n= the results are 2, 100, 1000 times as great as m. The 1 1 2 100' 1000' limit is infinity, which corresponds to n = 0: we perceive then, that a problem is absurd when the solution is a number infinitely great; this is represented by the symbol * here the two members of the equation are equal, whatever may be the value of a, which is altogether arbitrary. We perceive then, that a problem is indeterminate, and is susceptible of an infinite number of solutions, when the value of the unknown quantity appears under the form does not It is, however, highly important to observe, that the expression always indicate that the problem is indeterminate, but merely the existence of a factor common to both terms of the fraction, which factor becomes 0 under a particular hypothesis. Suppose, for example, that the solution of a problem is exhibited under the Now, if before making the hypothesis a = b, we suppress the common factor ab, the value of x becomes, an expression which, under the hypothesis that ab, is reduced to It must, however, be remarked, that there are questions of such a nature, that infinity may be con. sidered as the true answer of the problem. We shall find examples of this in Trigonometry, and in Analytical Geometry. ence of the common factor a b; but if, in the first instance, we suppress the common factor a―b, the value of x becomes, an expression which, under the hypothesis that a = b, is reduced to From this it appears, that the symbol in algebra sometimes indicates the 0 existence of a factor common to the two terms of the fraction which is reduced to that form. Hence, before we can pronounce with certainty upon the true value of such a fraction, we must ascertain whether its terms involve a common factor. If none such be found to exist, then we conclude that the équation in question is really indeterminate. If a common factor be found to exist, we must suppress it, and then make anew the particular hypothesis. This will now give us the true value of the fraction, which may present itself under one AAO B'0' 0 of the three forms In the first case, the equation is determinate; in the second, it is impossible in finite numbers; in the third, it is indeterminate. 156. We shall conclude this discussion with the following problem, which will serve as an illustration of the various singularities which may present themselves in the solution of a simple equation. The courier who sets out from A travels m miles an hour, the courier who sets out from B travels n miles an hour; the distance from A to B is a miles. At what distance from the points A and B will the couriers be together? Let C be the point where they are together, and let x and y denote the distances AC and BC, expressed in miles. We have manifestly for the first equation Since m and n denote the number of miles travelled by each in an hour, that is the respective velocities of the two couriers, it follows that the time required X y m n ; these two to traverse the two spaces x and y, must be designated by The values of x and y, derived from equations (1) and (2), are 1o. So long as we suppose mn, or m―n positive, the problem will be solved without embarrassment. For in that ease, we suppose the courier who starts from A to travel faster than the courier who starts from B, he must therefore overtake him eventually, and a point C can always be found where they will be together. 2o. Let us now suppose mn, or m―n negative, the values of x and y are both negative, and we have the solution, therefore, in this case, points out that some absurdity must exist in the conditions of the problem. In fact, if we suppose mn, we suppose that the courier who sets out from A travels slower than the courier who sets out from B; hence the distance between them augments every instant, and it is impossible that the couriers can ever be together, if they travel in the direction AC. Let us now substitute x for x, and -y for y, in equations (1) and (2); when modified in this manner they become in which the value of x and y are positive. These values of x and y give the solution, not of the proposed problem, which is absurd under the supposition that m▲n, but of the following: Two couriers set out at the same time from the points A and B, and travel in the direction B C', &c. (the rest as before;) the values of x and y mark the distances AC', BC', of the points C', where the couriers are together, from the points of departure A and B. 3. Let us next suppose m=n; the values of x and y in this case become that is to say, x and y each represent infinity. In fact, if we suppose m = n we suppose the courier who sets out from A to travel exactly at the same rate as the courier who sets out from B; consequently, the original distance a by which they are separated will always remain the same, and if the couriers travel for ever they can never be together. Here also the conditions of the problem are absurd, although the result is not susceptible of the same modification as in the last case. 4. Let us suppose m = n, and also a = 0; the values of x and y in this case become that is to say, the problem is indeterminate, and admits of an infinite number of solutions. In fact, if we suppose ɑ=0, we suppose that the couriers start from the same point, and if we at the same time suppose m=n, or that they travel equally fast, it is manifest that they must always be together, and consequently every point in the line AC satisfies the conditions of the problem. 5o. Finally, if we suppose a = 0, and m not = n, the values of x and y in this case become Ꮖ = 0 y = In fact, if we suppose the couriers to set out from the same point, and to travel with different velocities, it is manifest that the point of departure is the only point in which they can be together, ON QUADRATIC EQUATIONS 157. Quadratic equations, or equations of the second degree, are divided into two classes. I. Equations which involve the square only of the unknown quantity. These are termed pure quadratics. Of this description are the equations, they are sometimes called quadratic equations of two terms, because, by transposition and reduction, they can always be exhibited under the general form II. Equations which involve both the square and the simple power of the unknown quantity. These are termed adfected, or complete quadratic›; of this description are the equations, they are sometimes called quadratic equations of three terms, because, by transposition and reduction, they can always be exhibited under the general form presents no difficulty. Dividing each member by a, it becomes If be a particular number, either integral or fractional, we can extract its square root, either exactly, or approximately, by the rules of arithmetic. b a If be an algebraic expression, we must apply to it the rules established for the extraction of the square root of algebraic quantities. It is to be remarked, that since the square both of +m, and 2 2 b so, in like manner, both (+/-), and (-/-), is + / a a m, is + m2; Hence the above equation is susceptible of two solutions, or has two roots, that is, there are two quantities which, when substituted for x in the original equation, will render the two members identical; these are, for, substitute each of these values in the original equation a x2 = b |