ON MATHEMATICAL TABLES. 31 



eacli hundred, or century, in that million : it is similar to Part I. of last 

 year's Report, -which related to the fourth million. 



As the factor tables for the first five millions ai'e now published, so 

 that it is for the first time possible to extend the enumerations continuously 

 from to 5,000,000, it was thought desii'able to give here in a tabular 

 form the main facts relating to the distribution of primes over this range : 

 these tables form Section III. The results are given very briefly, because 

 it is hoped that by next year the series of tables will be complete as far 

 as 9,000,000, and a more detailed examination is deferred till it can be 

 rendered as complete as possible. 



One of the objects to which enumerations of primes are most directly 

 applicable is the examination of the degree of accuracy with which the 

 numbers of pi-imes in any given intervals are represented by certain 

 formute which have been proposed for the purpose. A formula of this 

 kind was proposed by Legendre, and another was independently obtained 

 by Gauss, TcbebychefF, and Hargreave. Certain comparisons between 

 the numbers of primes counted and the numbers given by these two 

 formulfe for intervals between and 5,000,000 are contained in Sec- 

 tion IV. 



I. State of the Factor Tahles for the Fourth, Fifth, and Sixth Millions. 



During the year the calculation for the three millions has been com- 

 pleted, and the printing of the tables has been steadily continued under 

 the direction of Mr. James Glaisher. The volumes containing the factor 

 tables for the fourth and fifth millions have been published, and twenty 

 pages of the volume containing the sixth million are now printed and 

 stereotyped. 



The fourth million was published in December, 1879, by Messrs. 

 Taylor and Francis. The table itself occupies 112 pages, and is uniform 

 with those of Biirckhardt and Dase. There is an introduction of fifty- 

 two pages, consisting of eight sections and an appendix. The titles of 

 the sections are (1) Manner of using the Table; (2) The Tables of 

 Burckhardt, Dase, and Chernac ; (3) Mode of Construction of the 

 Table ; (4) On Factor Tables ; (5) On the Distribution of Prime 

 Numbers ; (6) List of Writings on the Distribution of Prime Num- 

 bers ; (7) Results of the Enumeration of the Prime Numbers in the 

 Fourth Million ; (8) Application of the Table to the Calculation of 

 Logarithms. The appendix contains a list of prime numbers from 1 to 

 30,o41 with differences : this list was used in the determination of least 

 factors by the multiple method. There is also a specimen of one of the 

 lithographed sheets used in the calculation of the table, and from which 

 the sieves w^ere formed by stamping out certain of the squares. An 

 abstract of the third section, which relates to the mode of construction 

 of the table, appeared in the Report for 1878, and an abstract of the 

 seventh section, which contains the tables derived from the enumeration 

 of the primes in the fourth million, formed Part I. of last year's Report. 



The introduction to this million is intended to apply to the whole 

 three millions. 



The fifth million was published in July of this year. The introduction, 

 which contains eleven pages, consists of only two sections, the first of 

 which relates to the manner of using the table, and the second to the 

 results of the enumeration of the primes in the fifth million. An abstract 

 of the latter forms Section II. of this Report. 



