NUMBERS. 
That is, the produ& of the two formule 
(x? + avy + by’) and (#* + aa’ y' + by!) 
is itfelf alfo of the fame form; and, confequently, when «= 2", 
and y =y', we have 
te +axy+by)?=X?+aXY + 5Y° 
Hence we have a ready way of re a Ae of any 
fuch fond X’+a bY’, r+ars 
+ 6s*; which is done by - writing 
ra 
Sm Vary a 
in which pel « andy may be aflumed any integer 
numbers at pleafur 
Exam. 1 —Find the values of x als y in the equation 
P+ 3Zry toy =e 
Here a = 3, and 46 = §; therefore the general values of 
x and y are 
where, for diftinGtion ar we write ¢ and uw, in the above 
formule, inftead of x ye 
Whence, by saints faeceffively 
t= 3) 4) 5, 6, &e. 
u=tI, 1,1, 1, &¢ 
we fhall have the following correfponding values of « and y: 
% = 4, II, 20, 31, &c, 
¥ = Oo Il, 13, 15, &e. 
Exam. 2.—Find the values of # and y in the equation 
ee Pom! Mold 
Here, fince a= — 7, and 6= 3; the general values of 
# and y are 
cmf— 3x 
yrortu— pu 
And making now 
t= 4, 5, 6, 7, 8, &e. 
“=, 1, 1, 1, 1, &c. 
we obtain, 
% = 13, 22, 33, 46, 61, &e. 
Y= %y 35 5s Ts Gr Kee 
Each of which correfponding values of x and y anfwer the 
the equation ; ae it is manifeft that 
t be obtained by | 
e 
changing thofe of ¢andz, Barl ne ory of Numbers, 
On the Partition of Numbers. 
Prop. XX. 
Too find in how many different ways any propofed number 
may be divided into a given number of unequal parts. 
Let us propofe the following expreffion, viz. 
(1 $ x72). (1+ 0?x). (14 x°x) . (14 2%z). (1 +2°x) &, 
and endeavour to afcertain the form that it takes when ex- 
panded by multiplication. Aud, firft, let us fuppofe it to be- 
come 
1+ P24+QO2v+ R24 S24, &e. 
then it is evident, from = theory of equations, that P will 
be the fum of the powe 
xi + xt agua, &e. 
and Q, the fum of the preduéts, of ie the gee = 
nations of thefe powers taken two two; 
blage of the feveral powers of x, of which the pede ae 
the fums of two different terms of the feries 
a, b, ¢, d, e, f, &e. 
On the fame principle, R will be an caer oe of the 
powers of x, of which the exponents are the fums of three 
different terms of the fame feries; S will be an raeagice 
of all the powers of x, of which the exponents are = 
fums of four different terms of the fame feries ; and fo o: 
ow, it is manifeft, that the powers of #, which are 
comprifed in the values of P, Q, R, S, &c. will have 
unity for their co- Sgr 3 if their exponents can only be 
will this power have a co-efficient, that contains unity as 
many times. For example, if N x” be found in the 
the fe- 
rie n the develope. 
ment of i propofed fadtors the doe N. x" 2”, its co-effi- 
cient N, indicates, in how many differe Pilg er number 
a may be the {um of m, different terms af the fe: 
a; 6, ey d, “ef &e, 
Thus, the propofed produ, 
(1 + %7x).(E + a? x). (1 + eo u). (1 +22). (E+ 2%) &e. 
being ried developed by multiplication, the refult = ariel 
immediately in how many different manners a given num 
may be i fum of any propofed number of different tem 
“a cn feries a, 6, c, d, &c.: for exa i 
in how many different 
fonssl of m different terms of th: u 
tain the term a"*%™ in the expanded os and the 
oe of this term will be the number re 
order to render this the more evident, ie us pene this 
etna, compofed of an infinite number of factors, 
(It #mz).(1+x~?s). (i +a'2).(1 +4%2). (1 +952) &e. 
the real multiplication of which gives 
am 
P$a(deta2?t e324 ett wot et 74 x8 +2" + Re.) 
+ 2r(ae tet pox 420° 43K7 430° + 4H? + gah 4 ga" + &.) 
a 
+23 (2° + x7 +24? + 3%° + 4rd owl + 7? + Sat? 4 tax + &c.) 
tat (x pat t eet 4 ge3 4+ ga%4+ Oe + gx 4+ tae + 152% + &e.) 
tei (x5 4 ao tae tb ge + oe 4 76 4 lox + 138% + 1B a + Ke.) 
+ eS (2% +e + ae 4 Za + Se +e + IER + Iga" + 204% + Ke.) 
+27 (x +? + 23” + ga + Sa 4 7H% 4+ Ee + ge + 210% + Ke) 
+ ui (e? pet taht ge + Fe 4 
Vou. XV. 
+ 9a tire? 19 e 3 + ase + &e.) 
Ee 
