concerning Numbers. 46 



and finally. 



^=/, 



dx 

 log^r 



or the number of primes between the limits x and x' is li.r'— lia;. 

 If we take the more accurate value of Aa^, as deduced by the 

 above method, and make 



and use the approximate value already obtained for determi- 

 ning the last term, we shall get 



y=\\x-- log (27r)(log log^) +c; 



so that the correction is scarcely appreciable. 



In applying this formula, we must bear in mind, that as it is 

 a continuous algebraical expression derived from the applica- 

 tion of a principle of means, it represents not the absolute 

 number of primes which will actually be found between any 

 particular limits, but the average number which may be ex- 

 pected to occur within such limits, having regard to the general 

 law under which primes occur throughout the ordinal series. 

 The expression log x represents nearly the average distaiice 

 between two primes at the point x in the ordinal series ; and 

 with reference to that part of the ordinals which lies near this 

 point, that is, so near that log x has undergone no appreciable 

 change, it may be expected to indicate with tolerable accuracy 

 the average distance which a table of primes would show. The 

 larger the range over which we traverse, the more nearly 

 shall we be able to test the formula, or ascertain the nature 

 and extent of its deviation from truth ; and the largeness of 

 the range must be estimated with regard to the actual magni- 

 tude of the limits. 



The formula for determining the difference between two 

 logarithm-integrals is a series of powers of t, where t denotes 

 the logarithm of the ratio of the limits; the series being con- 

 vergent with considerable rapidity when / is less than one. 



If the lower limit be x^ the upper limit a?', and log — =^, and 



> — — =«;, we have 

 log^ 



\y^\ix={a^-'X)v- i^V^|l + ^"g*^^ ^ 

 3 — 3.2u + 3.2.1w2 4 — 4.3t;+4.3.2i;2_4 3 2.it;3 -| 



^ 3.4 3.4.5 J' 



of which a very few terms will suffice for our purpose. 



