AND DOUBLE THETA-FUNCTIONS. 
915 
34. For obtaining the foregoing relations it is necessary to observe that 
«-f 2 « 
© = © 
7 7 
by which the upper character is always reduced to 0, 1, \ or f ; and that for reducing 
the lower character we have 
0 0 1 1 
© _© .0 — —© ; 
7 + 2 7 7 + 2 7 
©' 
7 + 
3. 3. 3_ 
©- =—/©-, © a =;©-'; 
7+2 7 7—2 7 
by means of which the lower character is always reduced to 0 or 1 : in all these 
formulae the argument is arbitrary, and it is thus = 2u, or 2 u' as the case requires. 
The formulae are obtained without difficulty directly from the definition of the 
functions ©. 
35. 
As an instance, taking 
a a' 10 
7 ’ 7 ,= r i’ 
the product-equation is 
& l Au + u').&\i — u')= ©J(2m).© 2 (2w' )+ ®l(2u).® 2 (2u'), 
=i©J(2i 4 ).©J(2u / )-i@*(2M).©J(2M') > 
= bP.P' -fQ.Q', 
which agrees with the before-given value. 
36. The following values are not actually required, but I give them to hx the ideas, 
and show the meaning of the quantities with which we work. 
u —0 
X 
0/ s 
= © 0 (2w) = 
1+2(7 2 
cos 
27m-\-2q s 
COS 
47722 + . . , 
a =1 
+ 2 ++2++ • ■ 
Y 
= @J(2w) = 
2(f 
cos 
ttu-\-2<£- 
cos 
37 TU -j- . . , 
£ = 
2++2+ + . . 
X 
=e'’( 2 «)= 
1 — 2 q* 
cos 
2TTii + 2q s 
cos 
477" 21 
a , =1- 
- 2++2+.. 
Y / 
= ®J( 2 ") = 
— 2cf‘ 
sin 
7122 + 2+ 
sin 
37722+ . . , 
/3>2t7 
(-(/+ 3(7’-..) 
= yY for u = 0. 
an 
6 B 2 
