174 report— 1878. 



removed, and which may be called a screen or sieve. Place the sieve over 

 the first 13 columns of the first sheet of the fourth million ; then either 

 empty squares or squares containing a 7 or 11 will appear through the 

 holes of the sieve ; in each empty square write the number 13. Then 

 place the sieve over the next 13 columns and proceed as before, and so on 

 throughout the whole 44 sheets. 



The sieve for the next prime, 17, contains 17 columns, and is made in 

 the same way, viz., by cutting out the squares corresponding to the num- 

 bers between 3,000,000 and 3,000,000 + 17 x 300, which are divisible by 

 17, and not by 2, 3, or 5. Then this sieve is placed over the first 17 

 columns, and 17 entered in all the empty squares, then placed over the next 

 17, &c, and so on. 



The sieves for 13 and 17 are drawn in the plate (Plate TV.), the 

 shaded squares being those that are cut out. The 13-sieve is formed of 

 the first thirteen columns of one of the sheets, and the margin, con- 

 taining the figures 01,07,..., is retained in order to show the arrangement 

 of the form, which contains 77 columns. Of course in using the sieve 

 this margin is cut off as in the 17-sieve. The 13-sieve shows the num- 

 bers between 3,000,000 and 3,000,000 + 13 x 300 which have least factors 

 7, 11, or 13 ; thus, for example, from the third column we see that 



3,000,613 3,000,739 3,000,823 



3,000,641 3,000,767 3,000,851 



3,000,683 3,000,781 3,000,893 



3,000,697 3,000,809 



have 7 as their least factor ; that 



3,000,679 3,000,811 3,000,877 



3,000,701 3,000,833 3,000,899 



have 11 as their least factor, and that 



3,000,647 3,000,751 3,000,829 



3,000,673 3,000,803 3,000,881 



have 13 as their least factor. Of course the numbers such as 3,000,179, 

 for which 7 appears in a shaded square, have 7 as their least factor, and 

 are also divisible by 13 ; and similarly, when 11 appears in a shaded square, 

 the number has 11 for its least factor and is also divisible by 13. 



The 80 argument numbers 01, 07,. ..97 ; 01, 03,.. .99 ; 03, 09,. ..99 cor- 

 respond to the 80 numbers 1, 7,. ..97; 101, 103,. ..199; 203, 209,. ..299 that 

 remain when the numbers divisible by 2, 3, or 5 are thrown out from the 

 300 numbers 1, 2, 3, 4,. ..300. The numbers 01, 03... at the side are 

 lithographed on the form, but the headings of the columns of course are 

 different for each sheet and are written in. Each page in the printed 

 table contains 30 columns, and one advantage of this method of construc- 

 tion- is that the original sheets, when completed, are sent to the printer as 

 they stand, so that there is no copying required. 



The actual size of the form employed is 3P69 inches in length and 

 16 - 20 inches in width, exclusive of the argument numbers at the left. A 

 somewhat smaller form would have sufficed, but this gives ample space 

 in each square for four figures, and was not found to be inconveniently 

 large in use. The squares in the sieves were cut out by a punch made for 

 the purpose. The sieves drawn in the plate have been reduced to suit 

 the size of the page of this volume. 



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



