1901.] FENXELL— PURE CIRCULATING DECIMALS. 151 



12. But in other cases either 0, 9 



occur m + 1 times each, 3, 6 

 m + 2 times each, and the 

 D i other digits m times each; 



| 13. or 0, 9 occur m — 1 times 



each, and the other digits 



[ wi + 1 times each. 



; 14. If aw or p(g) = lOwi + 8 and q = 10/3 +1 or 10,5 + 3, 



then 0, 9, 3, 6 occur m + 2 

 times each, and the other 

 digits m times each. 



15. If an or c(q) = 10m + 8 and ^ = 10/3 + 7, then 3, 6 



occur ?ra times each, and the 

 other digits m + 1 times each. 



16. If an or <p(q) = 10m + 8 and o = 10/5 + 9, then 0, 9 



occur m times each, and the 

 other digits m + 1 times each. 



17. But if q = 10/3 + 9 = 115+1 



{e.g., 89, 199 or 419), then 

 3, 6 occur m — 4 times, and 

 the other digits m + 2 times 

 each, or some other excep- 

 tional distribution is found. 



(vi) The total number of values of q up to 401 is 1G0. 



A 1. Includes 18 primes (counting 401) and 22 multiples or 

 powers of primes. 



B 2. No primes; 15 cases with q = 10,3 + 1, only 3 cases with 

 q =z 10,3 + 7, and the exceptional case q = 357 = 3x 

 7 X 17. Beyond 401, q = 507 is regular. But the 

 limits of the invesiigation do not present sufficient data for 

 sound inference as to the cases where q = 10/5 + 7. 



B 3. Includes q = 243 and 20 primes with 3 for the unit digit 

 and 2 with 9 for the unit digit, namely 49 = V and 

 289 = 17 2 , the next number being 81 9=7X9X1 3. 



C 4. One case, q = 81. 



C 5. One case, q = 343. 



C 6. One case, q = 273. 



C 7. Five cases, q = 147, and 4 cases, q = 10,5 -f 9. 



I 



