90 Prof. J. C. Adams [Feb. 7, 



fractions may be reduced to a positive proper fraction, the value of 

 which in decimals may be taken from Grauss' Table, in the second 

 volume of his collected works, or calculated independently. 

 Thus I have found that : — 



_249 2 3 120 3 86 205 58 1 3 

 b500 ~ 256 + 81 + 5 + 343 + 121 + 169 + 289 + 361 + 23 + 29 

 21 30 11 15 26 32 _j_ 24 33 27 67 28 38 

 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 7T + 73" f 79 



73 72 33 61 ^ 45 [ 11 102 68 23 111 

 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 



116 25 126 27 28 29 85 _88_ _91_ 92 

 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 

 97 98 100 101 107 113 115 116 118 121 

 + 191 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239 



122 

 + 241 



+ (the sum of the reciprocals of the primes from 251 to 499) — 19. 



Similarly I have found that : — 



_249 310181J^_J>i 35 , 220 11 300 726 

 kiooo- 512 + 729 + 625 + 343 + 121 + 169 + 2g9 + 361 + 529 + 841 



32 34 21 10 40,^8,28,56 1_ 31 40 45 

 + 961 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 



25 49 44 69 82 90 104 12 67 84 121 

 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 

 85 144 10 26 141 | 83 | 34 53 132 171 

 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191 



102 196 125 90 95 21 212 138 22 223 

 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239 + 241 



211 216 221 226 47 48 236 49 246 53 

 + 251 + 257 + 263 + 269 + 271 + 277 + 281 + 283 + 293 + 307 



261 _54 266 _57_ 170 175 176 178 181 185 

 + 311 + 313 + 317 + 331 + 337 347 + 349 + 353 + 359 + 367 



188 191 193 196 200 202 206 . 211 212 217 

 + 373 + 379 + 383 + 389 + 397 + 401 + 409 + 419 + 421 + 431 



218 221 223 226 230 232 233 235 241 245 

 + 433 + 439 + 443 + 449 + 457 + 461 + 463 + 467 + 479 + 487 



247 251 

 + 491 + 499 



+ (the sum of the reciprocals of the primes from 503 to 997) — 43. 



