NOTE. That the supplementary fractions are correct in the last three quotients or developments, as they are called, is obvious from the general principle before established, as the remainder of the division is shown by that principle to be always 2an preceded by the sign of an in the dividend. But the accuracy of the fraction may be proved without reference to the theorem alluded to by aid of the complete developments above. Thus:

n-2

...

+a" x-a +

2a"

x+a

INVOLUTION AND EVOLUTION.

32. INVOLUTION is nothing but the multiplication of equal factors: it is the name given to the process by which powers of any quantity are obtained. The relation between power and root has already been defined: the latter generates the former by involution. Any quantity being proposed, we can always find, with comparatively little trouble, its second, third, fourth, &c. power; because all we have to do is to multiply the proposed quantity by itself, for the second power; this second power by the same factor, for the third power; this again, by the same factor, for the fourth power; and so on, repeating the factor till the required power of it is reached.

But if a quantity be proposed to determine its second, third, fourth, &c. root, the operation necessary, except in the simplest

case, that of the second or square root, is sometimes one of labour. Moreover, the processes for discovering roots are not uniform, like those for determining powers: in the latter, they are nothing but multiplications throughout; but, in the former, the operation for the cube root differs considerably from that for the square root; and in the case of odd roots especially, the difficulty very materially increases with the increase of the exponent of the radical sign. But whatever be the rule for extracting the root, the process comes under the general name of Evolution. There is another important distinction, too: whatever be the quantity proposed, whether algebraical or purely numerical, any power of it may always be accurately found, since the product of one quantity by another may always be accurately found: we know, therefore, that what is called the square, cube, &c. of any quantity actually exists; but a quantity may be proposed which, in strictness, has no square root, cube root, &c.; it may be such that no quantity whatever, squared, cubed, &c. shall accurately produce it.

The learner has already had some experience of this truth in common arithmetic: no number squared, cubed, &c. can produce the number 2, or the number 3, or the number 5, &c.; and it is the same in the expressions of algebra; yet the rules of evolution are applied indiscriminately, both in arithmetic and algebra, without any previous inquiry as to whether the root sought actually exists; in other words, without any previous inquiry as to whether the quantity proposed is a perfect square, cube, fourth power, &c. We, in like manner, apply the rule for division, without any previous stipulation as to whether the dividend accurately contains the divisor or not: we leave the operation itself to discover to us whether or not such be the case. If the divisor be not a factor of the dividend, this operation conducts to a remainder: the quotient is thus shown to be imperfect, and the remainder suggests the fractional correction of it. It is somewhat similar in the extraction of roots. If the operation terminates in a remainder, which no continuation of the process can exhaust, we conclude that the complete root is unattainable; that the expression proposed has no such root: that it cannot be produced from factors strictly equal. In actual practice, our business, in such a case, is to carry on the operation till our remainder, which can never be reduced to nothing, becomes too insignificant to be worth any attention in reference to the practical matter in

hand: the root evolved in such circumstances, though imperfect, because the power is imperfect, has been made to err by so small an amount, that the power, of which it is the complete root, differs from the imperfect power proposed by a quantity too insignificant to deserve notice.

INVOLUTION.

33. The involution of simple qualities will be easily understood from the following examples:—

(1) The third power of 2a is 2a× 2a× 2a=8a3; the square of 3a is 3a2x 3a-9a; the fourth power of ab is abx abx a2b×a2b=ab1: that is, (2a)3=8a3; (3a2)2=9a1; (a2b)*=a*b*.

(2) Also, (3a2b3)2= 3a2b3×3a2b3 =9a1bo; (a2bc3)1= a2bc3× a2bc3× a2bc3 abc3-ab1c12; (5a3y")3=5x3y3× 5x3y2 × 5x3y3= 125x3y; (3xy3z3)1=81x1y3z12.

It

And generally any power of a simple quantity is obtained by annexing to the power of the coefficient the proposed letters with their several indices multiplied by the index of the power. must be observed that every letter is considered to have an index or exponent, just as every term is considered to have a coefficient: unit-coefficients and unit-exponents, though never actually expressed, are always understood: axy, laxy, la'x'y', a'x'y', all mean the same thing, but the first form is that alone which is free from superfluous symbols. The fourth power of ax'y is therefore a1x2y, the exponents of the three letters of the quantity proposed being 1, 2 and 1, respectively; the fifth power is a313, and so on.

As involution is only multiplication, the learner will of course bear in mind the rule of signs; from which he will at once see that if the quantity to be involved be negative, the signs of the even powers must be positive, and those of the odd powers negative.

