148 FEXNELL — PURE CIECULATING DECIMALS. [Oct. 4, 



f p. p J Where go we, incl. Ngoondeeneennun 



. Where go we, excl. Ngoondeeneefiulla 

 j Second Person . Where go ye Ngoondeeneenoo 



Third Person . . Where go they Ngoondeeneeyoolung 



Adverbial meanings are sometimes conveyed by means of verbs, 

 as beetyballeemafi, he (or it) goes out of sight. Conjunctions 

 and interjections are few and unimportant. 



NOTES ON PURE CIRCULATING DECIMALS. 



BY C. A. M. FENXELL, CAMBRIDGE, ENGLAND. 



{Read October 4, 1901.) 



§ 1. The following properties of cyclic periods of decimals are 

 supplementary to those discussed by Prof. Glaisher in the Proceed- 

 ings of the Cambridge Philosophical Society, October 28, 1878, 

 Vol. Ill, Part v. 



§ 2. The following letters, definitions and theorem are taken 

 from p. 185 of Prof. Glaisher' s paper. The periods that arise 



p p ... 



from the series of fractions — , - being a vulgar fraction in its 



H 

 lowest terms, and p having all values less than q (which is prime 



to 10), are called the periods of the denominator q, or, more 



simply, the periods of q. Theorem: the denominator ^(a), 



which includes all the above values of p, has a certain number 



(?i) of periods, each containing the same number (a) of digits, n 



and a being connected by the relation, na = (f(q). 



§3. (i) The first inquiry relates to the distribution of the several 

 digits, 0, 9, 3, 6, 1, 8, 2, 7, 4, 5, over the n periods of a digits 

 which constitute Prof. Glaisher' s ?>(</). In this particular a 

 difference emerges between 0, 9, 3, 6, and the rest of the digits, 

 the observalion of which may prove important to the theory of 

 numbers. 



Of course there must always be as many 9s as 0s, 3s as 6s, Is 

 as 8s, etc., but as verified up to *<jt there are the same number, 

 say m, of each of the six digits, 1, 8, 2, 7, 4, 5, m being a posi- 

 tive integer. 



E.g., in the single period of }, viz., .142857, each of the six 



