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



where the induction has to be made from a comparatively small 

 number of instances and those all near the beginning of the series 

 of numerals. 



§ 12. The rules given in § 2 are not to be regarded as in any 

 sense a complete account of the properties of circulating deci- 

 mals ; they merely contain a brief statement of the principal 

 results which are illustrated by Mr Goodwyn's tables. Among 

 others of less importance may be noticed the following. If q be 

 a prime ending with 1, viz. = 10m + 1, then each of the digits 

 0, 1, 2, ... 9 occurs m times in the 10m digits which form the 

 periods of q. For example, if q = 41 the periods are 



■02439 -04878 -67317 -14634 

 -97560 -95121 -92682 -8536o 



and there are four O's, four I's, four 2's, &c. If q has only one 

 period^ (q — 61 is the lowest number of the given form for which 

 this is true) this period contains an equal number of O's, I's, 



2's, ...9's. 



It may be useful to refer briefly to the principal memoirs and 

 writings relating to the theory of circulating decimals. The most 

 considerable memoir that I have seen is one by John Bernoulli, 

 Sur les fractions decimales periodiques, which, with an addition, 

 occupies pp. 273 — 317 of the Berlin Memoirs for 1771. Besides 

 his own investigations Bernoulli gives a full account of the contents 

 of Chapter xil., Book i, of Euler's Algebra, of Chapter LXXXIX. of 

 Wallis's Algebra, and of Robertson's paper Of the theory of circidat- 

 ing fractions [Phil. Trans. 1768), all of which relate to the subject 

 of circulating decimals. In the addition to the memoir he also 

 gives an account of papers by Lambert, in the third volume of the 

 Acta Helvetica, printed in 1758, and in the Hova Acta Eruditorum 

 for March 1769. The memoir, with the addition, thus contains a 

 full explanation of what was then known on the subject : but its 

 interest is now chiefly historical. Arts. 308- — -318 of Gauss's Dis- 

 quisitiones arithmeticce {sectio sexta) relate, as is well known, 

 to circulating decimals. In vol. i. (1842), pp. 457 — 470, of the 

 Nouvelles Annales, M. Catalan stated and proved several of the 

 more elementary properties of circulating decimals. In Desmarest's 

 Theorie des nombres, already referred to, Art. 143 (pp. 289 — 296) is 



. . A . 



entitled " De la transformation d'une fraction ordinaire -^, dite 



XI 



ancienne, en fractions de I'ordre e, et plus parti culierement en frac- 

 tions de I'ordre decimal," and relates to the number of digits in 

 the periods. Mr W. H. H. Hudson's paper On primes and proper 



^ Messenger of Mathematics, First series, vol. ii. p. 4. 



