422 
SIR J. F. VV. HERSCHEL ON PARTITIONS. 
of lower periods), it is easy to see, that besides a non-periodic portion of 5 dimensions 
in X, and a periodic one with 60 constant coefficients, there will also be a circulating 
portion of the form 
whose coefficients may rise to the second dimension in x. In fact, if we execute the 
calculation of this portion by the foregoing processes, we find for the values of the 
coefficients 
j 4500a?2^ 1575037 3209;r 
0 ; — 1728OO ’ ‘^•*^“172800' 
With regard to the constant coefficients of the periodic portion, it is easy to see, 
from the manner of their formation, that they must all fall very far short, in nume- 
rical magnitude, of the half of 172800, so that the whole effect of this periodical part 
does, in effect, go to adjust the final value to the nearest integer of the rational frac- 
tion arising from the assemblage of all the terms in x, and a similar reasoning will 
apply in all cases, 
(42.) The number of partitions of which a given number x is susceptible, admitting 
0 into them as a component part, is the sum of the number of 1 -partitions, biparti- 
tions, tripartitions ....up to ^-partitions. It may therefore be found, by adding toge- 
ther all the values of n(cr), from.s=l to s=s inclusive. But it may also be obtained 
by formulae in all respects similar to those above demonstrated ; for if we take 
ns(x) to represent this species of partition, we have, if ^=1, 111 ( 0 ?) = ] as before. For 
.9=2 the partitions stand 0, 0 ?; l,o?— 1; 2, 0 ?— 2; ... ^-{-1^ terms, that is, 
112 ( 0 ?) = III ( 0 ?) -b III ( 0 ?— 2) -}-... r=+ 1 ) terms. 
Similarly, 
n 3 (o’) = 112 ( 0 ?) -f 112 ( 0 ?— 3) -f .... (^|-f 1 ) terms, 
and so on to 
n,(o?) = II,_i(o?) -f n,_i(o?— .9) -{-.... 0+ 0 terms, 
of which the formulae of (30.) and (31.), duly applied, give the value 
where 
9= — ^|.9— 1 .5^-, + 5— 2 ..9^,_2+ .... 1 
which, developed, affords a calculable value of the function in question. 
Collingwood, April 1“, 1850. 
