106 Mr Glaisher, On factor tables. [Feb. 11, 



I now proceed to describe the tables themselves. There are 

 three tables: Table A (one page), Table B (one page), a,nd the 

 principal table extending to 144,000 or 408,000, and occupying 24 

 or 68 pages respectively in the two copies. Table A contains a 

 list of primes up to 20,353 ; along the top line of the page run the . 

 Greek letters a, j3, ... X'> and down the extreme left-hand column 

 four alphabets in order, viz. small italic, small German, capital 

 italic and capital German (there being 100 lines) ; any prime con- 

 tained on this page is henceforth in the book denoted, as it were, 

 by its coordinates; thus 4,721 is denoted by ^m, 17,191 by t^, &c. 

 The first column has no Greek letter as a heading, so that the 

 primes up to 523 are denoted each by a single letter. 



Table B, and also each page of the principal table, contains 

 100 lines, the lines being numbered 1, 2, 3, ... 100 down a narrow 

 column in the middle ; the page is thus divided into two equal por- 

 tions, each of which contains eight columns headed a, b, c, d, e, f, g, h. 

 Table B (headed Tafel der aufjeder Seite wiederkehrenden En- 

 desziffern) contains the numbers not divisible by 2, 3 or 5, up to 

 6,000, arranged in order under the eight columns of each half- 

 page, viz. in the top line, left-hand side, 1 in column a, 7 in 

 column b, 11 in column c, 13 in column d, and so on up to 5,999, 

 which occupies the 100th line of the right-hand side in column h. 

 This table may be said to contain the arguments for the principal 

 table. 



The principal table consists, as it were, only of tabular results, 

 but each half-page may be divided into three blocks, each block 

 being denoted by an argument-number (and also, for a reason that 

 will appear further on, by a symbol involving a letter). These 

 argument-numbers of the blocks run consecutively from to 407, 

 the block denoted, say, by 77 referring to the numbers between 

 77,000 and 78,000. 



The mode of using the table is as follows : Suppose it required 

 to find the factors of 138,593. Divide 138 by 6, the remainder 

 is 0; enter table B at 0593, that is at 593; this number occurs in 

 column g, line 20. Now turn to the principal table and enter it 

 at block 138, column g, line 20; we find as tabular result e, g, ^x, 

 which interpreted by table A, gives 7, 13, 1,523 as the prime 

 factors of 138,593. As a second example, suppose the factors of 

 141,793 required. Divide 141 by 6 ; the remainder is 3. Enter 

 table B with 3,793 ; the result is column d, line 27. Enter the 

 principal table with block 141, column d, line 27, and we find 

 793 — S. The occurrence of the three last figures of the number 

 followed by the hyphen indicates that the number 141,793 is 

 prime, and as the block 141 is denoted by the symbol A), the 

 prime 141,793, when it occurs as a tabular result in the table, 

 will be denoted by A) 8. 



