154 FENNELL — PURE CIRCULATING DECIMALS. [Oct. 4, 



for it is obvious that the sum of the sections is equal to the sum of 

 the whole periods which begin with the respective sections. 

 E.g., for the period of ft .696 774 193 548 387 



.774 193 548 387 096 

 .i93 548 387 096 774 

 .548 387 096 774 193 

 .387 096 774 193 548 



1.998 998 998 998 998 =.9X2 

 + + + + 



1111 



(ii) If q be the product of primes or powers of primes or be a 

 power of a prime, then the summation of sections in some instances 

 gives results similar to those obtained when q is a prime. For in- 

 stance, the period of ft = jj^j = .610989, where the first half and 

 the second half of the period are complementary and .01 + 09 + 

 89 = 99. In other instances, however, variations occur, the gen- 

 eral nature of which is to be understood from the inspection of a 

 few examples. 



For the periods of 21, viz., .647619 and .952386 (ft = .695238), 

 .047 + 619 = 666, 952 + 380 = 1332 = 4 X 333. The two sums 

 together = 2X 999. But .04 + 76 + 19 = 99; 95 + 23 + 80 

 — 198 = 2 X 99. As in some cases in which 3 is a factor of q, 



TYl 



the sections when added give (10 d — 1), so when 9 is a factor 



17b 



of q they sometimes give (10 d — 1). E.g., for 117, 008 + 



o 



547 = 555, but 00 + 85 + 47 = 132 = 4 X 33, .99 + 14 + 52 

 — 165 = 5 X 33. 



For the period of 49 : 



3. i42854 == . i42857 X 22 154 = 7 X 22 



+ 

 3... = .142857 x 22 



