an are prime to each other (156); and therefore the fraction En cannot be equal to a whole number; hence the nth root PRODUCTS OF QUANTITIES-PRIME AND I. Products of quantities. (159.) The simple powers of all real algebraical quantities represent abstract numbers." For if x be any algebraical quantity, it is admissible to involve it to any power, as to 27. Now, x7 means that x is to be repeated x times (22); and therefore the factor x in the expression 2726 × x must be an abstract number (6); but it is not permitted to assign different meanings to the same symbol in the same expression; therefore the seven factors in 27 are abstract numbers. The same may be similarly proved of 7, when a represents a compound quantity. It is hence evident that the proposition is generally true. (160.) Since all real algebraical quantities, at least in their simple powers, denote abstract quantities, in order to represent concrete quantities, they ought to be multiplied by the unit of measure. (161.) The product of any two quantities is the same in whatever order the factors are taken." Let A and B be two quantities, then A taken B times is equal to B taken A times. 1st, When A and B are simple quantities. Since A and B represent numbers, the proposition is evidently true (145.) 2d, When A and B are compound quantities. In this case the quantities also representing numbers, it is evident from article (145), that the proposition is true respecting the numerical product; but the object here is to prove that the algebraical product is the same. Since in multiplying A by B, each term of A is successively multiplied by each term of B; and in multiplying B by A, each term of B is successively multiplied by each term of A; the terms of the two products must evidently be the same. (162.) The product of any number of quantities is the same in whatever order they are taken." Having proved the proposition for two quantities, it can be proved for any number exactly in the same manner as for the products of numerical factors in article (146.) II. Prime and composite quantities. (163.) All the propositions formerly proved respecting prime and composite numbers are evidently true in regard to prime and composite simple quantities.' For any two prime simple quantities must consist of different letters (97); and any two commensurable simple quantities must contain at least one common letter. (164.) A quantity is said to be a function of one or of all the letters it contains, because its value depends on their values. (165.) When the letters composing any function are connected only by the signs of the four fundamental operations, or of those of involution (176) or evolution (183), it is called an algebraical function." (166.) When an algebraical function contains no fractions, it is said to be integral; and if it contains only integral powers of the letters, it is said to be rational." (167.) When a function is arranged according to the powers of some letter, these powers being integral and positive, and their coefficients integral and rational, and either simple or compound, it is called an entire integral and rational function of that letter; and if the coefficients are any algebraical functions whatever, it is called an integral and rational function of that letter. Also, of two functions of the same quantity, that of the higher dimension is called the higher. (168.) If an entire integral and rational function of a quantity is divisible by another, the product of the former by any other quantity will be divisible by the latter.' For the product of the first function by any term of any quantity will evidently be divisible by the second function; and hence the sum of the partial products, that is, the whole product, is divisible by it. (169.) A measure of an entire integral and rational function of a quantity, which is independent of that quantity, is a measure of all its coefficients; and if the coefficient of the highest power of that quantity is unity, the coeffi cients have no measure." ... Let the function A + B + C x2 + Man have a divisor I independent of a, and let a + bx + cx2 + ... mx2 be the quotient, then is ... A+ Bx + C x2 + Mx"= D(a + bx + cx2 + ... mx1); and hence, by the principle of undetermined coefficients, (468), ... A=Da, B = Db, C = Dc, M=Dm; and therefore D is a factor of each of the coefficients A, B, C, ... M. (170.) If two entire integral and rational functions of a quantity are prime to each other, and the process for finding the greatest common measure be performed upon them, there will be a final remainder independent of the leading quantity.' This is evident from the process in article (101.) (171.) If two entire rational and integral functions of a quantity are prime, and the higher of them be multiplied by a quantity independent of the leading quantity, the product will be prime to the other function." For if D be the multiplier, the product will differ from the higher function only in having each of its coefficients equal to the corresponding coefficient of this function, multiplied by D (169); and therefore the terms of the quotient, and of the various products and remainders will be equal to those arising from the division of the given functions, multiplied by D; there will therefore be still a remainder equal to that arising from the last-mentioned division, multiplied by D. (172.) If an entire integral function of a quantity is prime to other two functions of the same quantity, it is prime to their product.' Let A, B, and P, be the functions, and if P is prime to A and B, it is prime to their product AB. For let B be a higher function than P, and perform upon them the same process as on B and P in_article (149); then since B and P are prime, there must be a remainder after the division is carried as far as possible, and this remainder must be independent of the leading quantity, a suppose. Let this remainder be R", then, since the divisibility of AB by P depends on that of AR" by P, as in (149), and as A is prime to P, and R" is independent of x, AR" is prime to P (171.) It appears evident from this conclusion, that the divisibility of AB by any factor of P, when it is a composite quantity, depends on that of A by this factor; therefore when P is prime to A and B, it is so to their product. (173.) Theorems respecting entire rational and integral functions, similar to those referring to numbers contained in the articles between (150) and (158) inclusive, may be similarly proved by means of the theorem in the last article, except the theorem in article (157.) (174.) The least common multiple of two entire integral and rational functions that are relatively prime, is their product. Let the functions be Am + Bxm-1 + ... = M, and ax2 + bxn−1 + ... N, then MN is their least common multiple, and is of the degree m+n. common some ... For if not, let SxP+ TxP-1+ P be their least multiple, where p⇒m+r and rn. Then, since P contains M, it will be equal to the product of M by polynomial R =Wx2 + Xx-1+...; that is, P= MR, and is also divisible by Ñ; and M and N being prime, R must be divisible by N (172), although N be of a higher dimension than R, which is impossible. Again, if pm +n, then r=n, and R being of the same dimension with N, and also divisible by it, it must be identical with it, or else the coefficients of R will be multiples of those of N, and exceed them in dimension. Therefore MN is the least common multiple of M and N. (175.) When the division of one integral and rational function of a quantity (167) by another produces a quotient, in which the coefficients of this quantity are not integral, the latter function is called a relative measure of the former; and a similar measure of two or more functions is called a relative common measure." Thus, the greatest common measure of 6x5 2x3 3x2 3 x ·4x2+x· 39 2 4x4-11x3 1 and 4x4 +223· 18x2+3x 5 is 1, whereas their greatest relative common measure is 203. 39 x2+ If in any case the leading quantity in the first term of any dividend or remainder, in the process for finding the greatest relative common measure of two functions, is divisible by that in the first term of the divisor, while the coefficient of the former term is not divisible by that of the latter, the coefficient of the resulting term in the quotient will be fractional; for it is not necessary in finding this measure, in any case, to multiply the dividend by such a quantity as will make the coefficient of its first term divisible by that of the divisor. INVOLUTION. (176.) The object of involution is to involve quantities, that is, to find any powers of quantities (31.) Powers are said to be odd or even, according as their exponents are odd or even numbers." It appears from article (31) that any power of any quantity is equal to the continued product of that quantity, repeated as a factor as often as there are units in the exponent of the power; but, generally, powers may be found by more concise methods. the CASE I. When the quantity is simple. (177.) Multiply the exponent of the quantity by that of power. |