50, 

 pp. 539—542. 



