44 Mr. C. J. Hargreave's Analytical Researches 



the X and a^+ A,r being properly the numerators of the expo- 

 nents. Now substitute for V{x-\-\) and r[x-\-iikX+l) their 

 respective algebraical approximate values 



(27ra;)^(-j and (27r(^ + A:r))*(^^j ; 



extract the a;th root of the first equation and the (o^-H- AA)th 

 root of the second ; and divide the latter by the former, re- 

 membering that the exponent ofj»y+i is in fact merely or 



1 , Py+i 

 r— ; and we get 



{w + ^xY^^'^(l+ ^\(27:{x + Lx)y^^'' {2'Kx) ^' ; 



an equation for determining Ax in terms of :t'. 



This equation becomes readily soluble if we avail ourselves 

 of an observed fact with reference to prime numbers, that Ax 

 is a small quantity as compared with x itself, at least in that 

 part of the series of ordinals which is at some distance from 

 the commencement. Taking then the logarithms of both sides 

 of this equation, and using the approximate expressions 



log {x + Ax) = log X H , 



and 



1 _ 1 Ax 



x-\- Ax X ~ x'^' 

 we have 



, Ax/ , 27r\ , I 



A^=loga;+— (^1+ log— j + 



2x 



The approximate solution of this equation obtained by rejecting 

 the terms divided by x is 



Ax= log j;; 

 and a more exact solution may be obtained by substituting 

 log X for Ax in the terms previously rejected ; which gives 



A^=log a:(l + ^ log (27re) j ; 



but the first solution is all that will ordinarily be required. 



Now j/ being the number of primes up to x, if ^=(py, we 

 have 



x + Ax=ip{y+l) = <py + <p'i/ 

 nearly ; and therefore 



Axov\ogx=i^'i/=—; 



