xy. 2 x-y I X X 3jxxxy I XY 4 This is called a refidual Root. -xy+yy 12 5xx-2xy+yy. The Square of x-y. I 38 xxx-3xxy+3xyy-yyy. The Cube of x-y. One may involve in this Manner to any Power whatever, by the Multiplication of any Power by its Root. Let x+y be involved to the 7th Power. The Powers from a Binomil have + before every Term, as you fee by the Involution of x+y. !1\x7+x+x+x++x3+x2+x 1+2 |3|x7+x3ý+x5y2+x+y3+x3y4+x2y5+xy°+y$ now these joined, the 7th Power of x+y stands in this Manner, without the Unica. N. B. The leading Quantity x, decreases in Arithmetical Proportion; the other Quantity y, increases in the fame Proportion. Το Method. IX. X To find the Unicæ, Sir ISAAC NEWTON has this mo M-I m2 M3 X 4. 5 To explain this Theorem, m is the exponent of the Power, that is m is 7 in the 7th Power, 6 in the 6th Power, &c. The Meaning is this, 1X7-0 7÷17, the first Unica. Then 7×7-1'=6=7 ×6=42, and 42÷221 the fecond Unicæ. Then 21X7-2=105÷3=35, the third Unica. Then 35×73=140435, the fourth Unicæ. Then 35X7 4 105+5=21, the fifth Unicæ. Then 21X7-5=42÷6=7, the 6th Unicæ, then 7x76=77=1. Now the Unica being united to the above Powers, will ftand in the following Man ner. *7+7x6y+21x5y2+35*4y3+35x3y++21*2y$+7xy6, A EVOLUTION. S Involution is a Multiplication, so Evolution is a Species of Divifion. If the Power given have no Coefficient prefixed to it, and its Index can be divided by the Index of the Root required, the Quotient will be the Index of the Root fought. EXAMPLE. N. B. The Figures 1 vv 2, I vu 3, &c. denote the Index of the Root to be extracted. If the given Power have Coefficients, then you muft extract their Roots, as in Arithmetic. But if the Root required cannot be truly extracted out of both the Coefficients and Indices of the given Power, then it is a Surd, and must have the Sign of the Root required prefixed to it. To discover readily the Roots of all compound Powers in general, mind that if either the Sum, or difference of feveral Quantities be involved to any Power, there will arife fo many fingle Powers as there are different Quantities. U EXAMPLE. EXAMPLE. *+y+n squared, it will be as follows: xx+2xy+2xn+yy+2ny+nn. Here is the xx, yy, nn; so that one may conclude the Root is x+y+n, as the Signs are affirmative. When you have extracted the Root, involve it again to fee whether it amounts to the same, if not, it is a Surd, or not right evolved. A Algebraick Fractions. LL Operations in Quantity are performed the fame as in Numbers, fo that an Inftance or two of every Sort will be fufficient. TR REDUCTION. O change Fractions into one Denomination. Rule, Multiply the Denominators for a new Denominator, and every Numerator into all the Denominators but its own, for a new Numerator. a b 3ay bx To bring Integers into Fractions. 33xy 3xy Rule, Multiply Integers into the Denominator for a new Numerator, and under place the Denominator. Example, reduce a+c to a Fraction, whofe Denominator is d, da+de it will be d To reduce any Quantity to the Form of an improper Fraction, as x+y, or, draw a Line of Seperation, and under it write 1, then it will ftand thus, I or To reduce Fractions to its, or their lowest Terms. Rule, expunge the Quantities that are alike in both Numerator and Denominator, and put down the Remainder. |