1878.] Mr Glaisher, On circulating decimals. . 185 



(2) Mr J. W. L. Glaismer, M.A., F.R.S., On circulating 

 decimals ivith special reference to Henry Goodwyn's " Table of 

 circles" and " Tabidar series of decimal quotients' \{Londo7i, 1818 — 



1823). 



§ 1. The chief rules relating to the conversion of vulgar frac- 

 tions into decimals are as follows, - denoting throughout a vulgar 

 fraction in its lowest terms, 



§ 2. (i) Ifqhe prime^ to 10, then -^ is equal to a pure circu- 

 lating decimal {i.e. is equal to a decimal which begins to circulate 

 from the digit immediately to the right of the decimal point) ; and 

 the number of digits in the period is equal to a divisor of ^ {q), 

 where ^ (g) denotes the number of numbers less than q and prime 



to it. Further, if - has a period of a digits {a being necessarily 



equal to j> (q) or to a submultiple of (q)), then every fraction 

 which, in its lowest terms, has q for denominator has a period of 

 a digits, and there are altogether n periods^ where na= cf) {q). 



If we define the periods that arise from the series of fractions -, 



p having all values less than q and prime to it, as the periods of 

 the denominator q or, more simply, as the periods of q ; the theorem 

 is that the denoniinator ^ (g) has a certain number {ii) of periods, 

 each containing the same number (a) of digits, n and a being con- 

 nected by the relation, na = (p (q). 



(ii) If q be prime, (j){q)=q-l so that in this case the 

 number of digits in the period must be equal to ^^ — 1 or to a 



submultiple of 3- - 1. If therefore - gives a period of q-1 digits, 



q must be prime ; but the converse is of course not true, viz. it 



does not follow that if q be prime the period of ^ will contain 



q —1 digits. Also nothing can be inferred from the fact that the 

 number of digits in the period is a submultiple of 5- — 1, for <f) (q) 

 and q — 1 may have common factors. Thus for q = 33, a = 2 which 

 is a submultiple of ^ - 1, = 32, and also of </> {q), = 20 ; for q = 91, 



1 Tlironghout tlie whole oi%2, q is supposed to be prime to 10. 

 * The periods a/37 ••• ^> (Sy ... ^a, 7 ... ^a/S, &c. are all regarded as the same period, 

 i.e. a period may be supposed to begin with any of the digits composing it. 



14—2 



