Mr, C. J. Hargreave on the Law of Prime Numbers. 115 



of a similar problem, by Professor Tchebycheff of the University 

 of St. Petersburgh, which appeared in the Transactions of the 

 Academy of Sciences of St. Petersburgh, having been read before 

 that learned body in the month of May, 1848 *. In that memoir 

 the same formula, lice, was arrived at by a process with which 

 the paper in the Philosophical Magazine has nothing in common; 

 and the formula was considered as denoting the limiting value 

 of the number of primes up to cc, as x increases without limit, 

 rather than as being a mode of determining the average number 

 of primes between given limits. 



The results given in that paper, (to which, as being a novel 

 and highly ingenious mode of applying analysis to a question in 

 the theory of numbers, the attention of the reader is particularly 

 called), may be thus stated. The author first demonstrates that 

 if <^^ be used to denote the exact number of primes between 

 and OS, then the expression 



(the summation with respect to x extending from the commence- 

 ment of the ordinal series 2 to £c= Qc) approximates to a finite 

 limit as p approximates to zero ; from which, it is readily inferred 

 that there are an infinite number of values of a? for which (j)X lies 



between \ix+ ■T^ — -^3 however small a be taken, and however 

 - (loga?)"^ 



large n be taken. 



The author then shows that any function of x which differs 

 from lia? by a quantity of the order of magnitude denoted by 

 X -T- (log xY, is incapable of approximating to (j>x within quan- 

 tities of the order of ^ -f- (log x)"^ ; so that, for example, the con- 

 jectural formula given by Legendre would ultimately give results 

 at variance from the truth by quantities of the order denoted by 

 X -T- (log x)"^ ; though it will not begin permanently to deviate 

 from the truth until x has reached a magnitude of about a mil- 

 lion and a quarter. 



The ultimate result is, that lia? expresses (jba? as accurately as 

 it can be expressed by means of x and its logarithmic and expo- 

 nential functions ; and the expression approximates to truth as 

 X increases without limit. 



The following new investigation of the formulae relating to 

 the occurrence of prime numbers has been suggested, partly by 

 a perusal of M. Tchebycheff's memoir, and partly by the theo- 



* " Sur la fonction qui determine la totalite des Nombres Premiers infe- 

 rieurs a une limite donnee," par M. le Prof. Tchebycheff. Lu le 24 Mai, 

 1848. 



12 



