the former value of x in the first case; and hence the solution in that case comprehends that in this, observing the conventional definition (42), and the principle in (56.) 2d, When b>d and ac, the result would be the same as the preceding, and would be found in the same manner, by taking a (ca) c for III. When b=d, and, at the same time, c nota, then as is evident from the original equation, which becomes IV. When b is not = d, but ac, then and since no finite value of a, multiplied by 0, can give a finite product b―d, x must = ∞. = V. When ac, and b = d, 0 x= 0' from which no particular value of a can be found, or its value is indeterminate. The equation in this case is and since any number, multiplied by 0, gives 0, therefore any value whatever of a fulfils the equation, or it has no particular value. (254.) The meaning therefore of the expression is merely the indeterminateness of the quantity whose value it expresses; it does not convey any sign of absurdity or impossibility in the equation. (255.) 'An equation, the members of which are just the same quantities expressed either in the same or in different forms, is called an identical equation." The last case of the preceding equation, namely, ax+b=cx+d or ax+b= ax + b since a =c and b=d, is an equation of this kind; and so are the equations 2x x (x — 2) = x2. (x — a) (x —b) = x2 — (a + b) x+ab 3 x (2-4)+36 = 6 x − 12 x +36 It is evident that whatever value is assigned to the unknown quantity in an identical equation, its two members are equal; so that the equation is satisfied by any value of this quantity, and therefore no particular value of it can be determined. (256.) The nature of a negative solution is very clearly illustrated by this question : Two couriers depart at the same time from two towns A and B, distant by a miles from each other; the former travels m miles an hour, and the latter n miles: where will they come together? There are two cases of this question : but A and B travel the same number of hours, therefore Whatever values be given to a, m, n, that of x will in this case always be positive, and no difficulty occurs. II. When they go in the same direction. Let, as before, M be the point of meeting, A and B travelling in A that direction, and let B M therefore a=AB then, as in the preceding case, It appears from this value of x, that, so long as m>n, or A's rate of travelling exceeds B's, the value of x is positive, and there is no difficulty in interpreting its value. But suppose that mn, the value of x is then negative, and the equation nx = mx -am This negative value implies some impossibility in the question understood literally, and the necessity of modifying its enunciation: but this will be found unnecessary by the following considerations: am To subtract is the same as adding n-m the rules of algebraical addition (56); and = am mn And this quantity being in this case negative as mn, so that the value of x is also negative, and being the numerical value of a line, its value must be measured in M' A B M a direction opposite to that of AM (44), namely, in the direction AM ́; hence AM′ and M' is the point where they would meet. Were this negative value of x to be measured from B, it would just be the same with that of x in the first case, m and n being interchanged. For in this case the value of will be measured from B, provided AB be added to it. But being in this case negative, AB must be taken negative, and a; then which is just equivalent to the interchanging of m and n in the first value of x in this case, and taking a negative. When the given quantities in a question are general, that is, when they are expressed by letters, interesting consequences may be sometimes deduced from the solution, by assigning particular values to the given quantities. Thus, from the preceding value of x in the first case, that is, each of the couriers just travels over half the distance, as is otherwise evident. or A has the whole distance to travel, or AB. or A does not travel any; B meets him at A, or B travels the whole distance BA, as is otherwise evident. am am In the second case, when m=n, x = mn 0 that is, A must travel an indefinitely great distance, or, in other words, he never can overtake B, as is manifest from the equality of their rates of travelling. or A travels in an opposite direction, that of AM', a distance = a, and B a distance therefore =2a. II. SIMPLE EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. (257.) Sometimes a question requires for its solution the determination of the values of two unknown quantities. In such a case two conditions are given in the question. These conditions are necessary, and they are sufficient; they must also be independent, that is, not deducible, by any process simple or complex, from one another. They must likewise be consistent, for if they be contradictory, the solution would lead to an absurd conclusion. The conditions, too, may be either implicit or explicit. Questions of this kind may also sometimes be solved by |