ON THE THEORY OF NUMBERS. 



(ii) 



(iii) 

 we obtain successively 



ac, x, 0, 



1, 1, 0, 



1, 1, 0, 



se, x, 0, 



0, 0, 0, 



1, 1, 0, 



k 2 + k' 2 =1, 



(15) 



K'6l (x)=6l 1 (x) + K el 1 (x) > (16) 



^l(*)+k^iO») (17) 



«'flW*) 



Again, attributing to the same elements the values 



we find 



x+y, te—y, 0, 



, 1,1,0 



1 , 1 , 0, 



=0i,i(^)«o,i(^)0o,oO/;0i,o(2/) 

 -M*)M*)«<.i(y)My)' 



Dividing by y, and diminishing y without limit, we obtain 

 1/Mf)\ WOjeyO) MaQflqoQe) 



Mm*)/ e liO (O)0 o ,o(O) eft,^) ' ' 



(18) 



Similarly, we might form the differential coefficient of any other quotient of 

 two Theta functions ; of these we require only the two following : — 



d A.o(g) \ MQ)e;.i(Q) fli.iO)flo.oO) 



dx\d QA {x)) e ljO (O)0cLi(O) 



rfa7 



fl&l(*) ' 



(0)0^(0) ft. (*)*i.i(*) 



(18 a) 



(18 6) 



o(O)0 M (O) 6l A (x) 



"We shall now attribute to a, which has hitherto been left indeterminate, 

 the value 2K, K being a constant, the square root of which is determined by 

 the equation 



+ GO OO 



^=0o,o(O)= S m ^=n m (l- 2 ='")(l+< ? 5 '"- 1 ) 2 ; . . . (19) 

 — oo 1 



we shall also write K' for ~. Attending to the values of »/k' and */ K , we 



find from (10) and (11), 



f 27K +co °° 



• -=0 O>1 (O)= 2 m (-iy"q m2 =n m (l-q"")a-q 2m ~ l y, 



/2K 



— 00 



+ GO CO 



^=0 1 ,o(O)=S M ^^) 5 =2 g i^ M (l-2 5 »O(l + 2 ^, ") 2 . 



— 00 1 



(20) 



(21) 



Multiplying together the infinite products (19), (20), (21), and reducing 

 by an identity of Euler's, 



00 00 



n(i-<f»-i)=n„ 

 l l 



1 + 2'"' 



(E) 



