188 il/r Glaisher, On circulating decimals. [Oct. 28, 



(viii) If ^ be a prime and the periods of q each contain an 

 even number of digits, the two halves of each period _will be com- 

 plementary ; for example, the periods of 13 are "OTG 923 and 

 •153 846. 



(ix) With regard to the number of figures in the periods of 

 the denominator q, the rules are as follows. If q = rst..., where 

 r, s, t, ... are primes (so that q has no squared factor), and if the 

 periods of r contain each p digits, of s contain each a digits, of t, t 

 digits, and so on ; then the periods of q will each contain w digits, 

 where m is the least common multiple of p, cr, t .... For example, 

 the periods of 13 contain six digits, of 31 fifteen digits, and of 41 

 five digits ; therefore the periods of 16523, = 13 x 31 x 41, contain 

 30 digits. 



(x) The reasoning by which the preceding theorem is 

 established requires that r, s, t... should be different primes. 

 Suppose now that q = r^, r being a prime whose periods contain 

 each p digits ; then 10" — 1 is a multiple of r. It does not how- 

 ever necessarily follow that lO'* — 1 is not a multiple of r^, i.e. 

 the periods of r may be divisible by r, in which case the period of 

 r^ would contain only p digits. In the general case however in 

 which IC — 1 is not divisible by r"^, the periods of r^ will contain 

 rp digits, and the periods of r^ will contain pr*"^ digits. 



Thus generally when q = r''s'f\.., r, s, ^, ... being primes, it 

 follows that the periods of q contain co digits, where co is the least 

 common multiple of /3*"V, a^'^s, t'^'H. . . ; the condition for the truth 

 of this theorem being that the periods of r, s, t, ... are not divisible 

 by r, s, i, ... respectively, or in other words that 10'' — 1 is not 

 divisible by r, W^ — 1 is not divisible by s, and so on. 



Exceptions. The case r = 3 is an exception to the general 

 rule : for, since 3 = '3, the periods of 3 are divisible by 3, and the 

 periods of 3^ contain only one digit. Thus the periods of 3* con- 

 tain 3*"^ digits. The case r = 487 is also an exception, for the 

 prime 487 has but one period which therefore consists of 486 

 digits, and this period is faund to be divisible by 487 (but not by 

 487^). It follows therefore that the periods of (487)* contain 

 486 (487)""' digits (see § 11). 



(xi) Desmarest states that, with the exception of 3 and 487, 

 there are no primes, up to 1000, which are divisors of their periods ; 

 so that if r, s, t, ...he any primes less than 1000, other than 3 and 

 487, the number of digits in the period of r^s't"'... is given by the 

 general theorem, while if r=3, s = 487, the quantities /o/'"\ as^'^ 

 are to be respectively replaced by 3*"^ 486 (487)^"^ 



(xii) If q have but one period, and if this period be not 

 divisible by q, then q'' will have but one period, for the period of 