(3) (5a2bc3x1)2=25a1b2c®x3. (6) (ab3xy2)3=a3b3x3y°. (4) (—3ab3cx)3——27a3b°c3x3. (7) (—ab3xy3)3——a3b3x3y®. (5) (-2a2xy3)=8ax1y12. (8) (-ab3xy)-a1b12x1y®.

These examples of the involution of simple quantities are evidently analogous to those given in Case I. of Multiplication, and might have been placed under that head. The following examples of the involution of compound quantities might also have

been introduced under Case III. of Multiplication: they require no additional explanation here.

(1) Required the cube or third power of a+b, as also of a—b.

(2) Required the cube of a+b+c, or the development of (a+b+c)3.

By last example, [a+(b+c)]3=a3+3a2(b+c)+3a(b+c)2+ (b+c)3

=a3+3a2(b+c)+3a(b2+2bc+c2)+b3+3b2c+3bc2+c3 =a3+3a2b+3a2c+3ab2+6abc+3ac2+b3+3b3c+3bc2+c3. The learner will bear in remembrance the remark made at page 56, namely, that when we are dealing, as in these two examples, with general symbols (a, b, c.), that is, with letters not restricted to any particular interpretation, they admit of any meanings we may choose to give them, and that, therefore, although we may at the commencement of any operation link together the symbols by particular signs, yet these signs must not be understood as necessarily limiting the character of the symbols, even as to sign. In the first of the preceding examples, for instance, b, although linked to a by the sign plus, is not necessarily a positive quantity; b may be —2, or —3, or —6, or minus anything, so that a+b may represent a-2, a—3, or a-6, &c. On this account, the results of our operations have a greater degree of generality than they seem to present to the eye; thus, since in the example just referred to, b may be either positive or negative, the development of (a+b)3 involves equally the development of (a-b)3, so that we have only to make b negative, in that development, to obtain the expression for (a-b)3, actually worked out above. It is plain, therefore, that having got the form for (a+b)3, the work alluded to, for (a—b)3 was quite unnecessary, since (a+b)3 becomes (a—b)3 by simply changing the sign of b. In like manner the single development

D

of (a+b+c)3, as exhibited above in Example (2), suffices also for the development of (a+b-c)3, and (a-b+c), and (a-b -c)3; for we have only to change the sign of c, in the form above, and we get that for (a+b-c)3, to change the sign of b, and we get the form for (a−b+c)3, and to change the signs of both b and c, to get the development of (a-b-c)3. It is of importance that the learner keep this principle in mind, namely, that algebraic symbols, not restricted by special conditions at the outset of any operation with them, are open to any interpretation whatever, as well as to sign, as in reference to value, at any stage of the process. And it is because of this that algebraists, when they have to deal with a compound quantity, the individual terms of which are general symbols, are in the habit of linking the terms together by the plus sign: they do this not only for the sake of uniformity, but because the additive sign has no influence on whatever sign we may suppose to be included in the interpretation of the term to which it is prefixed: thus, should a mean 6, b, -2 and c,-3; then a+b+c would mean 6+(−2)+(−3), which, by the rule for addition, amounts to 1, the additive signs having no influence on those really included in the symbols a, b, c: for if these additive sigus be removed, the expression is the same, namely 6—2—3—1.

EXAMPLES FOR PRACTICE.

Required the developments of the following expressions:

[Some of these examples are so framed as to bring the theorems at page 44

into operation to facilitate the work.]

(1) (x-3y)3.

(3) (a+2b-3c)2.

(7) (x2-2ax+3a2).

(2) (2a+3)3.

(4) (x2+2x-1).

(5) (1+x)3(1-x)3 or {(1+x) (1-x)}3.
(6) (x—-y) or {(x—y)2}3 or {(x—y)3}2.

(8) (a+b+c+d)2.

(9) {(1−x)2+x2(1+x2)}2.

(10) {(1—2x+x2)(1+2x+x2)}'.

(11) {(x2+2ax+a2)(x2-2ax+a2)}3.

(12) {(a—b)3+3a2b—ab2}2. (13) {(a+b)3+(a—b)3}?. (14) {2x−(3x-2)}3. (15) {(a+b+c)2—(a+b)2—c2 }3. (16) {(x+y)3—3xy(x+y)}2. (17) {(x2+a2)2—4a2x2}2. (18) ((3x-2)-(2x-3)} 3. (19) {(5x+1)2—(3x+1)2} 3. (20) {(4-x)—(3+2x)2}3.