TRANSACTIONS OF THE SECTIONS. 11 



and 4 X 7 = 28 is in the 6 + 8 = 14th place. Again, 12 and 6 are in the 7th and 18th 



12 X 6 



place, and — : gives a remainder 14, which is in the 7 + 18 = 25th place. 



2ndly. If the last index of a reciprocal repetend from a denominator ending in 9 be 

 made a multiplier, and multiplication be made from the last figure, the unit figures of 

 the successive products will be the left-hand following digit of the series. In 29ths the 

 last index is 3, and as the several products are placed over the series, this curious pro- 

 perty may be seen to advantage. 



As this property belongs to all repetends formed from denominators ending in 9, by 

 a knowledge of the last index, and the last figure in the series, any repetend may be 

 very easily calculated by multiplication, from the last figure of the series to the first; 

 while by a particular arrangement of the process, no multiplier need be above 9. 



The last index may be distinguished as the circulate multiplier, and is always one 

 more than the tens in any number ending in 9 ; the circulate multiplier for 19 is there- 

 fore 2 ; for 49 it is 5 ; and for 199 it is 20. 



Since every prime number terminates in either 1, 3, 7, 9, and the products of 1, 3, 7 

 by 9, 3, 7 respectively terminate in 9, the circulate multipliers for numbers ending in 

 1,3, 7 will be the circulate multipliers of such products ending in 9 ; for example, 

 7 X 7 = 49 gives 5 for the circulate multiplier for 7ths. 



7ths, or J 7, 21. 14. 42. 28. 35. 1 Indices to be divided 



I — i 1 4 2 8 5 7 f by 7 for 7ths. 



49ths. I J 



The last figure of the series, from denominators ending in 9, is always the unit figure 

 of the numerator. 



Having the circulate multiplier, and the last figure in the series, the repetend, as was 

 before observed, may be found by multiplying the circulate multiplier into the last figure, 

 then into the unit figure of the product, then into the unit of the next product, and so 

 on successively, until the whole of the series is produced from the end to the beginning. 



5 



Example 1. Required the repetend of — . 



Here 4 is the circulate multiplier, and 5 the last figure of the series. 

 39th -f Products - 5. 11. 32. 8. 2. 20. 

 iSeries. 12 8 2 5 



Example 2. Required the circulate of _. 



Here the circulate multiplier is 12, the last figure of the series 5 with 6 to carry into 

 the first product. 



Products. 116. 89. 57. 94. 107. 118. 109. 19. 71. 115. 79. 76. 26. 103. 78. 66. 

 Series. 



546 21 84 873 94 957 9 

 When the denominator is not a prime number, the numerator may be commensura- 

 ble with it, in which case all the products may be divided by the greatest common mea- 

 sure of the fraction ; the results will be the indices of a circulate from a prime denomi- 

 nator, of which the given denominator is the product by the common measure. 



56 



For example. Required the repetend of — — . 



In this question the circulate multiplier is 12, the last figure of the series 6, with 5 



56 8 

 to carry in. But — — = — ; consequently the series will be 1 7ths. Therefore, if the 



products be divided by 7 (the common measure), the quotients will be the indices of 

 the repetend of 17ths. 



