118 Proceedings of Royal Society of Edinburgh. [sess. 
OlfS 7ZS 
The number of integers lying between — and — - inclusive 
is then [fj - [jy . 
Giving s the series of values 1, 2, . . . , r, we find that the 
whole number S of fractions in this class is given by 
'] + [?] + - + [7]-{[m] + [rTl] + - + t?-.]} 
Suppose first that n is prime to r and to r + 1 ; then the 
remainders on dividing n , 2n , . . . , rn by r are 1, 2, . . . , r— 1,0 
(though not necessarily in this order), and the sum of these is 
J r(r- 1); and as there is one integer in the series, we must sub- 
tract J, hence 
w ”1 T 2 n~\ + + Vm~\ _ n + 2n + + rn _ Jr(r - 1 ) _ 1 
r J L r J L r J r r ' r r 2 
i-r(r+i) 
Iii the same way, n being prime to r + 1, the remainders are 
1, 2, . . . , r- 1, r, and there is no integer in the series, hence 
w 1 ^ d , ni 1 _n 2n rn |r(r + 1 ) 
r + 1 J Lr+lJ _r+lj r+ 1 r+ 1 r+ 1 r+ 1 
n 
r+ 1 
+ 1 ) - i-r . 
Hence S - J = J n(r + 1 ) - \nr — £ n , 
or S = J(ft+l). 
Next, suppose that n and r contain a G.C.M. k , so that n = kn, 
r — kr\ and ri is prime to r \ the remainders, on dividing 
n, 2 w, . . . , rn, by r, i.e. k(n, '2n, . . . , rn) by kr\ are 
k(l } 2, 1, 0) ; &(1, 2, 1, 0) ; ... (k periods), 
