134 Mr Glaisher, On factor tables.- [Feb. 11, 



including 307, and the multiple metLod for primes from 211 to 

 1,999. The numbers corresponding to the least factors from 211 

 to 307 inclusive were obtained by both methods. 



As the multiple method only gives numbers where the least 

 factor is the given prime p, it follows that every number so found 

 must correspond to an empty square, and the verification thus 

 afforded of the entries already made was very valuable. 



The sieve for 307 contains 307 columns, and therefore occupies 

 4 sheets all but 1 column : considered as a whole, therefore, it 

 has only to be moved 11 times for the million, while the sieve 

 for 13 has to be moved 257 times^. 



Before the calculation was begun, it seemed as if the excessive 

 length of the sieves (the 307-sieve would be 10 feet 6 inches in 

 length, and the 499-sieve 17 feet 1 inch) would be productive 

 of great inconvenience, and would also necessitate very great 

 accuracy and care in the lithographing and printing of the 

 sheets, so that the sqtiares should correspond exactly, over so 

 great a distance ; and it seemed surprising that Burckhardt should 

 have continued the sieve method so far. But this was on the 

 supposition that the portions of the sieve would be all fixed 

 together, so that it would consist of one long sheet. Experience, 

 however, soon showed that nothing was gained by fixing the sheets 

 together, and in fact that it was a positive inconvenience to do so. 

 The sheets forming the sieve were numbered 1, 2, 3, &c., and all 

 that was requisite was to use sheet 1 first, then sheet 2, then sheet 

 3, then sheet 1 again (if the sieve consisted of only 3 sheets), and 

 so on : in fact, the long sieves were found to be quite as easy to 

 use as the smaller ones. Above 307, however, it seemed to be 

 scarcely worth while to construct the sieves, as so little use was 

 made of them, and as the multiple method was preferable in con- 

 sequence of the verification afforded by it. 



The sieves were formed thus : Take for example 13 ; the first 

 uneven multiple of 13 exceeding 3,000,000 is 3,000,023 ; add 26 

 continually till 3,000,000 +13 x 300 is reached, and then throw out 

 the multiples of 3 and 5 ; there are thus left 80 numbers, 

 which correspond to the squares to be^ cut out from the sieve. 

 The accuracy of the 80 numbers that remain was verified by 



[In the fourth million the 13's were entered by a sieve consisting of 13 columns, 

 the l7's by a sieve of 17 columns, and so on. In the fifth and sixth millions now 

 in progress, the 13's are being entered by a sieve of 78 columns, equivalent to six 

 13-sieves fixed together. This is found to greatly facilitate the entries, as the 

 number of removals of the sieve is reduced in the proportion of 6 to 1, and there is 

 less risk of error. The saving of time effected by the use of the 78-column sieve 

 amounts to nearly one-half. For the 17's a sieve of 5x17, =85, columns will be 

 used, for the 19's a sieve of 4 x 19, —76, columns, and so on, the number of columns 

 being made as nearly as possible equal to the number of columns (77) on a sheet. — 

 July 22, 1878.] 



