152 



FENNELL — PUKE CIRCULATING DECIMALS. 



[Oct. 4, 



D 8. Nine cases. 



D 9. Twenty-three cases. 



D 10. Eight cases, 57, 87, 177, 237, 247, 267, 327, 387. 



D 11. Two cases, q = 3 2 and q = 131 



D 12. Three cases, q = 119 and q = 259, q = 329. 



D 13. One case, 9 = 399. 



E 14. Five cases, q = 273, q = 343, q = 133, q = 203, q = 353. 



E 15. One case, 5 = 27. 



E 16. Sixteen cases. 



E 17. Two cases, q = 89, q = 199. 



There is then a strong prima facie case in favor of a regular classi- 

 fication of the numerical distribution of the digits in various cases 

 of <p(q), but not a sufficient number of cases at present investi- 

 gated for a complete and certain induction, which would moreover 

 demand an explanation of the causes which lead to the observed 

 results. A complete investigation would probably supply eight or 

 ten more divisions of cases, as C 4, C 5, C 6, D 13, E 15, E 17 

 are probably susceptible of subdivision, and under B 2 the case 

 q = 357 may be the lowest case of a distinct division. 



The possibility of occasional exceptions must be frankly ad- 

 mitted, at any rate for the present. 



EXAMPLES. 



(vii) For q = 3 4 = 81, <p(q) = 54, a = 9, n = 6, the periods 



are 



0lz345v>79 containing all the digits except 8. 



.987654326 

 .624691358 

 .975308641 

 .649382716 

 .950617283 



1. 

 7. 

 2. 

 5. 

 4. 



Therefore obviously 0, 9, 3, 6 occur 6 times each and the other six 

 digits 5 times each. 



For q = 3 X 11 = 33, <p(q) = 20, a = 2, n — 10, the periods 

 are .03, .06, .12, .15, .24, .39, .48, .57, .69, .78, in which 

 every digit occurs twice. 



For q = 31, <p(q) = q — 1 = 30, a — 15, n = 2, the periods 

 are .632258064516129 and 



