1878.] Mr Glaisher, On factor tables. 135 



differencing them ; as the differences recur with a period of 

 eight \ 



The mode of work was as follows : The entries were made by 

 the sieves, and one multiple of p obtained from each position of 

 the ^j-sieve was divided out by p, in order to verify that the sieve 

 was always rightly j)laced : this verification was employed for each 

 position of every sieve. The numbers were then examined by my 

 father himself by the sieves. They were then examined a third 

 time by the sieves, and every number ticked. The least factors 

 obtained by the multiple method were read out and entered on 

 the sheets; and they were subsequently read out again in a dif- 

 ferent manner and ticked. Any numbers found unticked were 

 afterwards specially examined. The proofs of the table when 

 printed will be read with the original calculations of numbers 

 by the multiple method. 



On the whole the method of construction is a very perfect 

 one. I have explained it in some detail, because Burckhardt 

 contents himself with a very brief sketch occupying only two 

 paragraphs ; and the process is sufliciently interesting to deserve 

 a more complete account. Each sieve, as stated, has 80 squares 

 cut out, one in each line ; though of course, as there are only 80 

 squares cut out, whatever be the length of the sieve, many of the 

 columns on the longer sieves are left intact. The patterns formed 

 by the holes in the sieves were very curious, some being very 

 regular, while in others the holes were very scattered, and no two 

 were much alike. The sieves for 149 and 151 were remarkable, 

 the holes running steadily up in the one case and steadily down 

 in the other^ The reason for this is that these numbers are 

 nearly equal to the half of 300, the difference between two ad- 

 jacent squares in the same line, so that numbers distant from one 

 another by even multiples of 150 are in the same line. For a 

 similar reason the holes in the sieves for 59 and 61, and 29 and 81, 

 show a steady ascent and descent. The squares in the sieves were 

 cut out by a punch made for the purpose. 



It will be evident from this description that it would be just 

 as easy to enter all prime factors in the table as to enter only the 

 least; and if all the prime factors were entered the verification 

 would be far easier, and in the numbers entered by the multiple 

 method no error could occur, unless the same mistake were made 

 independently in entering both factors. 



The methods described in this section are no doubt practically 



1 It is easily seen that tliis must be so ; for form the miiltiples of the prime 

 p that are not divisible by 2, 3, or 5 ; these arep, 7p, 11jj,'13jj, IT/j, 19p, 23p, 29p, 

 then the next eight are obtained by adding 30^; to each of these and so on. Thus 

 the differences are 6p, 4p, 2p, 4p, 2p, Ap, (^p, 2p, recurring with a period of eight. 



2 Twenty-six of the sieves were exhibited to the Society when this paper was 

 read, inckiding these two. 



