472 report— 1878. 



divisor q. Thus, for example, suppose q = 39 : there are four periods, viz. dividing 

 1 , 38, 2, and 37 by 39, we have 



39) 1 (-0 39) 38 (-9 39) 2 (-0 39) 37 (-9 



10 2 29 7 20 5 19 4 



22 5 17 4 5 1 34 8 



25 6 14 3 11 2 28 7 



16 4 23 5 32 8 7 1 



4 1 35 8 8 2 31 7 



the remainders being written in the column in the middle and the corresponding 

 quotient digits at the side, e. g. 10 divided by 39 gives quotient and remainder 

 10, 100 divided by 39 gives quotient 2 and remainder 22, 220 divided by 39 gives 

 quotient 5 and remainder 25, and so on. It follows that 



3T = 025641, jj> - -256410, § = 564102 



I t <*«* m - 410256 ' m - - 102564 



and the numbers 1, 10, 22, 25, 16, 4 are such that if we divide any one of them 

 by 39, we obtain the others in this order, and all the fractions ^, |jf, . . . ^ give 

 rise to the same period, though the commencement is made in each case at a 

 different place. 



Considering periods in which the digits and their cyclical order are the same 

 (though the commencement may be made at a different place) as the same period, 

 we see that 39 has four periods, each containing 6 digits. In general, the number q 

 (supposed prime to 10) will have/ periods each containing a figures, a and/being 

 connected by the equation af= (f> (q), where <f> (q) denotes the number of numbers less 



than q and prime to it. Calling the period to which - belongs, the leading period, 



if the remainder q - 1 belongs to the leading period, the two halves of each period 

 will be complementary, while if it does not, the periods will form pairs, the periods 

 in each pair being complementary to one another. The theory is rendered much 

 more simple if all the periods be considered, than if attention be confined to the 

 leading period. 



^ The author had formed a table showing the values of a and/ for each number 

 prime to 10 up to 1000: this table was obtained by counting the number of 

 periods and the number of digits in each period in Henry Goodwyn's ' Table of 

 Circles ' (1823), which gives all the periods of the numbers prime to 10 up to 1024. 

 The table was verified by multiplying the values of a and /, which, in every case 

 was found to be equal to<ji(q). Other such tables that have been given have 

 usually related only to primes, and to the number of digits in the period. There is 

 a table of periods in vol. ii. of Gauss's ' Werke,' pp. 412-434, but it is less complete 

 than Goodwyn's. With regard to the number of figures in the periods of numbers, 

 it is known that if the periods of the primes N, P, Q . . . contain respectively 

 n, p, q . . . digits, then NPQ . . . has a period of a figures, a being the least com- 

 mon multiple of n, p, q . . . . The demonstration does not apply to the case of the 

 power of a prime. Generally, it is found that N* has a period of «N digits, N 3 of 

 nN 2 digits, &c, but for an obvious reason this is not true when N = 3, and also it 

 is not true when N = 487 ; for 487 has a period of 486 digits and 487 2 has also a 

 period of 486 digits. It is, however, true for all other primes less than 1000, so 

 that if N, P, Q . . . be any primes, each less than 1000 (3 and 487 excepted), the 

 period of N'P^Q*. . . contains a digits, a being the least common multiple of wN" -1 , 

 pV^- 1 , qQy- 1 , . . . The discovery that 487 divides its own periods is due to 

 Desmarest ('Theorie des Nombres,' 1852, p. 295), who seems to have determined 

 by actual division that no other prime up to 1000 possessed the same property. 

 In the event of a prime p being a factor of its period, we have 10p -1 = 1 (mo&.p 2 ). 

 By Fermat's theorem, we know that 10?-' = 1 (mod. p), but the theorem throws no 

 light upon whether 10" -1 = 1 (mod.^u 2 ). The question was proposed by Abel in t. iii. 

 p. 212 of Crelle's ' Journal : ' " Can a* -1 - 1, where p is a prime and x an integer 



