concerning Numbers. 39 



sites ofm primes, m — -— times, being the number of combi- 



nations of two things in m things. 



If we go through the series again, and strike out 2.3.5 and 

 all its multiples, and generally PiP^Ps and all its multiples, we 

 shall have struck out the composites of 3 primes, once ; the com- 

 posites of 4 primes, four times ; and generally the composites 



of 7» primes, m — — times, being the number of combi- 



nations of three things in m things. 



Repeating this process continually, alternately striking out 

 and restoring, the final result will be as follows :— The primes 

 will have been struck out once and never restored ; the com- 

 posites of 2 primes will have been struck out twice and restored 

 once ; the composites of 3 primes will have been struck out 

 three times, restored three times, and struck out once; the 

 composites of 4 primes will have been struck out four times, 



4.3 

 restored -^ times, struck out four times, and restored once ; 



the composites of 5 primes will have been struck out five times, 



* ,5.4 . ^ , ^ 5.4.3 . ^ , 5.4.3.2 



restored — - times, struck out times, restored 



times, and struck out once ; and generally the composites of 

 m primes will have been struck out in all 



m ^~ — + -^^ 273 ^-...±m+l times; 



that is simply once; since this expression is 1— (1 — 1)"» or 1. 

 The general effect therefore of the whole of this process is, 

 that the original series of natural numbers is exactly exhausted. 



This process is one of great simplicity, as it involves nothing 

 more than the mechanical process of determining prime num- 

 bers by exhausting the composites. 



Bearing these considerations in mind, it will be immediately 

 obvious that the proposed series wanting its first term, or 

 P«-l, is 



P (2 J V ^ I s ^ V 



and, deducting both sides of this equation from P„,and dividing 

 by P,j, we have 



=0-^)0-f)0-^)(-^) 



which is the theorem proposed. 



