130 MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS, 
therefore of the numerator of the fraction. Hence representing this numerator by 
Aj-j- A j.2?. . , , 
and putting a=hc, we have (corresponding to the case 6=1) 
Ao+ Ai+ A 2 .. . +Aa_i=0, 
and generally for the divisor b, 
Ao + Aa...+A(c_,)4 = 0 
Ai+Aa+i.. +A(£._j)j+, = 0 
Aa_i + A2A_i.. -j- Aci_, = 0. 
Suppose now that a.q denotes a circulating element to the period a, i. e, write 
Uqzzzl q=0 (mod. a) 
ag=0 in every other case. 
A function such as 
Ao« 5+ Aiaj_i . . . + 
will be a circulating function, or circulator to the period a, and may be represented 
by the notation 
(Ao, A,, ...A„_i) circlor a,. 
In the case however where the coefficients A satisfy, for each divisor b of the number 
a, the above-mentioned equations, the circulating function is what I call a prime 
circulator, and I represent it by the notation 
(Ao, A„ ..A„_,) pcr«,. 
By means of this notation we have at once 
coefficient Xg in^^l^] =(Ao, A...A„_i) pcra„ 
and thence also 
coefficient Xg in = 9 ’ (Ao, A,..A„_0 per a,. 
Hence assuming that in the fraction 
fa 
the degree of the numerator is less than that 
of the denominator (so that there is not any integral part), we have 
coefficient in^ = 2 q’'{Ao, A„ ...A«_i) per a, ; 
or, if we wish to put in evidence the non-circulating part arising from the divisor a= 1, 
coefficient Xg in^=Ag*~*-|-B 5 '*“^...-|-L 5 '+M 
+ 2 q’'{Ag, A,...A,_,) per a,; 
where k denotes the number of the factors 1 —x"" in the denominator fx, a is any 
divisor (unity excluded) of one or more of the indices m ; and for each value of a 
r extends from r=0 to r=k—\, where k denotes the number of indices m of which 
