270 THE REV. F. H. JACKSON 



which determines the form of the series expansion of the infinite product 



I + &% " 1)V ! + M<1 - })\y x + te(g-l) > * 



or, as it may be expressed in terms of the function E 9 (.x), 



E,K)E,(f )E,(- 



ad inf (50) 



W E,( 4 |)E,g) 



§ 6. Double Product Theta-Functions. 



Some interesting relations exist between the coefficients c^c^ and the 



generalisation of Bessel's Function denoted J [n] (^) in previous papers (loc. cit.). 

 From the relation 



^-JS'^) < 51) 



\{bqx) 



we form 



', axt\f, axt\ 2 f, , aa:f\ 3 



, g /V i-l^ £-^ = 1 + c.xt + c<,xH 2 + 



1+ ^)( 1+ -)' (l+ -y 



q A + f )\ + ^f) ,_,._ 



fcrf-Vt . bzt-W, bxt- 1 ^ -l + ejZt 1 + c. 2 xH 2 + 



0^'X 



(1+ ^ )(1+ ^ nl+ 



in which 



Whence 



(1 + 2axq- i cos 20 + a 2 x 2 g~ 2 )(l + 2axg~ 2 cos 20 + a 2 x 2 y- 4 ) 2 ( 1 + 2aa<T 3 cos 20 + a 2 x 2 q- 6 ) 3 



(1 + 2 toy" 1 cos 2(9 + l> 2 x 2 q- 2 )(l + 2bxg~* cos 20 + 6V 2 y" 4 )' 2 ( 1 + 2bxq~ 3 cos 20 + bh?q- f >) 3 



= 2<VZ r e 2r * 9 . x 2,^'e""" 9 = A o (:c) + 2A. 1 (x)cos20 + 2A. 2 (a;)cos40 + (53) 



or, by changing q l into q 2 , 



/ 1 • + 2ayry 2 cos 20 + aVt/ N VI + 2a.«/> cos 20 + a 2 / 2 / \ 2 /l + 2axg 6 cos 20 + a 2 xV 2 \ 3 ad jnf 



V 1 + 26ay/ 2 cos 20 + b 2 x 2 q* ') VI + 2fov/ 4 cos 20 + bWq* ) \1 + 21mf cos 20 + b 2 x 2 q™ ) 



= /a o (a;) + 2/^(a:)cos20 + 2/* 2 (a:)cos40+ . . • (54) 



