258 report — 1859. 



\P(i)=a + bi=p l ; \l>(—i)=a—bi=p 2 , so that p x p,,=p. The con- 

 je ^[yl^-D]]— 0> mod p, or a + £y^P -1 )=0, mod p, involves also the 



Let 

 gruence i//[y*^- I - , J=0, mod p, or a+Oy^p-'^^U, moap, involves also tne 

 congruence a + 6y* ( ' ,_1) = 0, mod p x ; i.e. yifo'-Dsst, mod j^ ; so that 



(Z_ j =e*. Hence we have, putting/ =&, mod p, 

 Pi/* 





From these formulae two cases of the law of Reciprocity are directly de- 

 ducible. 



a. Let q be a real prime of the form in + 3. Raising S to the power q, 

 we have 



k=p— l/i\, k=p— 1/1TS3 /„\ 



r*ii ^-i, (l>" s (?:), T "" M, ^ y(L) - 



Multiplying by S, we find 



5+i 2+1 j+i c-i , , 



S =(S l ) * =p * Pl 2 =(-1) * p (x.) ,'mod g; 



or, observing that /»,=/»/, mod #, aadp=p 1 p af 



*i 4 =(-1) 4 (^),rnod 9 ; 



«""» (tH^f), (A) 



which is in accordance with the law of Reciprocity. 

 /3« Again, let </ be a prime of the form 4?a + l ; 



then ! <! = (£. ) S, mod g; that is, S?-'^/ JL) , mod ff, 



\j»iA \/>iA 



or p*te-D/> t ite- 1 3^(^\^modyj 



V /'iA 



whence, if q=< J \q& 



\<h)\~<h)r\FJ* 



But, by changing i into -,", (A ) =(^) , and ($-($ 



-* (vHfX <?■> 



The symbolic equations (A.) and (B.) lead immediately to the conclusion 

 tint if a and b be any two primary uneven numbers, one, at least, of which is 



- 1 " •<&-©■-—-"— —- 



