294 A L G E 
from o, is increased by unity, in every fncceeding term. 
Alfo the coefficient of each term is found by multiplying 
the coefficient of the preceding term, by the index of x in 
that term, and dividing by the index of a increafed by unity. 
BRA. 
Thus, x-j-a] —x- s r6ax 5 -\- 
6 - 5 , 
6.5.4 
3 * 3 4 
6 - 5 - 4 - 3 . 
2.3.4 
6-5-4-3-a 
2 - 3 - 4 -S 
6 - 5 - 4 - 3 - 2 - 1 
2.3.4.5.6 
a 6 = x 6 -f Sax 3 
ax n '-j -aPx 11 &C. J 
are the fame feries; 
4 , 4 . 
15a 2 x A -\-20a 3 x 3 -{-1 $0.4 x * 2 * * -\-6a s x-\-a 6 
To invelligate this theorem, fuppofe n quantities, x^-a, 
x-\~b, v-f -c, Sec. multiplied together; it is manifelt that the 
firfl term of the produft will be x n , and that x" 
See. the other powers of x, will all be found in the remain¬ 
ing terms, with different combinations of a, b , c, d, Sec. 
Let x-\-b.x-\-c_.x-\-d. Sec. =x n '-j-Px n '- 4 - 2 *”'~~ 3 4 ' 
&c. and x-\-a.x-\-b.x-\-c.x-\-d. SiC.—x v -\-Ax n 2 
4 - &c. then x n -\-Ax n '-\-Bx n Sec. and x-)-aX 
x Il ~~'-\-Px ,L '*A r Qx n 3 -j- Sec. or 
* n -f Px n ~' 2 + Sec. \ 
figns, and retaining the common index. Thus, a n Xb"~ 
aD‘ ; 4/ 2 x 3/3 — ; 'a-\-bfe X <2— b | 2 —d‘ —Af 2 . 
If the furds have coefficients, the produff of thefe co¬ 
efficients muff be prefixed. Thus, a yj xy,b^y=zab \J xy. 
We muft obferve, that 3/—a'Xy 1 —a 2 , or the f'quare 
of 3/ —a 2 , is —a 2 ; becaufe it is that quantity whofe fquare 
root is y'—a 2 . 
Cor. Hence x ——ftx x — a —t/— Pzzix 1 — xax 
4 -a 2 4 -i 2 . 
If the indices of two furds have a common denominator, 
let the quantities be railed to the powers expreffed by their 
refpeflive numerators, and their produft may be found as 
before. 
: 24 ~F; a+Tl 
Example. 2 2 X3 2 =8 2 X3 2 
therefore A—P-\-a, B= 0 -\-aP, Sec. that is, by introdu¬ 
cing one faffor, jt-j -a, into the produft, the coefficient ot 
the fecond term is increafed by a, and by introducing x-^-b 
into the produft, that coefficient is increafed by b, See. 
therefore the whole value of A is a-\-b-\-c-\-d-\- Sec. Again, 
by the introduction of one faffor, the coefficient of 
the third term, Q, is increafed by aP , i. e. by a multi¬ 
plied by the preceding value of A, or by ay.b-\-c-\-d-\- Sec. 
and the fame may be faid with refpeft to the introduffion 
of every other faffor; therefore, upon the whole, 
£~a.b-\-c-\-d-{- Sec. 
■\b.c-\-d-\- &c. 
-\-c.d-\- Sec. 
In the fame manner, 
Czza.b .c-f-^ 4 ' Sec. 
-\-a.c.d-\- Sec. 
-\-b.c.d+ Sec. 
.and fo on ; that is, A is the fum of the quantities a, b, c, Sec. 
£, the fum of the produfts of any two; C, the fum of the 
produfts of any three, &c. 
If either term of the binomial be negative, its odd pow¬ 
ers will be negative, and confequently the figns of the 
terms, in which thofe odd powers are found, will be 
changed. 
Ex. a —x) 8 —a 8 —8 a y x-\- 2 8 a 6 * 2 —5 6a b x :, -\-joa‘x i — z,6a 3 x i 
4-28 a*x 6 — 8 ax'x 8 . 
The trinomial a-\-b-\-c may be raifed to any power by 
confidering two terms as one faffor, and proceeding as 
before. 
Thus, a-\-b-\-<\ n =ia n -\-n.b-\-c.a n —^.A 47 ) 2 .a K_2 
2 
4. Sec. and the powers of ^ 4 cma y^ e determined by the 
binomial theorem. 
The theorem by which a multinomial may be raifed to 
any power is given by M. Demoivre, Analyt. p. 87. 
On Surds. 
a-\-x.a—x | i 2 
If the indices have not a common denominator, they 
muft be transformed to others of the fame value, with a 
common denominator, and their produft found by the laft 
rule. 
. Example 
— n 1 —x ') 4 
4X3 f =8X^- 
X «—1= 
a — x-x a —.vi 
If two furds have the fame rational quantity under the 
radical ligns, their produft is foundby making the fum of 
the indices the index of that quantity. 
, , m n m+n 
Thus, a’ l x^"— aV,n 'X a ’' tn= i a m * • 
If the indices of two quantities have a common deno¬ 
minator, the quotient of one divided by the other is ob¬ 
tained by railing them, refpeftively, to the powers ex¬ 
preffed by the numerators of their indices, and extracting 
that root of the quotient which is expreffed by the com¬ 
mon denominator. 
i 1 m 1 1 
_ « 
£ ITT 
b n bn W[‘ 
For, -= 7 
and — =2 
1 
— 
2 _ 
1 4 
2 1 p 
HI y 
n ps‘ 
| yo 
II 
rS 
~ <1 
’ s 
~~r* 
Ex. 4 2 
If the indices have not a common denominator, reduce 
them to others of the fame value, with a common deno¬ 
minator, and proceed as before. 
Ex. a 2 —v 2 1 2 ' 4 - a 1 —x’l 4 = a‘—x',% ~ a 3 —* 3 4 — 
— sj_ 
a 1 — x\ 
If two furds have the fame rational quantity under the 
radical figns, their quotient is obtained by making the dif¬ 
ference of the indices the index of that quantity. 
4 j m n n —n 
Thus, a"-e r a m ~a mr - ~dpPn=.a mn . 
Ex. 2 2 -f- 2 _ 3 —2 V 4-2°=2 6 . 
A quantity may be reduced to the form of a given furd, 
by railing it to the power whofe root the furd expreffes, 
and affixing the radical fign. Thus, a—^J ed — \J a? Sec. 
m 
a-\-xz=.a+x\ m . In the fame manner, the form of any 
furd may be altered ; thus, a-\-x\ 2 =za-\-x?—Sec. 
The quantities are here raifed to certain powers, and the 
dame roots of thofe powers taken, therefore the values of 
the quantities are not altered. 
If two furds have the fame index, their produft is 
found by taking the produft of the quantities under the 
The fquare root of a quantity cannot be partly rational 
and partly irrational. If poflible, let n—a-\- 4/ m ; then 
by fquaring both fides, n—aS-\-2a \/ and 2 a-\/ m — 
1 71 - ■ " 12 2 ■ ■ -771 » 1 
n. —a 2 — nr. therefore, 1/ mzze -, a rational quanti¬ 
se 
ty, which is contrary to the fuppofition. 
In any equation, a 4 \Jy --®4 V confifting of rational 
quantities and quadratic furds, the rational parts on each 
iideare equal, and alfo the irrational parts. It vbe not equal 
to ei, let xzza-\-m } then a-\-m-\-i/y=za-\- \J b, or m-\- y=z 
