102 Mr Glaisher, On factor tables. [Feb. 11, 



It follows from the description giveu above that any method of 

 constructing a factor table affords all the prime factors, so that as 

 far as the actual calculation of the table is concerned, it is as easy 

 to give all the prime factors as to give the least. A little time, 

 in actual writing, may be saved by entering only the least factor, 

 but this is counterbalanced by the much greater ease with which 

 a table containing all the prime factors can be verified while in 

 course of construction. 



§ 2, There are several cases in which the factors of a number 

 may be of use in ordinary mathematical processes ; as for example, 

 in the calculation of logarithms : but the real importance of an 

 extended factor table lies in the fact tbat such a table is a funda- 

 mental one in the Theory of Numbers. The number of factors 

 of a number, their sum, &c., are elements which enter into many 

 results in the Theory of Numbers, and it is clear that, even on 

 this account alone, it is desirable to have a table by means of 

 which the resolution of high numbers into their factors is rendered 

 practicable, so that suitable examples, verifications, &c., of such 

 theorems may be obtainable. Also, conjectural theorems of this 

 nature could not readily be tested without such tables. Again, 

 the law of frequency of prime numbers has been the subject 

 of analytical investigation, and theoretical formulae giving the 

 average frequency of primes, and the approximate number of 

 primes between any given limits, have been obtained by Legendre, 

 Tchdbychef, Hargreave, Riemann and others, and it is essential 

 to have the means of comparing the numbers given by these 

 formulae with those found by actually counting the primes \ Such 

 asymptotic formulae cannot be satisfactorily tested by means of 

 enumerations made near the beginning of the series of numerals, 

 for the slightest consideration shows how anomalous is the distri- 

 bution of primes at first : in fact, if we possessed a complete factor 

 table from 1 to 10,000,000, this would be found to be only barely 

 sufficient to afford comparisons of real value. 



The practical use of a factor table in obtaining logarithms has 

 been alluded to at the beginning of this section. This, although 

 not the raison d'etre of such a table, is so important an application 

 as to deserve special notice, for by its means the number of 

 numbers whose logarithms are known is greatly extended. For 

 example, Abraham Sharp's table contains CI -decimal Briggian 

 logarithms of primes to 1,100, so that the logarithms of all 

 numbers whose greatest prime factor does not exceed this number, 



1 Eeferences to these investigations are given by Professor H. J. S. Smith 

 in his address, ' ' On the present state and prospects of some branches of pure 

 mathematics." Proceedings of the London Mathematical Society, Vol. viii. (no. 

 104) pp. 16—19, (1876—1877). 



