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XVII. On the Law of Prime Numbers, 

 By Charles James Hargreave, Esq,, LL.D., F.R.S.* 



IN a paper written by me in the year 1849, and published in 

 the rhilosophical Magazine, vol. xxxv. p. 36, I attempted, 

 by means of certain principles there laid down, to apply the pro- 

 cesses of analytical investigation to the theory of numbers ; and 

 I thereby arrived at certain conclusions relative to the occurrence 

 of prime numbers in the ordinal series which were expressed in 

 the following proposition : — " The average distance between two 

 successive prime numbers at the point x in the ordinal series is 

 . log X ; and the average number of primes which may be expected 

 to occur between x and x' is the logarithm-integral of x between 

 those limits, or \\x'—\ix." 



The nature and exact purport of this theorem will be readily 

 understood by any person who takes the trouble of counting the 

 number of primes between various limits, when he will discover 

 that, by taking ranges of sufficient magnitude, the rate at which 

 the primes occur appears to follow a very regular and uniform 

 law, though nothing can be apparently more irregular than the 

 particular places at which the individual primes are to be found. 



In the paper above alluded to, the law was verified for various 

 numbers under one million ; and it was found that the formula 

 always produced the proper number of primes within very mode- 

 rate limits of error. 



I propose in the present paper, first, to make a further inves- 

 tigation as to the exact nature of the formula, and of the demon- 

 stration upon which it rests ; and secondly, to point out a prac- 

 ticable method by which the number of primes can be counted 

 to a point in the ordinal series far beyond the limits of the exist- 

 ing tables of primes, a method which will of course enable us to 

 apply a more satisfactory test to the formula itself. 



Discussion of the Formula \ix. 



For the process by which this formula was orginally obtained 

 I must refer the reader to the paper above mentioned. It will 

 suffice for the present purpose to state, that the power of analy- 

 tically investigating such a question was made to depend upon 

 the substitution of an analytical equivalent for a quantity which 

 was necessarily from the nature of the problem indeterminate, 

 hut susceptible only of certain specified values. The equivalent 

 thus substituted was, as may be anticipated, the arithmetical 

 mean of the possible values. 



Since the publication of that paper, my attention has been 

 directed by my esteemed friend Mr. Sylvester to an investigation 



* Ck)mmunicated by the Author. 



