1901.] FENNELL — PURE CIRCULATING DECIMALS. 155 



To generalize, if a period has a number of digits which is a 

 multiple of the number (/) of digits in the period of a factor 



(r) of q, tben sections of kf digits when added give —(10*/ — 1). 



For the period of 221 (= 13 X 17), a = 48, n = 4 (i.e., two 

 pairs of complementary periods). The period of tV is .076923. 

 Sections of 6 digits. Corresponding remainders or numerators. 

 .604524... 1st— 1= 0X13 + 1 



.886877... 7th— 196 = 15 X 13 + 1 



.828054... 13th — 183 = 14 X 13 + 1 



.298642... 19th— 66 = 5x13 + 1 



.533936... 25th — 118= 9X134-1 



.651583... 31st — 144 = 11 X 13 + 1 



.710407... 37th — 157 = 12 X 13 + 1 



.239819... 43d— 53= 4x134-1 



4.i53842 = 54 X .076923 918 = 54 x 17 



+ " 

 4... 54 x. 076923 



Similarly — 



the 2d, 8th... 44th numerators are of the form ± m 13 + 10 

 andjthe 3d, 9th... 45th " " " " ± m 13 + 100 



and the 4th, 10th... 46th " " " " ±ml3 + 116 



and fc so on/ 



The halves of the period of zh = tt (10 24 — 1), and the quar- 

 ters = H (10 12 — 1), while the thirds = if (10 16 — 1) and the 

 numerators corresponding to the thirds =12 X 13. The sixths 

 = 2(10 8 — 1). The other periods yield analogous results. Note 

 that m = in the form of the first 6 numerators, and that the 

 minus sign only occurs for some values of q. Analysis of this kind 

 can be applied generally. 



The following partial exhibition of the relations to each other 

 and to 7 and 47 of the remainders of the period of yr~ = j^g may 

 perhaps prove suggestive. There is one period of 6 digits to 7 

 and one period of 46 digits to 47, and two periods of 138 digits 

 (the halves being complementary) to 329. 



