302 Mr Olaisher, On Legendre's formula for [Dec. 8, 



It thus appears that even when x is as large as 100,000,000 the 

 corresponding value of A is only reduced to 1 '06397, so that the 

 approach towards unity is extremely slow. Hargreave calculated 

 also the number of primes (excluding unity) inferior to 5,000,000 

 at 348,527; which gives to A the value 1-07890. The factor table 

 for the fifth million will shortly be published, and this will contain 

 the number of primes up to 5.000,000 as found by actual counting; 

 an enumeration of the fifth million already made, but not fully 

 verified, gives a result differing by about 15 from Hargreave's 

 value. 



§ 7. It may be observed that the value of A is very sensitive, 

 i.e. that a small difference in the number of primes counted 

 causes a considerable difference in the five-decimal values of A. 

 This is due to the fact that A is calculated from the formula 



l°g^-^) (1)' 



both terms of which, for the values of x considered, contain one 

 figure more than their difference. 



The ' sensitiveness ' of J. is shown by Tables I. and II. of § 3, 

 where for example a difference of only 2 in the value of {x) for 

 a; = 1,000,000 produces a difference of 0-00033 in the value of A, 

 and a difference of 49 in the value of ^ {x) for x = 2,000,000 pro- 

 duces a difference of 0-00442 iu the value of A. To illustrate the 

 loss of a figure by the subtraction in (1) we may take as examples 

 the extreme cases of ^ = 250.000 and x = 100,000,000. 



When X = 250,000, 



log a; = 12-4292162, —--=11-3404400, 

 * 4>{x) 



giving 



A = 12-4292162 - 11-3404400 = 1-0887762 ; 



and when x = 100,000,000, 



rp 



log X = 18-4208807, -j—, = 17-3567086, 



giving 



A = 18-4206807 - 17-3567086 = 1-0639721. 



§ 8. Since the limiting value of A when x is infinite is unity, 

 it seems worth while to compare the values given by the formula 



log x — 1 

 with the numbers of primes counted. The results of this com- 

 parison are shown in the following table ; in which the last column. 



