ON THE THEORY OF NUMBERS. 325 



It appears from the Additamenta to art. 306, X. of the Disq. Arlth., that 

 Gauss, at the time of the publication of that work, had already succeeded in 

 effecting this determination ; and the method by which lie effected it will at 

 length appear in the second volume of the complete edition of his works, 

 the publication of which is now promised by the Society of Gdttingen. 

 Nevertheless the originality of Dirichlet in this celebrated investigation is 

 unquestionable, as there is nothing whatever in the Disq. Arith. to suggest 

 either the form of the result, or the method by which it is obtained *. 



We propose, in what follows, to give as full an analysis as our limits will 

 permit of the contents of the memoir. Its first section contains certain prin- 

 ciples relative to the theory of series. 



(i.) " If ^1 ^ ^o — ^3 = ^4 • • • b^ ^ series of continually increasing positive 

 quantities; and if the ratio j- continually tend to a finite limit a (that 



is to say, if, S denoting a given positive quantity, however small, we can 

 always assign a finite value of n=N, such that for all values of w surpassing 

 N, the inequalities 



are satisfied), the limit of the expression p S , when the positive 



quantity p is diminished without limit, is a."t 



ForpS 7-r7-=p'^ ■mr+p'^ T-rr-' N denoting a finite 



71=1 ^n+O ' n=l kn'+P ^ = ^^+1^'+" 



number ; and by virtue of the inequalities written above 



„=N+1 "'+" n^N + l^"'^'' w=N + l "'^^ 



* The following is a list of the papers of Lejeune Dirichlet which relate to the theory of 

 quadratic forms : — 



1. Siir I'usage des series infinies dans la thcorie des nombies. — Crelle, vol. xviii. p. 259. 



2. Recherches sur diverses applications de I'analyse infinitesimale a la theorie des norabres. 

 — Crelle, vol. xix. p. 324, and xxi. pp. 1, 134. 



3. Auszug aiis einer der Akademie der Wissenschaften zu Berlin am 5 Marz 1840 vorge- 

 lesenen Abhandlung. (Crelle, vol. xxi. p. 98, or the Monatsberichte for 1840, p. 49.) 



This paper is an abstract of an unpublished memoir containing the demonstration of the 

 theorem that every properly primitive form represents an infinite number of primes. 



4. Untersuchungen iiber die Theorie der complexen Zahlen. (Crelle, vol. xxii. p. 375, or 

 in the Monatsberichte for 1841, p. 190.) An abstract of the following memoir. 



5. Recherches sur les formes quadratiques a coefficients et a indetermines complexes.— 

 Crelle, vol. xxiv. p. 291. 



6. Sur un theoreme relatif aux series. (Liouville, New Series, vol. i. p. 80, or Crelle, 

 vol. liii. p. 130.) 



7. Sur une propriete des formes quadratiques a determinant positif. (Monatsberichte for 

 July 16, 1855, or Liouville, Nevi- Series, vol. i. p. 76, or Crelle, vol. liii. p. 127.) 



8. Vereinfachung der Theorie der binaren quadratischen Formeu von positiver Determi- 

 nante. (Memoirs of the Berlin Academy for 1854, p. 99, or, with additions by the author, 

 in Liouville, New Series, vol. ii. p. 353.) 



9. Demonstration nouvelle d'une proposition relative a la theorie des formes quadratiques. 

 — Liouville, New Series, vol. ii. p. 273. 



10. De formarum binariura secundi gradus compositione. — Crelle, vol. xlvii. p. 155. 



The three last papers contain important simplifications of theories which appear in a very 

 complicated form in the Disq. Ariih. To two of them we have already referred (arts. 93, 94). 



t This theorem is a generalization of that in the memoir (Crelle, vol. xix. p. 326). It is 

 given by Dirichlet in No. 6 of the preceding list. 



