ON THE THEORY OF NUMBERS. 267 



The complementary theorems* relating to the unit p and the prime 1 — p 

 (which are not included in the preceding investigation), are 





,i{N/' i -U_ p « + ^ ! 



where p x is a primary prime, and a and /3 are defined by the equality 

 p x = 3 a, -l+3/3p. 



Eisenstein has observed t that a demonstration of the law of cubic reci- 

 procity, precisely similar to that analysed in Art. 33 of this Report, may be 



al jVcrb) ai 



obtained by considering the integral j , , :i and its inverse function, 



instead of the Lemniscate integral and Lemniscate function. He has not, 

 however, entered into any details on this interesting subject (which is the 

 more to be regretted, because there appears to be no published memoir 



treating specially of the integral S —j ,. _ 3 ^ ); although his latest proof of 



the Biquadratic Law (see Art. 35) has been exhibited by him in such a form 

 as to extend equally to Cubic Residues, and even to residues of the sixth 

 power. 



38. The first enunciation of the law of Cubic Reciprocity is due to Jacobi, 

 and the demonstration of it which we have inserted in the preceding article 

 is doubtless the same wilh that which he gave in his Kdnigsberg Lectures. 

 In one of his earliest memoirs ("De residuis cubicis commentatio numerosa," 

 Crelle, vol. ii. p. 66), which was composed after the announcement, but before 

 the publication, of Gauss's memoirs on Biquadratic Residues, Jacobi had 

 already arrived at two theorems relating to Cubic Residues, which involve the 

 law of Reciprocity, and which he seems to have deduced from his formulas 

 for the division of the circle. But, as it had not occurred to Jacobi, at the 

 time when this memoir was written, to introduce, as modules, instead of the 

 prime numbers themselves, the complex factors of which they are composed, 

 the law of Cubic Reciprocity in its simplest form does not appear in the 

 memoir. 



To complete the present account of the Theory of the Residues of Powers, 

 or of Binomial congruences, we should have in the next place to review the 

 recent investigations of M. Kummer on complex numbers, and on the reci- 

 procity of the residues of powers of which the index is a prime. But the 

 consideration of these investigations, as well as of the other researches be- 

 longing to our present subject, our limits compel us to postpone to the second 

 part of this Report. 



* Eisenstein, Crelle's Journal, vol. xxviii. p. 28 (the continuation of the memoir cited in 

 the preceding note). 

 t In the memoir, "Application de I'Algebre," &c, already referred to. 



