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x--y. This is called a residual Root. 12|xy
I X X 3XXXY
1 & 2 5**---2xy+yy. The Square of x-y.
5 X x 16 xxx-2xxy + xyy 5 X y 17 xxy+2xyyyyy I @ 3 18[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 see by the Involution of x + y.
y+ya +93 +94 +ystyø +y? 1 + 2 13x7+x6ý+*$y* +x+y3+x3y4+**ys +xy • ty? now these joined, the 7th Power of xty stands in this Manner, without the Unicæ.
N. B. The leading Quantity x, decreases in Arithmetical Proportion; the other Quantity y, increases in the same Proportion.
Το To find the Unicæ, Sir Isaac NEWTON bas this Method.
m2 M_3 -4. m-5 IX
X - X 3 4
To explain this Theorem, m is the exponent of the Power, that is m is 7 in the 7th Power, 6 in the 6th Power, &c. The Meaning is this, 1X7-03 7=1=7, the first Unicæ. Then 7x7-'=6=7 X6=42, and 42 = 2=21 the second Unicæ, Then 21X7--2=105= 3=35, the third Unica. Then 35x7-3=140+4=35, the fourth Unicæ. Then 35X7-43105+5=21, the fifth Unicæ. Then 21X7-5=42-6=7, the 6th Unicæ, then 7X76=7+7=1. Now the Unicæ being united to the above Powers, will stand in the following Man.
*7+7*®y+2 1*$y2 +35**y3+35*344 +21**y$+7*y +37
S Involution is a Multiplication, fo Evolution
is a Species of Division. 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.
N. B. The Figures. I vu 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.
E X A M P L E. 8 174
129638 38 I VU 22
36x+y+ 3* But if the Root required cannot be truly extracted out of boch 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 several Quantities be involved to any Power, there will arise so many single Powers as there are different Quantities.
E X A M P L E.
*x+2xy+2xn+yy+any+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 see whether it amounts to the same, if not, it is a Surd, or not right evolved.
LL Operations in Quantity are performed the
same as in Numbers, so that an Instance or two of every Sort will be sufficient.
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.
b 3ay bx To bring Integers into Fractions, * 39=3*y 3*y Rule, Multiply Integers into the
Denominator for a new Numerator, and under place the Denominator. Example, reduce atc to a Fraction, whose Denominator is d,
da+de it will be
To reduce any Quantity to the Form of an improper Fraction, as *+), or *, draw a Line of Seperation, and under it write 1, then it will stand thus,
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.