OX THE THEORY OF NUMBERS. 161 



form, and at the same time the necessity for constructing a system of qua- 

 dratic forms of determinant — D is avoided by the following modification of 

 it: — By a known property of quadratic forms, whenever the congruence 

 r 2 +D=0, mod P, is resoluble, the equation mP=r+Dy is resoluble for 



some value of m < 2*/?. By assigning, therefore, to m all values in suc- 

 cession which are inferior to that limit, and which satisfy the condition 

 ( f)) = ( f))' anc * tIien obtaining (by Gauss's method) all prime representa- 

 tions of the resulting products by the form a^-j-Dy 5 , we shall have r~± — 



x" 



r = ± -ff> mod P, x', y', x", y" etc. denoting the different pairs of values 



of a: and y in the equation mP=x 2 + Dy 2 . 



69. General Theory of Congruences — We may infer from several passages 

 in the Disquisitiones Arithmeticse, that Gauss intended to give a general 

 theory of congruences of every order in the 8th section of his work, and 

 that, at the time of its publication, he was already in possession of the prin- 

 cipal theorems relating to the subject*. These theorems were, however, 

 first given by Evariste Galoist, in a note published in the Bulletin de Ferus- 

 sac for June, 1830 (vol. xiii. p. 438), and reprinted in Liouville's Journal, 

 vol. xi. p. 398. An account of Galois's method (completed and extended in 

 some respects) will be found in M. Serret's Cours d'Algebre Superieure, 

 lecon 25. The theory has also been independently investigated by iVL Schoe- 

 nemann, who seems to have been unacquainted with the earlier researches of 

 Galois (see Crelle's Journal, vol. xxxi. p. 269, and vol. xxxii. p. 93). In 

 several of Cauchy's arithmetical memoirs (see in particular Exercices de 

 Mathematiques, vol. i. p. 160, vol. iv. p. 217; Comptes Rendus, vol. xxiv. 

 p. 1117; Exercices d'Analyse et de Physique Mathematique, vol. iv. p. 87) 

 we find observations and theorems relating to it. Lastly, in a memoir in 

 Crelle's Journal (vol. liv. p. 1) M. Dedekind has given (with important 

 accessions) an excellent and lucid resume of the results obtained by his pre- 

 decessors. 



In the following account of the principles of this theory, the functional 

 symbols F, cp, \p, . . . will represent (as in general throughout this Report) 

 rational and integral functions having integral coefficients; we shall use j9 

 to denote a prime modulus, and x an absolutely indeterminate quantity. As 

 we shall have to consider the functions F(x),f(x), ^(x), etc., only in relation 

 to the modulus/*, we shall consider two functions F, (x) and F 2 (x), which 

 differ only by multiples of p, as identical, and we shall represent their identity 

 by the congruence ? x (#)=F 2 (x), mod p, which is equivalent to an identical 

 equation of the form F 1 (x) = F 2 (x)+pp(x). The designation "modular 

 function," which has been introduced by Cauchy (Comptes Rendus, vol. xxiv. 

 p. 11 18) will serve (though, perhaps, not in itself very appropriate) to indicate 

 that the function to which it is applied is thus considered in relation to a 



* See Disq. Arith. art. 11 and 43. 



t Galois was born October 26, 181 1, and lost his life in a duel, May 30, 1832. He was 

 consequently eighteen at the time of the publication of the note referred to in the text. His 

 mathematical works are collected in Liouville's Journal, vol. xi. p. 331. Obscure and frag- 

 mentary as some of these papers are, they nevertheless evince an extraordinary genius, un- 

 paralleled, perhaps, for its early maturity," except by that of Pascal. It is impossible to read 

 without emotion the letter in which, on the day before his death and in anticipation of it, 

 Galois endeavours to rescue from oblivion the unfinished researches which have given him a 

 place for ever iu the history of mathematical science. 



I860. M 



