ON THE THEORY OF NUMBERS. 261 



tions to the arithmetic of real integers. The function <f>(v) also satisfies the 

 equation f(i k v) = i k <p(v), whence 



•A,+2fr,+3fc,_ V Pi / _(9x\ /I \ 



_ ncr u 



the sign of multiplication II extending to every residue r included in the 

 group (0). Similarly, if q v like p x , be a prime, 



®. 





(2.) 



s denoting the general term of a quarter-system of Residues for the mo- 

 dulus a,. 



By an elementary theorem in the calculus of Elliptic Functions, ■ ^ is 



for every uneven value of k a rational and fractional function of x=q>(v). 



/2raA 

 If p l be primary, as we shall now suppose, and if we put a r =(M ), we have, 



by the principles of that calculus, 



<t>( Pl v)_ n(x*-«* ) rs x 



0(r) ~n(l-«V)' V *' 



the sign II extending to all the different values of o. r ; and similarly, 



0(y,tQ_ n(V-/3*) , 4) 



K«) ~n(i-/3V)' w 



if /3 S =^>( — ). Combining the equations (3.) and (4.) with (1.) and (2.), 

 we find 



W,~n(i-« v /' 4 ) 



(Pi\ n(/3 4 -"' ) 

 W«~n(i-* 4 /37 



the sign of multiplication extending to the |(p— lX? - 1) combinations of 

 the values of a and (i; whence, evidently, 



D.(D=(- i >^ ,x - ,) - 



The priority of Eisenstein in this singularly beautiful investigation is 

 indisputable. 



34. In a later memoir (Beitrage zur Theorie der Elliptischen Functionen, 

 Crelle, xxx. p. 185, or Math. Abhandl. p. 129), Eisenstein has put this proof 

 into a slightly different form. He shows, by a peculiar method, that if p 1 be 

 an imaginary and primary complex prime, every coefficient in U(x 4 — a*) ex- 

 cept the first is divisible by p v and that for every primary uneven value of p x 

 (whether prime or not) the last coefficient is p v so that ( — ]) iip ~ i) p l =na\ 



