4-6 Mr. C. J. Hargreave's Analytical Researches 



The following Table was constructed, for convenience of 



computation, with the value t=--\ so that the numbers pro- 



ceed in a geometrical series whose common ratio is si. This 

 proceeds regularly from 100 to 180,804'; the limits following 

 this were selected in order to compare the results with the 

 number of primes counted from the tables by Legendre : — 



It will be seen from this Table, that at the commencement 

 of the numeral series, the formula is in excess to the extent of 

 about 1 in 600 ; but from the character of the investigation, 

 I am disposed to think that this error diminishes as we ad- 

 vance in the series. On the whole the coincidence is of a re- 

 markable character, having regard to the nature of the sub- 

 ject; forjudging merely from the apparently irregular occur- 

 rence of primes, it might be thought impossible that their law 

 should be represented in any sense by a continuous analytical 

 fo lunula. 



Assuming the logarithm-integral to represent the number 

 of primes between its limits, it is easy to see how it be- 

 comes possible to frame a formula like that of Legendre's 



•os^fifi/' ^^^'*^^ ^'^^ g^^^ approximately true re- 

 al least over a small range of the series. 



( ^ 



Vloff a:*— !•( 



Jog 

 suits, 



The variation 



of the logarithm is so slow as compared with that of the num- 

 ber, that it may be treated as constant within certain limits ; 

 and the numerical correction becomes necessary in conse- 



