1901.J FENNELL — PURE CIRCULATING DECIMALS. 149 



occurs once, and the same with the six periods of ?, viz., .i, .2, 

 .4, .o, .7, .8. In the 5 periods of n, viz., .09, .i.8, .27, .36, 

 . 45, every digit occurs once. 



(ii) As might be expected, when an or <p(q) is an exact mul- 

 tiple of 10, each of the 10 digits occurs an equal number of times. 

 In other words, if an or <p(q) = 10m, then each digit occurs in 

 times. 



(iii) Prof. Glaisher writes: "Among .... results which are 

 illustrated by Mr. Goodwyn's tables .... of less importance 

 may be noticed the following : If q be a prime ending with one, 

 viz., = 10??i + 1, then each of the digits 0, 1, 2 . . . . 9 occurs 

 m times in the 10m digits which form the periods of q. " This is a 

 partial statement included under the statement in my immediately 

 preceding paragraph and only embracing the cases in which an or 

 <?(q) = q — 1 =10 m. 



It seems a safe inference that my more general statement and 

 its place in the methodical distribution of digits in the periods of 

 q, which is based on the forms both of q and of an or <?(q), were 

 not known when Prof. Glaisher wrote as above, and I have reason 

 for believing that they have not been discovered since, or at any 

 rate published since. 



(iv) The said methodical distribution of the several digits, so 

 far as traced at present, comprises at least seventeen distinct 

 divisions of cases which fall into five groups, A, B, . . . . E. 



The results have been verified for all values of q from 3 to 401 

 inclusive, and for sundry higher values, e.g., 419, 423, 487, 507, 

 603 and 621. 



(v) 

 A 1. If an or <p(q) = 10m, then for all values of q each of 

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

 period or periods of q. 

 f 2. If an or <p(q) — 10m + 2 and q = either 10/? + 1 or 10,3 



+ 7, then 0, 9 occur m + 1 

 times each, and the other digits 

 m times each. [But for q = 

 B 357 (em = 192), 0, 9, 3, 6 



occur 18 times (»i — 1) each, 

 and the other digits 20 times 

 (m+1)]. 



