124 REPORT—1882. 
to 10,000,000 and to 100,000,000, by a method which is equivalent to 
actually counting them, so that his numbers should be exact. Hargreave, 
also, in the ‘ Philosophical Magazine’ for 1854, had obtained by means of 
a similar process the number of primes inferior to 10,000,000. In the 
case of 10,000,000 Meissel’s number is 664,580, and Hargreave’s 
664,633 ; for 100,000,000 Meissel’s number is 5,761,461. Meissel is so 
accurate a calculator that his results are entitled to be accepted with 
confidence ; and, taking his numbers to represent the actual numbers of 
primes counted, we have the following results :— 
Number of Primes 
aleulated by 
3 Counted by ee 
ee Riemann Tchebycheft Legendre 
10,000,000 664,580 664,667 * 664,918 665,140 
100,000,000 5,761,461 5,761,551 5,762,209 5,768,004 
Deviations of the three formulx from Meissel’s counted numbers 
Riemann Tchebycheff Legendre 
10,000,000 + 87 + 368 + 560 
100,000,000 + 90 + 748 + 6,543 
The great accuracy with which Riemann’s formula represents the 
number of primes, both at 10,000,000 and 100,000,000, is very remark- 
able. At 100,000,000 the function li still affords a good approximation ; 
and its superiority to Legendre’s formula, which gives a result differing 
widely from the truth, is very apparent. 
Assuming a formula of the form , and supposing the con- 
av 
oga —A 
stant A to be determined by making the value given by this formula agree 
with the actual number of primes counted for a given value of a, it would 
follow that Legendre’s value—viz. A = 1:08366—was determined from 
# =1230.! The Introduction contains a table showing the variations 
in the value of A, according as it is determined from «# = 50,000, 
«= 100,000... . and so on, at intervals of 50,000, up to ze = 9,000,000, 
and also certain results connected with the value of A. The diminution 
of the value of the constant as 2 increases is very slow, as it only varies 
between 1-090 and 1:072 in the whole nine millions. Taking Meissel’s 
values for the numbers of primes counted, it is found that the value of A, 
as determined from # = 10,000,000, is 1:07110; and, as determined 
from # = 100,000,000, is 1:06397. 
Denoting by ¢ (z) the number of primes inferior to a, it was shown 
by Tchebycheff that if loge — —“~ have a limit when z is infinite, that 
oe 
] 
limit must be unity. It follows, therefore, that if the number of primes 
be represented by a formula of the form the limiting value of 
Vv 
log 2 — A’ 
A is unity. It appears from the results just given that the approach of 
1 It is not unlikely, however, that Legendre assigned the value 1:08366 to the 
constant in order to represent, as nearly as possible, the results of the entire enume- 
rations that he had then made. 
ee 
