NUMBERS. 
of = if we fuppofe Nx* to reprefent the general 
e co-efficient N will indicate in how m 
numbers 1, 2, and 3; and the general term of the 
foregoing — being N«x"t’, we hence draw the fol- 
lowing theorems. 
Tuerorem II. 
¢ divided into three unequal 
Any number, n+ 6, may b 
n may be formed by the 
3- 
ports, in as many ways as the number 
addition of the three numbers 1, 25 and 
xe 
(1 —2). (1 — x"). (1 —x).(1— x") 
when developed in a feries, gives 
wp et pon? + 3x84 gx + Ox + x + 1107 + &e. 
the co-efficient of each term indicates in how many 
different ways the exponent of the correfponding power may 
be feparated into four unequal parts; = the transforma- 
The fourth expreflion, 
tion of the expreffion Ga) a). “Gi —¥).G— 2) 
into a feries, or the divifion of the one above by x", pro- 
duces 
Latta + 3x34 gxt + 6205+ ox’ + 11%7+ &e.; 
of which, fuppofing the Seely term to be Nx”, it follows, 
that the co-efficient N how many different ways the 
number 2 may - formed iy the addition of the four num- 
bers I) 2, 35 43 and hence, again, we deduce the fol- 
lowing 
dae Il. 
Any number, 2 iz > may be divided into four unequal 
arts, in as many w nies ver n may be formed by the 
addition of the four ake Ty 25 3, and 4 
In general, therefore, if the sein 
be con- 
I 
(1 — x). (1 — x). (1 — &*). &e. (1 — wx”) 
d into a feries, and of which we take N x" to reprefent R 
al 
term; the co-efficient N will always indicate in 
med by 
verte 
the genera 
how many different ways the number 2 a4 a for: 
the addition of the numbers 1, 2, 35 4) 
But if the expreffion 
m.omth 
(a —a#).G — x"). (1 — #5). (1 — #) &e. (1 — ae”) 
be converted into a feries, the general term will be 
Ne ate s and of which the co-efficient N thews in how 
m+t 
many ways the number a cae may be divided into 
m unequal parts; and = we draw the following 
general 
Tuerorem IV. 
= = may be divided into m un- 
Any number, n oe 
equal parts, in as raged ways as the wut n — be formed 
by the addition of the 4> 
Having thus ee iad the law for the pacnoad of num- 
bers into unequal parts, we fhall proceed to the inveftigation 
of propofition, which includes both equal and unequal 
arts. 
Prop. XXIII. 
ra isa of the developement by divi- 
of the powers that were deduced by this 
To oo 
ion, the ferie 
method in Pro on XXI Ta 
To this effet, let there be — the expreffion 
Zo 
(I~—a#x).(1 ay — #2). (I —xte).(1— xa) 
and fuppofe that, from aétual divifion, it becomes 
Pe+Qe?4 R234 Set4+ Te 4 &e. 
and here, it is evident, that : we put w 2, inftead of x, in the 
above fraGtion, we hall hav. 
Q—#2)Z= 
a I 
(I — #*2).(1 — az) .(1— #2) (1 — %) &e. 
and the fame fubftitution having been made in the fore- 
going feries, there will refult 
eg ee x? 
Sat zt + 2 
aa nae the firft feries by (1 — x2), we fhall 
7 +4 Rxei2d 4 
(ren) Eas eee on 
z+ Rex? + &e. 
And hence by comparifon, we have 
P x R 
= 2 Rat, = =, &e, 
— we I— 
ah gives for 2. 2 R, S, &c. 
seepage the following inde- 
pendent values ; 
= (7—«).(1—#)’ 
rx 
= (Gs) G—*). = 
S= 
(1-2). (1 =a (1 —*). (7 — +) 
hefe expreffions differ from thofe found in the foregoing 
propofition, only in this; that the numerators in thefe have 
t from a comparifon of a 
a fets of ee which we lee deduced at Prop. XX 
and Prop. XXII. And hence, without a repetition of aa 
meer we may deduce the following a a analogous 
to thofe derived from Prop. XXIII. 
THEOREMS. 
1. Any number, n + 2, may be divided into two parts, in as 
many different aad as the number n may be formed by the ad- 
dition of the numbers 1 and 2 
2. Any ede vg n+ 3; may be divided tnto three parts, in 
many different ways as the number n can be formed by the 
addition hi the numbers 1, 2, 3. fi 7 
umber, n + 4, may be divided into four parts, in as 
many di Prot ways as the number 2 may be formed by the ad- 
dition o go the numbers 1, 2) 35 4. eee = 
4. And, 
