158 FENNELL— PURE CIRCULATING DECIMALS. [Oct. 4. 



§ 5. No explanation is here proposed of the following curions 

 property of periods for which q is prime and a = q — 1, and is 

 also divisible by 4; so that its universality is not deduced or 

 assumed. 



Let a section of m digits of a period be represented by G ( 1 , 2, 3. . . 

 ??i),and G(4. . .m)represent part of the section from the 4th digit to the 

 ??ith or last digit, and G (1.. . [m — 6] ) represent part of the section 

 from the first to the (m — 6th)digit, and G (#. . . [m — y\ ) represent 

 a middle portion of the section from the #th digit to the (m — ?/)th 



digit. Let A(l...i=i), B <L.X=±), C (l...*=i), D 



(1...^— t — ), be the four sections of the period of ■- in order. 



Arrange A (1...X—T — ) followed by C (1...— -r — ) over 



B (1...^-— — ), followed by D (1...^-— — ), making two ranks of 



digits, and add; then the sum E (1...^— - — ) will contain in order 



^—^ — of the digits of the period. If, however, q — 1 be a multiple 



of 10, E (1...^-^ — ) [will contain only — ^ 2 of the said 



digits. 



EXAMPLES. 



For tV 05889411 For h 03448274137931 



23527647 58620689655172 



29417058 62068963793103 



For -h 016393442622950180327868852459 

 819672131147540983606557377049 



836065573770491163934426229508 



As this property is not shared by periods of q when n does not 

 = 1, it cannot be altogether due to the halves of the periods being 

 complementary. It appears to be due to the arrangement of all the 

 periods of q under one cycle of digits. . 



