116 Mr. C. J. Hargreave on the Law of Prime Numbers. 



terns given in the latter part of my former paper. In that paper 

 the term average is applied in its ordinary sense, which is well 

 understood as meaning the rate at which primes occur at and 

 about any particular points of the ordinal series. 



It is easy, however, to suggest other and more algebraical 

 meanings of the term average as applied to this subject. Let 

 Pn denote the wth actual prime number ; and let us compare the 

 expression 2<^(j9„)from ;?„=2 to ;;„= oc with another expression 

 S<^(P„) between the same limits, where P„ represents one of a 

 series of terms P,, Pg, P3 . . . vvhich are connected with each other 

 by the law Pn+i = P„ + V^(Pn). Then if 2</)(P„) be equal to 

 S<^ (/>„), or differs therefrom only by quantities of an order lower 

 than 2^(j9„),we may say that the two series pY,p^,p^,;V^,V2,l^s'** 

 run pari passu with each other, though there may be no such 

 thing as an absolute equality between any term of one series and 

 the corresponding term of the other. The law of formation, 

 which is strictly true with regard to the latter series, may be 

 regarded as possessing a species of average truth with reference 

 to the former series. 



Lemma. The expression \p , where 



V = H- 2TT; + 3rrp + 4rrp + • • • c^d infinitum 



approximates to 7, or '57712 ... as /o diminishes without limit. 

 For 



r tp ^'\^ dt=\ tp{6-^ + e-''^-\-6-^^+...)dt=T(p-\-l)\p. 

 Jo ^""^ Jo 



But, since • 



1 1 1 j_,_2._f_ LJl 



l-€-'~ t'^2'^12 30[4]^43[6] '"' 

 we have 



£t<'j^,dt=Tp+lT{p+i)+hr{p+z)-±r(p-\-3)+...; 



whence 

 and 



From this it follows, that as p diminishes, the logarithm of \p 

 approximates to — log /3 4- 7/3. 



Now, by referring to my former paper, it will be seen that 



