it must also be divisible by b + c − a, b + a − c, a + c − b without remainder.

+

44. If x2a 2bc, y = 2a + 2cb, z=2b + 2c - a ; prove that xy + xz+yz = 9(ab + ac + bc); and find the value of x2 + y2+z2.

45. Find the value of (a - b) (a + b −c) + (c − a) (a − b + c) + (b −c) (b+c-a).

46. If x=√2 + 1 find the value of x2 - 2x + 3.

47. If x = √2 - 1 find the value of x2 + 2x + 3.

N

48. If x = √√3 + 2 find the value of x3 3x2 3x + 1. 49. If x = √√3 - 2 find the value of x3 + 4x2 + x + 4. 50. If x=3+ √2 find the value of x-2√3(x+3)+7. 51. If x=√3-2 find the value of x2+2√2(x-2)+7. 52. If x be an odd number, the product (x + 1) (x + 3) is divisible by 8 without remainder.

53. And the product (x+1)(x+3) (x+5) is divisible by 24.

54. Prove that any number or fraction being increased by 1, and the reciprocal of the number or fraction being also increased by 1, the quotient of the former sum by the latter is the number or fraction itself.

55. Prove that if 1 be divided into any two parts, the difference of their squares is equal to the difference of the parts themselves.

56. Prove that if 2 be divided into any two parts the difference of their squares is equal to twice the difference of the parts.

57. Prove that the difference of the squares of two consecutive numbers is equal to the sum of the numbers.

CHAPTER III.

INVOLUTION, LAW OF INDICES, INDUCTION, AND EVOLUTION.

55. INVOLUTION is the continual multiplication of a product by the quantity from which it originated. We have already stated that the notation a2 is adopted to signify the product of a by itself; a the product a×a×a×a×a and so on. Thus,

43 = 4 x 4 x 4

=

64

75 = 7 x 7 x 7 x 7 x7 = 16807.

To express such an operation as is indicated by a×a×a ×a×a, we shall adopt the phraseology a multiplied by itself to five times.

We have also stated that the law of indices is an immediate consequence of the definition (Art. 12), viz., a" × a"=a"+", when m and n are whole numbers. Now, when the index is

not a positive whole number it has no signification of itself. We are at liberty, therefore, to give it any signification we please, provided our definition involves no necessary inconsistency. Accordingly, we define an index generally from its property as a whole number, thus:

DEFINITION-LAW OF INDICES.-The index of a product is the sum of the indices of the factors in all cases.

Fractional and negative indices are now capable of interpretation from their connection with integral and positive ones;

4a × 4a = 4a +# = 43 = 64:

for example, 4 is interpreted thus: 4 x 4 = 4

i.e., 4 is such a quantity that when multiplied by itself the product is 64. Now, the arithmetical definition of the square

root of 64 is the same; therefore, 4* = √64 = 8.

56. The following axiom will be useful in proving the elementary properties of indices from the definition.

AXIOM.-If two simple or single operations performed on the same thing produce the same result the operations are equivalent, or the quantities operated by are equal.

PROP. 1. a = a.

For a' x a2 = a1+2=a (Def of index, Art. 55);

(aTM) × (aTM), × etc., to q times = (a"); ++.

It has been assumed that two quantities are equal, because either of them, when multiplied into itself to q times, produces

amp

It is necessary to guard against misinterpretation relative to this assumption. That it is not applicable in all its generality is manifest when it is remembered that either + 2 or 2 when multiplied by itself produces 4. We are safe, however, if we limit the proposition to numerical quantities, for there can be no hesitation about its truth with this limitation. No number or fraction but 2 can, when multiplied by itself, produce 4. PROP. 5. a/a.

For each produces a by multiplication n times.

17. Multiply a

18.

1

(64a*2*)*

=

1

8(a2V2c')}

2ax+ 3x by a +2ax-3x.

a2 - 2a3x-1+3α'x-2 by a

19. Divide a 6ax + 5a3x+2ax1- 2a

by x-2ax + a!.

50xу3 — 6y+ 25x1ył – 45x3y – 41x*y* + 20x

by 5x-4x3y + 5x3y‡ — 3y.

57. As involution is merely multiplication, it is not necessary to repeat the process, which has, in fact, been fully given in the chapter on multiplication. It will suffice if we write down a few results and prove that the coefficient of the second term in a power of a binomial is the index.

6. (a + b + c)2

7. (a + b + c)3

= a + 5ab + 10a3b2 + 10u3b3 + 5aba + b3. = a* — 4a3b + 6a2b2 - 4ab3 + b2.

= a3 - 5a1b+10a3b2 – 10a2b3 + 5ab1 — b3.

2

= a2 + b2 + c2 + 2ab + 2ac + 2bc.

= a3 ÷ b3 + c3 + 3a2b + 3a2c + 3ab2 + 3ac2 +36°c + 3bc2 + 6abc.

A convenient mode of finding the square, cube, etc., of a quantity consisting of many terms, is to divide it into two parts; it being always easy to remember the powers of a binomial, thus:

· 3y+a-b.

8. To find the square of 2x (2x-3y+ a − b)2 = { (2x − 3y) + (a−b)} 2

2

= (2x-3y)2 + 2(2x-3y) (a - b) + (a - b)2 =4x-12xy +9y2+2(2ax-2bx-3ay + 3by) + a2 - 2ab+b2

=

= 4x2 - 12xy + 9y2 + 4ax 4bx-6ay + 6by + a2 — 2ab+b2.

m

M

58. PROP. To prove that (a + b)m = a + maTM−1b + etc.

We have shewn that this law holds true for many small values of the index: in order to complete the demonstration, it only remains to prove that, provided it hold true for one value, it will hold true for the next greater value.

m-1

Suppose, then, that (a+b)"1 = aTM-1 + (m − 1) aTM-28 + ...