120 REPORT — 1860. 



CORRIGENDA ET ADDENDA. 



Page 53, line 22 from top : for " 1596 " read 1596.*. 



Page 59, line 18 from top : for " Sarthe " read Sarthe. 



Page 62, 1804. Apr. 15. Geneva, fireball : add, s. to n. ; also, followed by a train of smaller 

 balls. 



Page 64, line 18 from top : for " Aug. 10 " read early part of Aug. 



Page 64, line 12 from bottom : for " Iron-fail" read Stone-fall. 



Page 65, line 7 from bottom : fireball at Gottingen ; add, followed by many smaller balls. 



Page 67, top line : for " 1819. June 13. Jonsac " read and add, 1819.* June 13. Jonsac, 

 Chare nte, &c. &c. 



Page 70, line 5 from top : Gorlitz ; fireball ; add aerolitic ?. 



Page 71, line 11 from bottom : replace the (*) before May 12, by a (?). 



Page 71, line 12 from bottom : insert the (*) before May 19. Ekaterinosloff, &c. 



Page 72, line 6 from top : read February 27 or February 16. 



Page 72, line 11 from bottom of Notes : for " Summer co." read Sumner co. 



Page 73, line 3 fro:n top : add Vouille near " Poitiers." 



Page 74, line 21 from top : for " Okaninak " read Okaninah. 



Page 82, line 6 from top : for " Nuremberg " read Nurenberg. 



Page 92, after line 5 from top : insert, Apr. 12. Berne. Fireball. 



Page 94, line 10 from top : for " Columb " read Columbus. 



Page 96, after line 16 from bottom: insert I860.* Feb. 2. Alessandria, Piedmont. A stone- 

 fall. Also omitted in the Tables. 



Report on the Theory of Numbers. — Part II. By H. J. Stephen 

 Smith, M.A., F.R.S., Savilian Professor of Geometry, Oxford. 



39. Residues of the Higher Powers. Researches of Jacobi. — The principles 

 which have sufficed for the determination of the laws of reciprocity affecting 

 quadratic, cubic, biquadratic, and sextic residues, are found to be inadequate 

 when we come to residues of the 5th, 7th, or higher powers. This was early- 

 observed by Jacobi, when, after his investigations of the cubic and biqua- 

 dratic theorems, he turned his attention to residues of the 5th, 8th, and 12th 

 powers*. It was evident, from a comparison of the cubic and biquadratic 

 theories, that in the investigation of the laws of reciprocity the ordinary 

 prime numbers of arithmetic must be replaced by certain factors of those 

 prime numbers composed of roots of unity ; and Jacobi, in the note just re- 

 ferred to, has indicated very clearly the nature of those factors in the case 

 of the 5th, 8th, and 12th powers respectively. He ascertained that the two 

 complex factors composed of 5th roots of unity into which ever)' prime 

 number of the form 5n+l is resoluble by virtue of Theorem IV. of art. SO 

 of tbis Report, are not prime numbers, i. e. are each capable of decomposi- 

 tion into the product of two similar complex numbers; so that every (real) 

 prime number of the form 5n-\-] is to be regarded as the product of four 

 conjugate complex factors ; and these factors are precisely the complex primes 

 which we have to consider in the theory of ;quintic residues, in the place of 

 the real primes they divide. To this we may add that primes of the forms 

 5?i + 2 continue primes in the complex theory ; while those of the form 5n — 1 

 resolve themselves into two complex prime factors. Thus 



7 = 7; ll = (2 + a)(2 + ar)(2 + a. 3 X2 + a 4 ); 13 = 13; 



19 = (4-3(a + a 4 ))(4-3(<r + <)) ; 29=(5-(a + a*))(5-(a a + a 8 )) ; 



31 = (2-a)(2-a 2 )(2-a 3 )(^-a 4 ), &c, 



* See a note communicated by him to the Berlin Academy on May 16, 1839, in the 

 ' Monatsberichte ' for that year, or in Crelle, vol. six. p. 314, or Liouville, vol. viii. p. 268, 

 in which, however, he implies that he had not as yet obtained a definitive result; nor does 

 he seem at any subsequent period to have succeeded in completing this investigation. 



