ence of the common factor a-b; but if, in the first instance, we suppress the common factor a-b, the value of a becomes, an expression which, under the hypothesis that a = b, is reduced to 2 a = = From this it appears, that the symbol in algebra sometimes indicates the 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 equation 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 A A O of the three forms B 0' 0 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 to traverse the two spaces x and y, must be designated by x y n m ; these two periods, moreover, are equal, hence we have for our second equation The values of x and y, derived from ean tions (1) and (2), are. (2) 1o. So long as we suppose mn, or m―n positive, the problem will be solved without embarrassment. For in that case, 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 A C. 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 mn, 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 modifica tion 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 a=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. 5. Finally, if we suppose a = 0, and m not =n, the values of x and y in this case become 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. L 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 quadratics; of this description are the equations, they are sometimes called quadratic equations of three terms, position and reduction, they can always be exhibited under t presents no difficulty. Dividing each member by a, it de co a If be a particular number, either integral or fractional, . square root, either exactly, or approximately, by the rules a be an algebraic expression, we must apply to it the rules extraction of the square root of algebraic quantities. It is to be remarked, that since the square both of +m, an 2 so, in like manner, both (+ √1⁄2-)2, , and (-/-) is + above equation is susceptible of two solutions, or has two roots, t two quantities which, when substituted for x in the original equar the two members identical; these are, for, substitute each of these values in the original equation a r |