Mr. Herschel on circulating functions, &c. 161 
now such a one presents itself at once, viz. — J because 
we have obviously — J—— - = 1 . Hence then we obtain 
2 A, = f + i - i 
3 A == - -I- i — L 
x n * n 
and so on, from which we obtain for the general value of u x 
U x = Const. + -A X .S, + *VS_„ +I 
= Const. + { S, + S x _ , + . . . S,_. +I } (I + *) 
— s * + ,s *-i + (»— 1 ) S M)+I } 
But by (6) we have S x -|- S ;c _ I -f . . . . S x __ n+l — l, and if 
we include the independent unit in the arbitrary constant, we 
find 
« = 2 S — — ^ S X-,- a - S X-2---("— 1 ) S »-» + . ■ C 
U x x n ' 
There is something remarkable in this expression. If x -£- l 
be put for x, and the integral made to vanish when oc == o, it 
expresses the sum of the series 
S i + S 2 s 3 + $ x 
this sum will therefore be represented by 
°- s x- ( m ~- i ) s ,_„ + , 
n 
Now it is evident that this series of terms will contain as 
many equal to unity, as there are units in the integer part of 
|, and all the rest are zero. Here then we have an analytical 
expression for the integer part of the quotient in the division 
of any one number, a?, by any other, n, without in any way 
specifying their numerical form ; whence also a similar ex- 
pression for the remainder in the same division is easily 
Y 
MDCCCXVIII. 
