ON MATHEMATICAL TABLES. 123 
Taste II. (continued). 
Difference between numbers counted and calculated by 
; Number 
z of primes 
counted Riemann’s Tehebycheff’s Legendre’s 
formula formula formula 
5,400,000 374,364 —54 +141 4+ 161 
5,500,000 380,802 —4AT +150 +174 
5,600,000 387,204 —11 +187 +215 
5,700,000 393,608 +16 +215 + 247 
5,800,000 399,995 +52 + 253 +289 
5,900,000 406,431 +32 + 235 +275 
6,000,000 412,851 +22 + 226 +270 
6,100,000 419,248 +27 + 232 + 280 
6,200,000 425,650 +21 +228 +280 
6,300,000 432,075 —15 +193 +249 
6,400,000 438,412 +31 + 240 +300 
6,500,000 444,759 +60 +271 +335 
6,600,000 451,161 +29 + 240 +309 
6,700,000 457,499 +55 +268 +340 
6,800,000 463,874 +38 + 252 +329 
6,900,000 470,285 —22 +194 +275 
7,000,000 476 650 — 40 +177 + 262 
7,100,000 483,019 —69 +150 + 239 
7,200,000 489,325 —40 +180 +273 
~ 7,300,000 495,673 —58 +162 ‘+260 
7,400,000 501,972 —34 +188 +291 
7,500,000 508,273 —16 +207 +314 
7,600,000 514,578 — 8 +216 +327 
7,700,000 520,925 —AT +178 +294 
7,800,000 527,170 +10 + 237 +357 
7,900,000 533,534 —56 +172 +296 
8,000,000 539,808 —37 +192 +320 
8,100,000 546,058 0 +231 +364 
8,200,000 552,359 ~18 +214 +351 
8,300,000 558,642 — 23 +210 + 352 
8,400,000 564,927 —35 +199 +346 
- 8,500,000 571,172 —Ii11 + 224 +375 
8,600,000 577,498 —73 +163 +319 
8,700,000 583,779 —95 +142 + 303 
8,800,000 590,078 —139 +100 + 264 
- 8,900,000 596,298 —108 +131 +301 
9,000,000 602,568 —132 +108 + 282 
_ The mean deviations for the three formule are respectively— 
—9, +1638, +4171. 
The great superiority of Riemann’s formula is at once apparent; it is 
more accurate than Legendre’s even for the smaller values of «, and it 
represents the numbers of primes over the whole nine millions most satis- 
factorily. It seems scarcely possible that a continuous formula not involy- 
ing periodic terms could more accurately represent numbers which exhibit 
such great irregularities. 
Tt may be remarked that Legendre’s and Tchebycheff’s formule are 
coincident for « = 4,850,000 :, beyond this point they steadily diverge. 
Tn the second and third volumes of the ‘Mathematische Annalen’ 
(1870 and 1871), Meissel has determined the numbers of primes inferior 
