262 



REPORT — 1859. 



Representing therefore by p l an imaginary and primary prime, by q 1 any 

 complex prime, the equation 





♦(E) 



assumes the form 



or 



\9i' 

 (£)=(-,)™f) 



which establishes the law of Reciprocity for every case except that of two 

 real primes, when the value of the symbols I - 1 ) =( — ) =1 is at once appa- 

 rent from their definition and from Fermat's Theorem. 



35. A third, and no less interesting application of the theory of elliptic func- 

 tions to the formula of Biquadratic Reciprocity, occurs in the memoir, "Ge- 

 naue Untersuchung derUnendlichen Doppel-Producte, aus welchen die Ellip- 

 tische Functionen als Quotienten zusammengesetzt sind " (Mathematische 

 Abhandl, p. 213, or Crelle's Journal, vol. xxxv. p. 249). The elliptic function 



F(*)= n n (i — g-A 



tx 



which is considered in this memoir, and in which the factor 1 — r- is to be 



replaced by tx\ coincides (if we disregard a constant factor) with the nume- 

 rator of (l>(v), when that function is expressed as the quotient of one infinitely 

 continued product divided by another. This may be seen by comparing F(x) 

 with the expression of the general elliptic function <j>(a) given by Abel, viz. 



*C 



jU=CO 





n 



m = l 



n 



2. .21 ^ 



M 



(x + mu))- 



-w 



1 + 



(a — fflu) 2 



u. 'US' 



i>+o-ivr ' 1+ [>-o— i>i 



j 





1 + 



(See Abel, CEuvres, vol. i. p. 213, equat. 178.) 



If we particularize this expression, by putting w=ot (which changes <j>(cc) 

 into the Lemniscate-function) and then write cot a? for a, we shall find that 

 the function of x which appears in the numerator is precisely Eisenstein's 

 function T(x). This function (which is, consequently, a particular case of 

 Jacobi's function H in his 'Fundamenta Nova') is only singly periodic; so 



that F(x)='p(x-\-~), if /x denote any real integer; but F(*' + T") ' s equal 



