( 607 ) 
though wholly empirical formula communicated by LrGENDRE !). 
According to LEGENDRE we shall have 
= p-! = log (loge — 0.08366) — 0.22150, 
pe 
but LEGENDRE not taking 2 for a prime number, as we did, 0,5 
should be added to his result, so that we must read 
E p-! = log (log ce — 0.08366) + 027850, 
pse 
or ultimately for e = oo 
Lim | = p—! — log log | = 0.27850. 
c= p<e 
Thus it appears that the error in LEGENDRE’s formula for ec — oo 
amounts to 0.017, this error diminishing somewhat, when the for- 
mula is used for large though finite values of c. 
In the preceding we have considered almost exclusively the inte- 
grals Gs(e); we now proceed to show that we may deal similarly 
with the sum 2 p=“, though by so doing the formulae lose in 
; ome p<e 
simplicity. 
From 
bee (— 
zip) = gl te log € (hz + hs) 
we derive 
Bite 
h=e (—1)+, 1 oe eh! , ! 
Esen 0 ( Jen mf = - logo(h:--hs)de = = ( Es Gps (c*). 
pee t=! h oni hes h 
a—in 
But since 
G:(b)== 0, 
if the limit b be less than 2, the foregoing infinite series actually 
contains only a finite number of terms and the summation letter 
h throughout satisfies the inequality 
log c 
log 2 
1) “Essai sur la théorie des nombres,” 1808, p. 394. D'une loi remarquable observée 
dans l’énumération des nombres premiers. 
44 
Proceedings Royal Acad. Amsterdam. Vol. II. 
