concerning. Numbers. 43 



't> 



\py 



character; for (-^) vanishes whenever—- is less than 1. 



\py p^ its 



Now in applying to the equation the principle of means in the 

 manner above suggested, it is proper to consider it in this ex- 

 tended form ; for if we suppose the series to stop so soon as 



p^ exceeds ar, then in deducting I — ) + ( — 2 ) + • • fro^ 



j + ( 2 — ) + • • J we lose sight of the circumstance 



that the number of terms in the latter expression is not neces- 

 sarily the same as that in the former expression ; or, in other 

 words, that the mean number of terms in the latter is neces- 

 sarily greater than the mean number of terms in the former. 

 Acting upon this principle of means, we equate 



(^)+e-^)--((|;)-(~7)-") 



to 



^"(^+^-*-")' 



which amounts in effect to this ; that if we write the first part 

 in the form 



{(jXih-)- 



(x+Ax) 

 and the other part in the form 



then for the purpose of subtraction the ( — j + (— -2} + . • niay 



be regarded as meaning the same thing in both. 



Prop. 3. To investigate the law which regulates the occur- 

 rence of prime numbers in the ordinal series. 



Let j: be a prime number occupying the place py in the 

 series of primes; and let x + Ax be the next prime number 

 occupying the place jo^+i in the series of primes. We have 

 then 



T{x+l)^(p(^F)^-pJ^y- .,.p(jy)^y^ 



T{x + Ax + 1) 



