concerning Numbers. 51 



of the square roots of the primes from y exclusive to x inclu- 

 sive, diminished by - ( s/x— Vy)i is the number of the primes 



from 5/^ to ^^; and a similar theorem may be expressed for 

 any other roots, or for roots of povi^ers. 



As an example, the sum of the square roots of the primes 

 from 1789 exclusive to 2399 inclusive is 3638*5; from which 

 we deduct 3*4 for half the diflf'erence of the roots, leaving 

 3635 ; and it will be found by counting, that the number of 

 primes from 1789^ or 75,668 to 2399^ or 117,502 is 3631 ; 

 a degree of accuracy which arises from the circumstance that 

 between 1 800 and 2400 there is no great excess or defect of 

 primes from the average number. 



The sum of the cube roots of the primes between the same 

 limits, duly corrected, is 1013; and the number of primes 

 between 1789^ or 21,717 and 2399^ or 32,115 is 1008. 



Case 3. Let <^x= -; then the sum of the reciprocals of the 



primes up to x inclusive is 



Pdx 1 1 log a? 



kJ xlogx 2x 12 a?* 

 or 



c + log log ^ + 5— nearly. 



u X 



The sum of the reciprocals of the primes, therefore, from y 

 exclusive to x inclusive, is 



° log^ 2\x y/' 



It follows from this that the sum of the reciprocals of the 

 primes from any number to the prime next to its wth power is 

 nearly log w, whatever the number may be ; the corresponding 

 proposition for ordinals being that the sum of the reciprocals 

 from X to nx is nearly log7^, and is independent of the value 

 of a-. We also see that the sum of the reciprocals of the 

 primes up to a large number x differs only by a constant from 

 the sum of the reciprocals of the ordinals up to log x. 



The constant c would require, like the corresponding con- 

 stant of the ordinal series (y or '5771213), to be determined 

 by computation. By employing values of a? up to about 900, 

 the constant appears to be nearly '267; but as at the com- 

 mencement of the series the actual primes differ sensibly from 

 what may be called the theoretical primes, it is difficult to 

 determine the constant thus with accuracy. 



E2 



