concerning Numbers. 49 



appears in some cases to give results more closely correspond- 

 ing with the Tables than are derived from the logarithm-inte- 

 gral. Thus for the primes from 100,000 to 1,000,000 it gives 

 68,853, the actual number being 68,901. 



Prop. 4. To find the sum of any function (f) of the primes 

 up to A\ the wth prime. 



If this sum (exclusive of the function of ^ itself) be denoted 

 by rI/«, we have {x being a function of w, say ft,n) 



f(2) + f(3)+f(5) + ...-f<p(a(«-l)) =4'//^ 



f(2)+f(3)+f(5) + ... + <p(|^(n--l)) + 9(f*(«))=4'(«+l); 

 whence 



and 



^ J^ ^^ Q 2 dn 30 2.3.4 dn^ 



Now by the last proposition 



,. J dx 



n—i\x or dn=, : 



log a? 



whence 



Cases of the above Theorem. 



Case 1. Let (^x—x] then the sum of the primes up to x 

 inclusive is 



or 



c-\.\\{a'^)+-x+ —logx + ..; 



that is, the sum of the primes from 3/ exclusive to x inclusive, 

 corrected by deducting half the difference between y and ^, 

 (for the other terms may be neglected), is equal to the number 

 of primes between j/^ and x'^. 



This remarkable theorem, which connects in an unexpected 

 manner the values of the primes in one part of the series with 

 their number in another and remote part of the series, will 

 enable us to verify the theorem as to the number of primes, 

 and will confirm the conclusion that the relative error in de- 

 termining the number of primes diminishes as the numbers 



Phil. Mag. S. 3. Vol. 35. No. 233. JuIj/ 1849. E 



