INVOLUTION.

Involution is the raising of powers from any proposed root; or the method of finding the square, cube, biquadrate, &c. of any given quantity.

RULE.

Multiply the quantity into itself as many times as there are units in the index less one, and the last product will be the power required. Or,

Multiply the index of the quantity by the index of the power, and the result will be the same as before.

Note. When the sign of the root is +, all the powers of it will be ; and when the sign is -> all the even powers will be +, and all the odd powers

The nth power of any product is equal to the nth power of each of the factors, multiplied together.

And the nth power of a fraction is equal to the nth power of the umerator, divided by the nth power of the denominator.

EXAMPLES FOR PRACTICE.

1. Required the cube or third power of 2a2.

2. Required the 4th power of 2a2x.
3. Required the third power of -8x2y3.

Ans. 8a6.

Ans. 16a8x4.

Ans. -512x6y?.

5. Required the 5th power of a-x.

Ans, a5-5a4x+10a3 x2-10a2x3+5ax4—-x5.

SIR ISAAC NEWTON'S RULE for raising a binomial or residual quantity to any power whatever*.

1. To find the terms without the co-efficients. The index of the first or leading quantity begins with that of the given power, and decreases continually by 1, in every term to the last; and in the following quantity the indices of the terms are 0, 1, 2, 3, 4, &c.

2. To find the uncia, or co-efficients. The first is always 1, and the second is the index of the power: and, in general, if the co-efficient of any term be multiplied by the index of the leading quantity, and the product be divided by the number of terms to that place, it will give the co-efficient of the term next following.

* This rule, expressed in general terms, is as follows:

n-1 n-2 n-3 2 3

a

n-1

n-1 n-2,

=a+n.a

Note. The whole number of terms will be one more than the index of the given power; and when both terms of the root are +, all the terms of the power will be +; but if the second term be all the odd terms will be +, and the even terms —.

EXAMPLES.

1. Let a+ be involved to the fifth power. The terms without the co-efficients will be a5, a1x, a3x2, a2x3, ax4, x5,

and the co-efficients will de

5X4 10X3 10x2 5X1

And therefore the fifth power is
a5+5a4x+10a3x2+10а2x3+5ax4+x5.

2. Let x-a be involved to the 6th power.
The terms without the co-efficients will be
x6, x3α, x1a2, x3a3, x2aa, xa5, a6,

and the co-efficients will be

And therefore the 6th power of x-a is

x6-6x3α+15x4a2—20x3a3+15x2a4—6xa5+a6.

3. Required the 4th power of x—ɑ.

Ans. x44x3a+6x2a2 4xa3+a1.

4. Required the 7th power of x+a.

Ans. x7+7x6a+21x5a2+35x4a3+35x3aa + 21x2a5 + 7xa+a7.

The sum of the co-efficients, in every power, is equal to the number 2, raised to that power Thus, 1+1=2 for the first power; 1+2+1=4=22 for the square; 1++33+1-8=22 for the cube, or third power,

EVOLUTION.

Evolution is the reverse of involution, or the method of finding the square root, cube root, &c. of any given quantity, whether simple or compound.

CASE I.

To find the roots of simple quantities.

RULE*.

Extract the root of the co-efficient for the numerical part, and divide the index of the letter, or letters, by the index of the power, and it will give the root required.

EXAMPLES.

1. Required the square root of 9x2, and the cube root of 8x3.

Ans. √9x2=3x=3x; and 38x3=2x3=2x.

* Any even root of an affirmative quantity may be either + or; thus the square root of + as is either +a, ora: for (+a)x(+a)=+a2, and (—a)X(—a)=+æ2.

And an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of a3 is a, and the cube