ON MATHEMATICAL TABLES. 311 



b) a i)rime, for the several prime values ai=3 to 1229 in the Table IX. (one 

 page) at the end of the work. 



14. A table of frequency is given 



Glaisher, J. AV. L., British Association Report for 1872, p. 20. This 

 gives for the second and ninth millions, respectively divided into intervals 

 of 50,000, the actual number of primes in each interval, as compared with 

 the theoretic value li.r' — ILt' ; and also deduced therefrom, by the formula 

 log h{x' -\-x)^ a table of the average interval between two consecutive primes ; 

 this~average interval increases very slowly : at the beginning and end of the 

 second million the values are 13' 76 and 14"5S (theoretic values 13'84 and 

 14"50) ; at the beginning and end of the ninth million 16-02 and 15-95 

 (theoretic values 15-90 and 16-01), 



15. Coming under the head of Divisor Tables, some tables by Reuschle 

 and Gauss may be here referred to. These are : — 



Reuschle, Mathematische Abhandlung, zahlentheoretische Tabellcn 

 sammt einer dieselben treffenden Correspondenz mit der verewigten C. G. J. 

 Jacobi, 4°, pp. 1-61 * (1856). The tables belonging to the present subject are 

 A. Tafeln zur Zerlegung von «»— 1 (pp. 18-22). 

 I. Table of the prime factors of 10" — 1, viz. 

 (a. pp. 18-19). Complete decomposition of 10»— 1 (m=1 to 42) and 10" + 1 

 (n=l to 21). Some values of n omitted. 

 A specimen is 



10" -1=3% 53 . 79 . 265371653, 



10" + 1 = 11 . 189 . 1058313049. 



(b. p. 19). List of the specific prime factors/ of 10" — 1 (or the prime 

 factors of the residue after separation of the analytical factors) 

 for those values of n for which the complete decomposition is 

 unlinown, and omitting those values for which no factor is known, 

 «=25 to 243. 

 A specimen is n f 



25 2141. 



The meaning seems to be, residue of 10''— 1 is 1 + 10' + 10" + 10" + 10-''', 

 and this contains the prime factor 21401 ; but it is not clear why this is 

 the " specific prime factor." 



II. Prime factors of a" — 1 for different values of a and n. 



(a. p. 20) gives for 41 values of a (2, 3, &c. at intervals to 100) and 

 for the following values of n the decompositions of the residues 



* Titlepage missing in my copy ; but I SbcI from Prof. Kuminer's notice of the work, 

 ' Crelle,' t. liii.(1857), p. 37'J, that it appeared as a Programm of the Stuttgart Gymnasium, 

 Michaelmas, 1856, and was separately printed by Liescliiug and Co., Stuttgart. 



