e +ƒ − g, we may use any one of the following expressions, viz., a − b + c xe +ƒ−g, a−b+c.e+f−g, (a − b + c). (e+f-g), (a − b + c) (e + f — g). 9. When we have an expression which shows that quantities are to be multiplied together, the multiplication is said to be expressed, signified, or indicated; and when the multiplication is completed, it is said to be executed, developed, or expanded. 10. It is clear, from Axiom V., that any multiplication, when indicated, ought to be regarded as identical with the multiplication when executed; in other terms, the expression which shows that any quantities are to be multiplied together, forms an identical equation with the product of the quantities. Thus, (a + b) (c + d) : ac+be+ad + bd, is an identical equation, which is clearly true for any possible values that a, b, c, and d can be conceived to have; since (a + b)(c + d) and ac +be+ad + bd are only different forms of the product of a + b and c + d. In a similar way it may be shown that (a + x) (a — x) = a2 — x2, is an identical equation. We are now prepared to give the following rule, which is applicable to every case of multiplication. RULE. 1. For convenience, place the first (or left-hand) term of the multiplier under the first term of the multiplicand, and the second term of the multiplier under the second term of the multiplicand, and so on, until all the terms of the multiplicand and multiplier, with their proper signs, are put down. If any of the terms of the multiplicand and multiplier involve the same letter, then (for simplicity) the terms of the multiplicand and multiplier ought to be set down so that the multiplicand and multiplier may each be arranged, either according to the ascending or descending powers of the letter; observing that any term which may be wanting, either in the multiplicand or multiplier, ought to be supplied. After the multiplicand and multiplier have been put down as required, draw a right line or bar under the multiplier. 2. Then (by the rule for the multiplication of monomials) multiply the first, second, third, etc., terms of the multiplicand successively by the first term of the multiplier, and put each product beneath the bar, and under the corresponding term of the multiplicand. After each term of the multiplicand has been multiplied by the first term of the multiplier, then multiply the first, second, third, etc., terms of the multiplicand successively by the second term of the multiplier; putting the first of these products under the second of the former products, and the second of these products under the third of the former products, and so on. Proceed in like manner to multiply each term of the multiplicand by the third, fourth, etc., terms of the multiplier, until each term of the multiplicand has been multiplied by each term of the multiplier; always putting the product of the first term of the multiplicand, and any term of the multiplier, under the term of the multiplier and below the preceding products that correspond to it, and the product of the second term of the multiplicand and the same term of the multiplier one term or place to the right, and so on. 3. Having multiplied each term of the multiplicand by each term of the multiplier, draw a bar below the products thus found, then the products added according to their signs, and their sum, written below the bar, will be the complete product of the multiplicand and multiplier, as required. 4. When the multiplicand and multiplier have been arranged according to the powers of some letter in each, it is easy to see that the products which involve the same power of the letter will stand in vertical columns under each other; 80 that when the coefficients of any power which stand under each other in any column, are numbers, and are added according to their signs, their sum will be the coefficient of the power of the letter; but if the coefficients are letters, or if some of them are letters and others are numbers, then they must be added by the rules of addition, and their sum, when written in a parenthesis, must be placed before the power of the letter for its coefficient. Proceeding in like manner for each vertical column, we readily get the sum of all the partial products; and, of course, we have the complete product of the given multiplicand and multiplier, as required. 5. When the multiplicand and multiplier are arranged according to the powers of a letter, the multiplication will be greatly simplified by detaching the coefficients, and using them only in the multiplication, and afterward supplying the powers of the letter that correspond to the sums of the partial products of the coefficients that stand in any vertical column of the complete product of the coefficients. 6. It may be added that, in all cases of multiplication, the rule of signs must be observed; viz., that like signs give +, and unlike signs give; so that when any term of the mul tiplicand is multiplied by any term of the multiplier, if the signs of the two terms are both +, or both —, their product must be +, but if one of the terms is + and the other their product must have the sign — prefixed to it. 7. It is to be further noticed, that if the same letter enters the product of any term of the multiplicand by any term of the multiplier more than once, then we must write the letter once, and give it an exponent equal to the sum of the exponents of the letter in the product; in other terms, powers of the same quantity are found by adding their exponents. Always observing, when a letter or any quantity has no exponent expressed, that it is understood to be unity (or 1). It is easy to see that our rule of multiplication is in conformity to the methods which we used in doing the examples in multiplication that preceded it, so that they may serve to show its correctness and use. The rule is clearly, in some respects, similar to the multiplication of whole numbers in Arithmetic, especially when the multiplicand and multiplier are arranged according to the dimensions of a common letter. For in multiplying the multiplicand by the first, second, third, etc., terms of the multiplier, from left to right, the partial products will be so placed that the coefficients of the same power of the common letter will stand in a vertical column under each other, in the same way that the figures of the same local value stand under each other in the partial products when we multiply one whole number by another in Arithmetic. We will now give some examples for further illustration of the rule; some of which will be of use to us hereafter. a, x-b, and x Ex. 3.-To find the product of x-a, xor to develop (x − a) (x — b) (x — c). By the last example, we have (x − a) (x — b) = x2 —(a + b) x+ab; and of course we shall have (x − a) (x — b) (x — c) = [x2 - (a + b)x + ab] (x — c), or we must multiply (a + b)x + ab by x C. 213 − (a+b+c)x2 + (ab+ac+bc)x — abc for the sought product. Ex. 4. To develop (xa) (x —b) (x — c) (x — d). By Ex. 3, we have (x − a) (x — b) (x — c) = x3 → (a + b+c)x2 + (ab + ac + be)x - abc; therefore this product, when multiplied by x-d, will be the development required. Hence we have 23—(a+b+c)x2+(ab+ac+bc)x -abc x-d x+−(a+b+c)x2+(ab+ac+bc)x2—abcx - -da3+(ad+bd+cd)x2-(abd+acd+bcd)x+abcd x-(a+b+c+d)x+(ab+ac+be+ad+bd+cd)-(abc+abd +acd+bcd)x+abcd for the product required. REMARKS. It is easy to see, from these examples, that the product (x − a) (x — b) (x — c) (x — d) (x − e) (x — b), etc., can be written down without any formal multiplication. For the index of a in the first term equals the number of factors that are multiplied together; and the index of x in the second term is smaller by a unit than the index of x in the first term; and the index of x in the third term is less by a unit than it is in the second term, and so on, decreasing by a unit from each term to the next successive term, until we come to the last, which does not contain x. - a, b, Also, the coefficient of the power of x, in the second terin, is the sum of all the quantities, c, -d, etc., without repetition. The coefficient of the power of x, in the third term, is the sum of the products of every two of the quantities, a, b, c, d, b, c, d, etc., without repetition. The coefficient of the power of x, in the fourth term, is the sum of the products of every three of the same quantities, etc., without repetition. The coefficient of the power of x, in the fifth term, is the sum of the products of every four of the same quantities, -a, — b, — c, -d, etc., without repetition. b, c, d, In like manner, the coefficient of the power of x, in the sixth term, is the sum of the products of every five of the same quantities, without repetition; and so on, until we come to the last term, which does not contain æ, and is called the absolute term; which is equal to the product of all the quantities, a, b, c, d, e, etc. Hence, if the positive integer n is equal to the number of factors in the product, and we put A, for the sum of all the |