306 REPORT— 1875. 



D. Algebraic constants. 



9. Accurate integer or fractional values. Bernoulli's Nos., A'jO'", d'c. 

 Binomial coefficients. 

 10. Decimal values auxiliary to the calculation of series. 



E. 11. Transcendental constants e, tt, y, &c., and their povrers and func- 



tions. 

 The present Eeport (1875) comprises the tables included under the head- 

 ings : — 



F. Arithmological. 



12. Divisors and prime numbers. Prime roots. The Canon arithme- 



ticus, &c. 



13. The PeUian equation. 



14. Partitions. 



15. Quadratic forms ct?-\-h' &c., and partition of numbers into squares, 



cubes, and biquadrates. 



16. Binary, ternary, &c. quadratic and higher forms. 



17. Complex theories, 



which divisions are herein referred to, for instance, as [F. 12. Divisors ikc.]. 

 Report 1873 consists of six sections (§) divided into articles, which are 

 separately numbered (see contents, p. 174) ; the present Rejiort 1875 forms 

 a single section (§ 7), divided in liie manner into articles, which are sepa- 

 rately numbered ; but besides this the paragraphs are numbered, and that 

 continuously, through the present Report 1875, so that any paragraph may 

 be cited as Report 1875, No. — (as the case may be). 



Art. 1. [F. 12, Divisors Sfc.'] Divisors and Prime Numhers. 



1. As to divisors and prime numbers see Report 1873, art. 8 (Tables of 

 Divisors (factor tables) and Tables of Primes), pp. 34^40. The tables there 

 referred to, such as Chernac, Burckhardt, Dase, Dase and Rosenberg, are 

 chiefly tables running up to very high numbers (the last of them the ninth 

 million), wherein to save space multiples of 2, 3, 5 are frequently omitted, 

 and in some of them only the least divisor is given. It would be for many 

 purposes convenient to have a small table, going up say to 10,000, showing 

 m every case all the prime factors of the number. Such a table might be 

 arranged, 500 numbers in a page, io. some such form as the following ; — 



Factor Table 1 to 500 



12345 6 78 9 



39 2.3.5.13 17.23 2^7^- 3.131 2.197 1 5.79 2^3M1 397* 2.199 3.7.19 



where the top line shows the units, and the left-hand column the remaining 

 figures, viz. the specimen exhibits the composition of the several numbers from 

 390 to 399 : a prime number, e. g. 397, would be sutfieiently indicated by the 

 absence of any decomposition, or it may be further indicated by an asterisk. 



It may be noticed that in the theory of numbers the decomposition is spe- 

 cially required when the next following number is a prime, viz. that of p — 1, 

 p being a prime : and that this is given incidentally, for prime numbers ^ 

 up to 1000, in Jacobi's ' Canon Arithmeticus,' post, No. 20, and up to 15,000 

 in Reuschle's Tables, V. (a, b, c) post, No. 22. 



2. It may be proper to remark here that any table of a binary form is 

 really a factor-table in the complex theory connected with such binary form. 



