NUMBERS. 
may be produced eleven diferen ways, by the addition of 
whole numbers, as follow: 
6=6 
6=s5+1 
6=4+2 
6=44+141 
6=3+4+3 
6=3+2+1 
6=3+1+14+7 
6=2+2+2 
6=2+2+14+1 
6=>2+1%+1+1+1 
6=-=1+1r+14+14+ 141 
ré we may remark oe seed the propofed omnes 
being contained in the feries ers I, 2) 35 4) 5) 6, 
&c. is itfelf one way of formaing it, 
Prop. XXII. 
To find, as ieee area of the developement by multipli- 
cation, the feries the powers x, that were deduced by 
that method in Pro X. 
Let there be prioia to this effect, the following ex- 
preffion 
Z=(1 deel Cr hed Z) + (1 + 232). (14 x42). 
(1 + 25s) + 
which being aieel sy peso cam a arranged ac- 
cording to the powers of z, gives this 
Z=1+P24+Q2’ nee : 
and it is here required to find an oS method of ob- 
taining the fundions P, of «; for we fhall 
ave, by this means, the folution os the queftion pro- 
ofed 
fed. 
Now it is evident, that if we write «2 for z, we fhall 
ve 
{r+ x2). (1 +232). (1+ at2). (1 +42) &. = 
% 
I+ xe 
Therefore, in fubftituting wz for x, the value of the 
produ, which was x, is changed into : 3; and, con- 
+ 7% 
fequently, fince 
Z=1+P24+Q24 R234 Se2t4+ &e; 
we fhall have 
sie a Sa +Px24+Qe's? + Rat23 + Satzt + &e, 
1+ 242 
Multiplying of which by 1 + +z, we thall obtain 
Z=14+Pr24+QOHr 2? 4+ Rais? +S x 21+ &e. 
+ #@+ Pz? + Qviz3 + Ret2t+ &e. 
And this value of z, — with the former, will give 
Qe = R «x 
*.g= 2", »R= Poa S=- _—s &e., 
We hall say therefore, for P, Q,; R, S, &c. the fol- 
= value 
P= ; 
i-s 
P=; 
5 # * 
(1 —x).(1 — «’)’ 
Pid 
Q= 
“(—s)-(—#).(1—#y’ 
s = ae 
(I—x).G—#).aQ—#)-G—a)* 
x's 
LTE ey eee) 
en mays a obtain oa each of the feries 
of the powers o w in how many dif. 
ferent ways a propofed number may “be formed, by the 
addition 4 any’ given number of integers: and it is evident 
t when converted by divifion recurrin 
feries, been they refult fon a fradtional salacie of x. 
Thus, the firft expreffion, P = 
2 gives the pects, 
cal progreffion 
wp eb oF 4 ot vd + 2 + 2? + Kes 
which indicates that every number is contained once in the 
feries of integers 1, 2, 3, 4, &c. as ia otherwife evident 
from firft principles. 
The fecond expreffion, x’ gives the 
oe 
(1 —«).(1 —« 
feries 
wf at ba + 20° + 327 + 30° + 42? + 4a? + Ke; 
in which the co-efficient of each term fhews in how many 
ways the exponent x may be parted into two unequal parts. 
For example, the term 4.2° fhews that the number,g ma 
be feparated, in four different ways, into two unequal parts. 
If we divide this feries by «3, we — have that which is 
» as follows: 
derived from the fra@tion 
(1 ane — x’) 
I+xu4+2e°4 2x + 308 + 3x + 4a® + 4a + &e.5 
of which we will fuppofe the general term = N x”. Now 
from the generation of this feries, we know that the Ges 
efficient N indicates in how m: 
nce the Saper term of the firft feries is 
"+3, we may thence draw t 
THEOREM. 
Any aumber, n+ 3, may be feparated into two unequal 
parts, in as many ways as the a er n may be formed by the 
addition of the numbers 1 and 2 
rad 
The third expreffion, G-s Goe) Gas, be- 
ing reduced into a feries, will give 
pa bax + ga + 4+ ca + 7 4 8x34 &e. 
And the co-efficient of each term, ‘in this feries, fhews in 
how many different ways the exponent of the ce 
power of w may be feparated into = e 
But the pel peli of the fration 
1 
(a — a). (1 — x’). (1 — #5) oe 
Tpat+ ae + 3x8 + 4x t sa’ +7 a8 4 Sal + &e.5 
Cca of 
» gives the feries | 
