TRANSACTIONS OF THE SECTIONS. 19 



and 



»r2 22.32... 



6 (22_1)(.32-1)...' 

 60 that 



_ 2r(2«+l) (22-lf(3^-l)"(5'-l)». . . 

 "~ (24)™ (22''-l)(3-»-l)(52"-l) . . . ' 



2, 3, 5, ... , being the series of prime numhera. 



On the Laiu of Distrihution of Prime Numbers. 

 By J. W. L. Glaisher, 'B.A., F.RA.S. 



In the 'Philosophical Magazine ' for July 1849 the late Mr. Hargreave proved two 

 results of great interest in the theory of minibers, viz. that the average distance 

 between two primes about the point x of the ordinal series was log^ x, and that the 

 number of primes between x' and x was very nearly lia;' — lij?, li.r being the 

 logarithm-integi'al of x, ^iz. li x=z C — — . A result practically the same was 



also arrived at by Tchebycheif, Petersburgh Transactions, 1848 (see ' Philosophical 

 Magazine,' August 1854). 



The general truth of these results was verified by IlargTeave for a number of 

 ranges among numbers less than a million ; but iu only one case did he compare the 

 numbers given by the formulae with the numbers counted above this limit. The 

 means for making this comparison are afforded bj' Burckhardt's Tables, which give 

 the least divisor of every uiunber not divisible by 2, 3, or 5 from unity to three 

 millions, and Dase's Tables, which do the same for numbers between six millions 

 and nine millions. The intermediate three millions, although existing in manu- 

 script in the libraiy of the Berlin Academy, have not been published. Burckhardt's 

 Tables were published in 1814-17, and were therefore accessible to Hargreave ; but 

 Dase's have only been published since 1862. By means of these Tables, of course 

 all the primes included within their limits can be found, as their "least factors" being 

 themselves, they are denoted in the Tables by a bar. I have therefore had all the 

 primes in every hundred of the six millions over which the Tables extend counted, 

 and have also calculated the numbers given by the formidfe ; and the results, 

 arranged in groups of 50,000 for two millions (viz. the second and the ninth), 

 are given in the two Tables below. The second million was chosen iu preference 

 to the first for insertion in this abstract, partly because results derived from the 

 counting of primes in the latter have been exhibited by Legendre, Hargreave, and 

 others, and partly because the distribution is very anomalous near the commence- 

 ment of the series of numerals. 



The numbers in the millions were divided iato groups of 50,000, and x' is written 

 for brevity for x-{- 50,000. In the first Table the numbers iu the " Primes counted " 

 column are the numbers of primes between x and x' ; thus there are 3635 primes 

 between 1,000,000 and 1,050,000, &c. In the second Table the logarithm of the 

 middle number of the group of the 50,000 was taken as the logarithm for the 

 group, and the " Average interval between the primes " was found by dividing 

 50,000 by the corresponding number in the " Primes counted " column of the fii-st 

 Table, the average intervals between two primes in the group from 1,000,000 to 

 1,050,000 being 1376, &c. 



The logarithm-integral is only a transformation of the exponential integral, the 

 relation between the two being li e''=:Ei .r ; and by the use of Toylor's theorem we 

 find 



a; 1 . 2 V.r .r- / 1 . 2 . 3 Va; x' x^ 1 



