1901.] FENNELL — PURE CIRCULATING DECIMALS. 153 



.967741935483876, in which every digit occurs 3 times, each 

 pair of complements of 9 contributing 3 digits to each period. 



§ 4. The phenomena noted and illustrated in the following para- 

 graphs can be doubtless fully classified and explained by special- 

 ists in the theory of numbers : 



(i) If when q is prime its period is divisible into sections, each 

 of which contains an equal number of digits— the number being 

 greater than 1 — the sum of the sections arranged in column 

 amounts to 10 d — 1 or a multiple of 10 rf — 1, where d is the num- 

 ber of digits in each section, and the sum of the numerators 

 corresponding to the periods which begin with the several sections 

 is q or a multiple of q. 



E.g., for the period of 31, a = 15 and n = 2, and written in 

 column of 5 sections of 3 digits each the period of tt is 



.032 

 258 

 064 

 516 



129 = 999; 

 and in column of 3 sections of 5 digits each is 



.03225 

 ' 80645 



16129 = 99999; 

 while the five enumerators answering to the sections of 3 digits 

 are 1, 8, 2, 16, 4 = 31, and those answering to the sections of 5 

 digits are 1, 25, 5 = 31. 



For the period of 7, which js .i42857 (for which figures see 

 § 6), 14 + 28 + 57 = 99; 85 + 71 + 42 = 198; while 142 + 857 

 = 999. In the latter case the first half and the second half of 

 the period are complementary. This is an instance of the simplest 

 and most obvious case of the sum of sections of a period being 

 = 10 d — 1, and this case must occur, whether q be prime or not, 

 whenever a complementary remainder occurs in the division of p 

 by q. This particular case of complementary halves of a period is 

 not brought under a general theorem relating to sections of periods 

 by Prof. Glaisher. 



This property of sections of a period containing an equal number 

 of digits each depends upon the property of the corresponding 

 numerators, viz., that their sum is equal to q or a multiple of q ; 



PROC. AMER. PHILOS. SOC. XL. 167. K. PRINTED FEB. 8, 1902. 



