196 Mr Glaisher, On circulating decimals. [Oct. 28, 



table annexed to this paper. Adopting the notation of § 2 (i) the 

 table shows the values of a (the number of figures in each period) 

 and n (the number of periods) for every number q, prime to 10, from 

 1 to 1024. The last column in this table gives the value of na, 

 found by multiplying n and a : this should be equal to (f){q), and in 

 fact I have in every case calculated (g) independently by means 

 of the formula 



a,b,c,... being the prime factors of q, and found that the value of 

 (j) (g) so obtained was the same as the value of na. 



Extended tables (which are referred to in the next section) 

 have been published of the values of a when g is a prime, and 

 from these by means of the rules in § 2, the values of a and 7i can 

 be readily found for values much exceeding the limit of the table 

 in this paper. It appears to me however that it is very interesting 

 to have, exhibited in a table, the values of a and n, for the numbers 

 from 1 to 1024, and that this interest is greatly enhanced by the 

 fact that they were obtained hy actual counting from the periods 

 themselves, and were only verified by the rules in § 2. It would in 

 any case be of interest to tabulate the values of a and n for the first 

 thousand numbers — though I do not think it would be worth 

 while to proceed much beyond this limit — but there is a distinct 

 increase of value in such a table when the periods themselves 

 have been actually found in all instances by direct division, and 

 the results obtained from the periods by counting, without the 

 employment of any rules derived from theoretical considerations : 

 in fact I should scarcely have communicated the table to the 

 Society, but for the importance which it seemed to derive from 

 the fact that it was the result of actual observation, and only veri- 

 fied by the theory. 



§ 8. Several tables have been published of the numbers of 

 digits in the periods of the reciprocals of primes. Burckhardt at 

 the end of his Tables des diviseurs pour tous les nomhres da premier 

 million (Paris, 1817) gave such a table for all primes up to 2,543 

 and for 22 primes exceeding this limits Desmarest's table on 

 p. 308 of his Theorie des nomhres (Paris, 1852) includes all primes 

 up to 10,000. Reuschle's Mathematische Ahhandlung enthaltend 

 neue zahlentheoretische Tahellen (1856)"^ contains a similar table up 

 to 15,000. Mr Shanks has extended the table to 60,000; the 



^ This is the table referred to on pp. 127, 128 of the present volume of Pro- 

 ceedings. 



' For the justification for this date see the British Association Report for 1875 

 (Bristol), p. 311. My copy, like Prof. Cayley's, has no title-page or date. 



