NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 401 
have the following general relations, in which and denote any circulating func- 
tions, such that 
Qj- = Aj, . -p Bj, . , -f- C^.s^_2 -1- &-C. 
/(P^, Q,)=/(«^, AJ .s,+f{b,, B^) &c (13.) 
of which, particular cases are 
/(Pj=/K) (14.) 
+ &c (15.) 
q^=q:^^.+q'%-i+q:^^.-2+ &c (le.) 
P^Q*=«^.A^.a’^H-6^.B^.^^_i+ & c (17.) 
(7.) As special relations to which we shall refer, we have 
•^^+'^^-1+ (18.) 
and since also {y being any other index) 
ty “h ^y— 1 ”1” • • • * ty^t^ 1 1 . 
Therefore, multiplying and denoting by S the sum of all terms so originating, 
S{^,_,.^,_,.} = 1, (19.) 
i having all values from z=0 to i=s— 1, andj all values from^'=0 toj=#— 1. And 
the same holds good for any number of indices x, y, &c. 
(8.) If n and s be prime to one another, we shall also have 
( 20 .) 
For if the series of numbers 0, s, 2s, (w— 1)^ be divided by n, they will leave s 
remainders, all different, and all less than n, so that among them will be found, 
though not in the same order, all the numbers 0, 1, 2, (w— 1), whence, since 
77, . = 
'‘.r— i '*a?— nm— i? 
the truth of the equation (20.) is apparent. 
(9.) If w be a multiple of s, or n=ts, then 
%+w^-*+w^_2»+ (21.) 
But if n and s have a common measure v, so that n=tv, s=qv, then 
n^+‘n,-s+---‘n,_t,+,=v, ( 22 .) 
Thus, for example, 
6i -l- 6 ^_ 2 + 6^_4 = 2^ ; 64,4-64.-3=3, 
15 , 4 - ] 54 _ 64 ' — 164 _ 24 = 34 - 
(10.) To find the product or other functional combination of circulating or periodic 
functions having different periods of circulation, they must be reduced to a common 
period. Thus, if m represent the product of « and t divided by their greatest com- 
MDCCCL. 3 F 
