A 'treatise of Algebra. 49

Also, if there were given the following Fraction*4 — •lax'i — 8 a 2 * 1 + 1 8 a 3 x — 8 a 4

- -—--- ; by proceeding as

ar 1 —ax 1 — 8a 2 * 4 6a 3

above, it will be reduced to

Process.

x 1 -— 5 ax +« 2

x—^a

See the

x J —ax*—$a*x-{-6a i )x* — ^ax *—8a 2 * 2 4i8 a 3*—8a4(*_

*4— ax 3—8a 2 x 2 -f- 6a3j*

— 2 aAr 3 -f- o -f i 2a*x — 8 a 4—2axi-\-2a i x 1 416 a 3 x — 1 2 <j 4Remainder — 2 a 3 x 2 — 44**4 4 « 4 iwhich divided by — za 2 , gives at 2 4 xa * — 2a 2 for thenext Divisor.

**■)■ 2 ax—za i )x i — ax 1 — 8a 2 v4 6a3 ( *—3 ax 1 4 za * z ~~ 2a 2 x

— 3 ax 2 — 6 a 2 x 4 6a3

— 3 ax x — f a x x 4 6aJ

o 00

x*4 tax— 2 a 2 )x 3 —zax'—8a 2 x 2 418a3x-8a4(x 2 —5 ax-\- 4 a*

x* -j- 2 ax 3 —2a x x x _

——5 ax i — 6a 2 * 2 4 I 8a J x

—5 ax 3 — roa^-j-iog**

44 a z * 2 48 a*x— 8 a 4

44 a 2 x 2 -j- 8 a S x—8a4

o OO

Now if, by proceeding in this Manner, no compoundDivisor can be found, that is, if the last Remainder beonly a simple Quantity, it is certain the Cafe proposeddoes not admit of any, but is already in it’s lowestTerms. Thus, for Instance, if the Fraction proposed

a 34 ta r x 4 3***4 4 * 3 . - , . , T c A- „

was-- i it is plain, by inspection,

a 2 4 ox 4

that it is not reducible by any simple Divisor, but toknow whether it may not, by a compound one, I pro-ceed as above, and sind the last Remainder to be a lim-

E pie