ON MATHEMATICAL TABLES. 119 
tables for groups of 100,000 and for the complete millions. There are 
also tables giving sequences of 100 or more consecutive composite num- 
bers in the nine millions. 
A short account of the results of this enumeration was given in the 
Report for 1881 (pp. 305-308), and it is perhaps worth while to supple- 
ment that account by giving the following list of sequences exceeding 
130 in the whole nine millions, arranged in the order of their length. 
0 to 9,000,000. 
Sequences exceeding 130. 
Lower Limit Upper Limit Sequence 
4,652,353 4,652,507 153 
8,421,251 8,421,403 151 
2,010,733 2,010,881 147 
7,230,331 7,230,479 147 
6,034,247 6,034,393 145 
7,621,259 7,621,399 139 
8,917,523 8,917,663 139 
3,826,019 3,826,157 137 
7,743,233 7,743,371 137 
6,371,401 6,371,537 135 
6,958,667 6,958,801 133 
1,357,201 1,357,333 131 
1,561,919 1,562,051 131 
3,933,599 3,933,731 131.. 
| 5,888,741 5,888,873 131 
8,001,359 8,001,491 131 
The three formulz which have been proposed for the approximate 
representation of the number of primes inferior to any given number 
#@ are :— 
(i.) Legendre’s formula— 
, x 
log « — 1:08366 
(ii.) Tchebycheff’s or Gauss’s formula— 
xv 
da 
0 log # 
liz, where liz = | 
Gii.) Riemann’s formula— 
lie —dliat—dlivs—lliat+tliet— &e, 
1 
‘mn which the general term is:liz”, where m denotes any number not 
divisible by a squared factor, namely, any number of the formabec... 
where a, b,c, . . . are different primes; the sign of the term is positive 
when the number of the prime factors a, b, c,. . . is even, and negative 
when it is uneven. 
The Introduction contains comparisons between the numbers of primes 
counted and the values given by these three formule, and also by the 
formulee 
(iy.) Sey ee 2d 
log ee. 4 bs 
log « 
(v.) z 
log «—1 
