310 REPORT — 1875 . 



viz. in the interval 1,000,000 to 1,010,000, 100 hundreds, there is 1 hundred 

 containing 1 prime, 2 hundreds each containing 4 primes, 11 hundi-eds each 

 containing 5 primes, ... 1 hundred containing 13 primes, 



Ix 1= 1 

 4x 2= 8 

 5x 11= 55 



13'x 1= 13 



100 752 



or the whole 10,000 contains 752 primes ; the next 10,000 contains 719 

 primes,and so on; the whole 100,000 thus containing 752 + 719 + &c. . . = 7210 

 primes, which number is at the foot compared with the theoretic approximate 



value i — (Umits 1,000,000 to l,010,000)=7212-99. The integral in 



question is represented by the notation Li. x or li. x. 



p. 443. "We have the like tables 1,000,000 to 2,000,000 and 2,000,000 to 

 3,000,000, showing for each 100,000 how many hundreds there are contain- 

 ing prime, 1 prime, 2 primes, up to (the largest number) 17 primes. 



13. It is noticed that 



the 26,379th hundred contains no prime, 

 the 27,050th hundred contains 17 primes. 



It may be observed that if N=2 . 3 . 5 . . .p, the product of aU the primes 

 up tojf), then each of the mimbers N + 1 and N+2 (if g' be the prime next 

 succeeding p) is or is not a prime; but the intermediate numbers N+2, 

 !N" + 3, .. .N+5?— 1 are certainly composite; viz. we thus have at least q—2 

 consecutive composites. To obtain in this manner 99 consecutive composites, 

 the value of N would be =2 .3.5... 97, viz. this is a number far exceeding 

 2,637,900 ; but in fact the hundred numbers 2,637,901 to 2,638,000 are all 

 composite. 



Legendre, in his 'Essai sur la Theorie des Nombres' (1st edit., 1798; 

 2nd edit., 1808 ; supplement, 1816 : references to this edition), gives for the 

 number of primes inferior to a given limit x the approximate formula 



log a;- 1-08366 ' 



and p. 394, and svipplement, p. 62, he compares for each 10,000 up to 100,000, 

 and for each 100,000 up to 1,000,000, the values as computed by this for- 

 mula with the actual numbers of primes exhibited by the tables of Wega and 

 Chernac. Thus .r= 1,000,000, the computed value is 78,543, the actual 

 value 78,493. 



He shows, p. 414, that the number of integers less than n, and not divi- 

 sible by any of the numbers d, X, /u, ... is approximately 



=»C-^)(-.0(-3- 



and taking d,\, f.i .. . the successive primes 3, 5, 7, , . . he gives the values of 

 the function in question, or, say, the function 



2 4 6 10 0,-1 



3*5 "7 "IT ZT' 



