NUMBERS. 
And by means of this feries we may afcertain at once in 
how many different ways a propofed number = be formed 
of any determined number of terms of the feri 
15 25 39 49 5» 6, 7, 8, &e. 
Suppofe, for example, it were required to find in how 
many a lifferent ways the cal 35 may be the fum of {even 
different terms of the fe 
I, 25 39 4y 5> 5, 7, 8, &c. 
Find in af bara that has the atl =’, the powe 
x35 and its ent, 15, indicates that the number 
may be rae “fifteen aii acs by the fums of feven 
terms of cal — feri 
But if w = a and thus unite together all the 
ual powe which is the fame thing, if we de- 
logs by multiplication the following infinite produ&, 
(P42). ($8). (0 pa). (1 bat). (1 $2). (1-f 09 Ke. 
we fhall have the feries, 
T+x+ 42034 2444+ 305+ 40°54 5074+ 62°+ &e. 
in which each co-efficient indicates in how many ogee 
ways the exponent of the correfponding power of x 
refult ‘from the additio on of different terms of the rae 
Iy 29 35 45 5 6, 7, &c. without regard to the number of 
them 
Thus it t appears, that hate are fix different manners 
of forming the nu 
umber 8, by the addition of different 
ninbens as follows ; 
8=8 
S=a7+1 
§=6+4+2 
9 8=5+3 
&=5 +2 
8=4+3+1 
It fhould be ali here, that we muft include the 
number itfelf, as one way of forming it; becaufe, the num- 
er of terms to be flected i in the ahove feries being inde- 
terminate, it neceffarily sree that we may confider a 
fingle term as one of the feleCtions. 
Cor. —From what has been faid, ne e learn how many ways 
a number may be produced by the addition of different 
But this condition, se Fic si different num- 
bers, will no longer have place, if we tranfpofe thefe faGors 
to the denominator. Let us therefore se this cafe. 
Pror. XXI. 
To find in how many different waye any given number 
may i ahi =— into a propofed number of equal or unequal 
integral part: 
Let there be propofed this expreffion, 
I 
G—#"2).G —x'x).(1— #2). (1— 2%). (1 — sx) 
which, being developed by divifion, gives the feries 
1+ P24+Q2°+R2?4+Set4 &c. 
and from the firft principles of algebra, it : evident that P 
ie the fum of all the powers of x, of which the exponents 
are contained in the feri ies 
a, b, c, d, e, f, &e. 
ro Ww the’ exponents are formed by the 
addition of three terms; an the fum of the powers, 
of which the exponents are formed by the addition of 
four terms, comprifed in this fence and bo on of the other 
co-efficients 
Confequently, if we fuppofe that the i ea have 
been aQually developed, and that we have c 
the fimilar terms, we fhall f 
ed 
? dq, €y J» XC. 
us feek, for example, in the developed expreffion, 
its — which we will fuppofe 
N;; in fhort that the whole eggs be = Nx" &”; then 
will the co-efficient, N, fhew ay oe many different ways 
m may be formed. i the addition of m terms 
And hence we 
io 
iave been confidering, except that in this 
they are not _ different terms, which was a condition 
in ae sole 2 
a 
re 
wana aint has been faid to a particular cafe, 
aoe this expreffion, 
1 
(I—asz).(1—#°2).(1— 4° 2). (1 —atz).(1—2 2) &e. 
the aGtual developement of which, by divifion, gives 
Te (t+ wr 3 + ott ad 4 eS 4 x7 + 22 + 2? + &c.) 
+2? (ae? +eF + anett2e 432% +3607 + 40? + 4H9 + Fe + &C.) 
+23 (e3 + rt +245 + 3 x 
+ at (et 4 e582 254 3 x? 
+4e7 +58 + 7H? + 8x + 10K" + Ke.) 
+529 +649 + gave? + 112" 4 15 #4 &e.) 
$e (wi fen + ae74 30% +5? +74 4+ 10K" + 13 4% 4+ 18 3 + &e.) 
+ eo (ei te? taxi + 3? 
+5 e+ 7 e+ Ie + 14 e+ 20%% + &e.) 
tal (e7 pe 420 $324 gett pet rie + 15 we + 214 + Ke.) 
So Ce Oe Aas oe cae Oh OO hil oy I a © Ig x + 22 #* + &c.) 
We may, pgaban ae means of this _ a i 
ately in how many diffe manners 
formed by the addition of a? propofed camber of t ai 
of this feries, 1, 2) 3, 4, > ce Su 
a5, and the co- 
» 18, us, that the number in queftion, 
may be eighteen aes ways formed by the addition 
af five integer numbers 
If we fuppofe z = 1, and unite in ong {um all the fimi 
lar powers of #, this expreflion is transformed into this 
feries, 
Ipat 24° 388 4 fat 7a + ra 415 K%7 + 228°-+ &c, 
in which each co-efficient indicates in how many different 
ae the exponents of ee correfponding power can be 
formed by the addition of integers, without regard to the 
number of them, or whether they be equal or unequal. 
For example, the term 41.%° thews that the number 6 
may 
