i 6 q Mr. Hersci-iel on circulating functions } &c. 
obtained : a proposition which seems likely to be of some 
service in the theory of numbers. 
The same expression may also be obtained by the follow- 
ing considerations. If x be a multiple of n , the integer part 
of ~ n will evidently be represented by if a: be of the form 
in i, or a multiple of n increased by unity, the same 
integer part will be represented by ; if of the form in 2 
by and so on. If then we can devise such a function 
f(x), that when x = in we shall have f (x) = o, when 
x = in -f* 1 ,f(x) = 1, when x = in -j- 2 ,/(#) = 2, and so 
on, i being any integer whatever, it is evident that 
will represent in general the integer part of Now 
o. s'") + ,.sW, + 2 .s j M 2 + (« - 1). S«„ + , 
is such a function. I thought it right to mention this, because 
the observation of this fact (so deduced ) was among the first 
things which led me to the general consideration of circula- 
ting equations in the form I have here presented it. 
(11). The next case I propose to consider is that of cir- 
culating equations, in which the period of circulation is not 
the same in all the coefficients. This, however, will not 
detain us long. Suppose 
«. + pS°- + pf • + pf 0 - = p^ J 
n, p, q, ..... it, v, denoting the respective periods of circu- 
lation in each of the coefficients. Take N to represent the 
product of all the numbers n. p.q t,v, divided by their 
greatest common measure, and may be regarded as a 
circulating function, whose period of circulation is N, but the. 
