TBANSACTIOXS OF SECTION A, 581 



gives rise to m recurring decimals, and any projtei- fraction with ^for denominator 



must give the same numlei- of decimals. For if, say, - gave fewer places, it could 



P . . 



be converted into a fraction -witli denominator of form 9„ r being less than m, \n 



which case p would divide 9,. contrary to the hypothesis above. 



Now in the conversion of - into m recurring decimals there are m different 



1' he.. 



remainders, say h.c.d. . . a; then the m fractions -, -, &c., give rise to the 

 •^ p p 



same cvcle of digits. Thus the fractions -, -, -, &c., ■?-^- must be capable 



p p p p 



of arrangement in groups, m at a time, giving the same cycle : 



i.e. m is a factor of jo — 1 



' • P !> >} '^p-l 



i.e. of lOP-i - 1. 



Repeating the reasoning in a different scale of notation we have Fermat's 



theorem : the radix must -clearly be prime to p, otherwise the fraction - would 



give rise to pure as well as recurring decimals. 



In connection with recurring decimals a curious property may be noticed of 



the equivalent to ^ , viz., -012345679 (nine different digits out of the usual ten), 

 81 



the 8 being absent. If we multiplj^ this by a number less than 81, and prime to 



it, a similar result holds, the absent digit being the defect of the multiplier from 



the next greater multiple of 9, thus : 



^ = -09876543^, 1 absent 

 81 



4 = -049382716, 5 „ 

 81 



1- = -086419755, 2 „ 



|i = -34567901:^, 8 „ = 4 x 9 - 28. 



Hence a new, so-called, recreation. If the number 012345679 be multiplied 

 by a single digit, and the digits only of the product be given in any order, the 

 multiplier can be detected. 



3. On the JRelations between Orbits, Catenaries, and Curved Bays. 

 By Professor J. D. Everett, F.B.8. 



If the same curve be regarded — 



I. As the orbit of a particle under a force of intensity P directed to a fixed 

 point. 



II. As the curve assumed by a string under a force of magnitude F per unit 

 length, opposite in direction to P. 



III. As the path of a curved ray of light in a medium in whioh the absolute 

 index of refraction /j is a function of distance from the common centre of the 

 forces P and F. 



Then, in passing from point to point of the curve, the ratio of P to F will vary 

 directly as fi. 



In fact, from the well-known formulae 



vp = h,Tp = C,t.p = A,V = ^l^±- 



