1878.] Mr Glaisher, On factor tables. 131 



and sixth millions, have been calculated and are ready for entry 

 on the sheets. 



I now give a description of the method (Burckhardt's) adopted 

 in the formation of the tables. 



§ 20. A form was lithographed, having 78 vertical lines and 

 81 horizontal lines (besides several other lines used for head- 

 ings, &c.) ; it is thus divided into 77 x 80 oblong spaces, which 

 may for convenience be called squares. The eighty rows are 

 numbered, at the extreme left of the sheet, 01,07,. • .97 ; 01,03,. . .99 ; 

 03,09,... 99; there being three horizontal broad lines separating 

 the hundreds^ This is the same as in Burckhardt's tables (see 

 § 15), each column representing 300 numbers. The advantage 

 of having 77 columns is that the 7's and ll's are lithographed on 

 the form and have not to be determined and inserted by hand. 

 Thus if 77 consecutive columns of Burckhardt's tables be taken, 

 and ail the headings and tabular results except 7's and ll's be 

 supposed to be removed, we have a representation of the form. 

 The form actually used was constructed to begin from 3,000,000, 

 so that for the exact representation of it we are to commence 

 with the column headed 201 on p. 3 of Burckhardt's table {i. e. 

 the 68th column). 



Since each sheet corresponds to 77 x 300 numbers, a million 

 occupies about 43J sheets, and as on each sheet the number of 7's 

 lithographed is 880, and the number of ll's is 480, it follows that, 

 by adopting a form which permits the 7's and ll's to be litho- 

 graphed, about 59,000 entries are saved in each million ; and, 

 what is even more important, the accuracy of these 59,000 tabular 

 results is assured. 



The squares to which the least factor 13 belongs were ob- 

 tained as follows : Find the numbers between 3,000,000 and 

 3,000,000+13 X 800, which are divisible by 13, but not by 2, 3 

 or 5. Take 13 consecutive columns of any blank form and cut 

 them off from the rest of the form ; then, supposing the first 

 column to correspond to the column headed 3,000,000, make a 

 mark in the squares that correspond to the multiples of 13, pre- 

 viously found, and cut out the squares so marked. We thus have 

 a group of 13 columns, from which a number of squares (80) 

 have been 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 



1 The form is 31'69 in. long and 16'20 in. wide, esclusiye of margins and the 

 exterior argument members at the left. A somewhat smaller form would have 

 siif&ced ; but this gives ample space in each square for four figure?, and has not 

 been found to be inconveniently large in use. 



YoL. III. Pt. IV. 10 



