1878.] Mr Glaisher, On circulating decimals. 187 



(v) The periods arising from the different denominators may 

 therefore be divided into two classes as follows : (1) Jf a com- 

 plementary remainder does not occur in a division there will 

 be an even number of periods, which may be arranged in pairs, 

 each pair containing complementary periods : for example, the 

 periods of the denominator 21 are 



•047619 

 •952380 



and the periods of the denominator 41 are 



•02439 -04878 -07317 -14634 

 •97560 -95121 -92682 -8536o 



each pair of complementary periods being printed in the same 

 column. 



(2) If a complementary remainder does occur in a division, 

 each period will contain an even number of digits and the first 

 half and the second half will be complementary. For example, 

 the periods of 73 are 



•0136 9863 -0273 9726 -0410 9589 

 •0547 9452 -0684 931o -0821 9178 

 •1232 8767 -1643 8356 -3424 657o 

 and in each period the two halves are complementary. 



(vi) If there be but one period of the denominator q, or, in 

 other words, if the period contain {q) digits, then, since all the 

 possible remainders must occur in the division, the complementary 

 remainder must occur and therefore the period must belong- 

 to the second class, i.e. it must consist of two complementary 

 portions. For example, the denominator 17 has but one period, 

 viz. •05882352 94117647, the denominator 49 has but one period 

 containing (^ (49), = 42, digits, viz., 



•020408163265306122448 



979591836734693877551 

 the complementary digits of the second half of the period being 

 printed under those of the first half. 



(vii) It may be observed that when a given denominator q 

 has only one period, this period must be such that when multiplied 

 by each of the ^ {q) numbers less than q and prime to it, the 

 resulting products are the same period, though, commencing at a 

 diiferent place. Thus the first 16 multiples of -0588235294117647 

 consist of these same figures in the same cyclical order. Similarly 

 if the period of the denominator 49 be multiplied by the 42 

 numbers 1, 2, .3, 4... 48 which are less than 49 and prime to it, 

 the products reproduce the same digits in the same cyclical order. 



