DISCUSSION OF FORMULAS FURNISHED BY THE GENERAL EQUATIONS OF THE FIRST DEGREE, WITH TWO OR MORE UNKNOWN QUANTITIES. When the common denominator of the general values of the unknown quantities reduces to zero, it is not readily seen how the given equations are to be verified. We shall examine here the particular cases of this kind which may occur. Resume the two equations, ax-\-by=k [1] a'x+b'y=zk' [2] from which we derive the formulas _kb'—bid _ak'—ka' X ab'—ba" TM ab' — ba'' First particular Case.—Suppose the denominators to be zero and the numerators not; then we have kb'— bk' ak'—ka' ab'—ba'=0, z= - , y = -. The values of x and y are then infinite; that is to say, in order to satisfy the two given equations, they must surpass every assignable magnitude. ab' From the equality ab'—ba'=0, we derive a'=-g-, and, consequently, the equation [2], by putting in it this value, becomes ab' -£-x-{-b'y=k', .•. b'(ax-\-by)=zbk'. The first member is tho first member of [1] multiplied by b'; the same relation must subsist between the second members, in order that the value of x and y may verify at the same time equations [1] and [2]. Hence 6Jf=kb', or, kb'—bk' = 0; i. e., the numerator of x would be equal to zero, which is contrary to hypothesis.* In this way tho impossibility of finding values of x and y, which satisfy at the same time the two given equations, is made apparent; but this impossibility is still better characterized by tho infinite values, which, at the same time that they indicate the impossibility, show besides that it arises from the fact that the values of the unknown quantities are too great to be assigned. If we suppose ab'—ba' to be at first a very small quantity, the values of x and y will be vory great, but they will always satisfy the equations until the instant ab'—ba' reduces to zero, when, if we can not effect in a direct manner the verification of the equations, it is solely because x and y then surpass all assignable magnitude.f Second particular Case.—Suppose the denominator to bo zero at tho same time as one of the numerators; for example, that we have ab'—ba'=0, kb' — bk'=0. I maintain that the other numerator will be also equal to zero; for the two equalities above give * The note to art 154 explain) this anomaly. The finite quantities W and Af are equal when compared with infinity. t Considered in relation to the question, the conditions of which are expressed by the problem, infinite values may be sometimes a true solution of the question. The application of alcrehra to geometry furnishes numerous examples of this kind; ajnoiit? others may be cited thut where nn antrle is unknown, and we find fur its tangent an iidinite value. It u clear, then, that the angle must be right. ab' , kb' a =T, k'=T, and, consequently, the other numerator becomes akb' akb' If at first wo had supposed this numerator equal to zero, we could have proved in a similar manner that of x to be so also. The present hypothesis then gives 0 0 Of themselves these symbols indicate indetcrmination; I shall prove, by going back to the equations, that they ought, in fact, to be indeterminate. For this purpose, substitute in equation [2] the values of a' and k', found above, and it becomes ab' , kb' b' , b' xI+6'y=TT' •'• b{eu+brt=jk Thus we see that it can be formed by multiplying the two members of equa6' tion [1] by y; then all values of x and y which satisfy one of the two equations will also satisfy the other. But if wo give to x values at pleasure in equation [1], we can, by resolving it afterward, find corresponding values of y; and as these same values satisfy the second equation, we conclude that the proposed equations admit an infinite number of solutions. Let it, however, be observed, that the indetermination in this case does not permit us to take whatever value of y, and, at the same time, of x, we please, because the abovo explication shows that, when one of these unknown quantities is assumed, the value of the other is determined. The case before us comprehends that in which A-=0, &'=0, ab'—ba'=0, 0 because then x and y become -. If we return to the equations proposed, they reduce to these, ax-\-by=0, a'x-\-b'y=0. They give respectively a a' y=—bx'y=~b'x a a' ^ But upon the hypothesis of ab' — ba'=0, we derive ^=-p; then the two values of y are equal, whatever be that of x, and there is veritable indetermination. Yet it is to be observed, that, if we take the relation of y to x, this relation is determinate, because we have y a a' If the condition ?=r; had not existed, the two values of y above could not oo have been equal, except we suppose x=0; y would have been then zero, and the relation of x and y no longer determinate, but indeterminate. A similar discussion to the above might be given to a system of threo or more equations, with ns many unknown quantities. It would, however, be more difficult to investigate the cases of impossibility and iudetermination, and it is not worth while to delay upon them. We shall content ourselves with setting down here some observations intended to caution the student against certain hasty conclusions to which he might naturally be led. We have seen, in the case of two equations with two unknown quantities, that x and y become infinite and indeterminate simultaneously. The first error which might be committed would be that of supposing from analogy that, in the case of several equations, the unknown quantities would all become infinite or indeterminate together. Suppose, for example, there :u-e under consideration the three equations ax -\-by -\-cz =^> a'x -\-b'y -\-c'z =k/, a"x-\-b"y-\-c"z —k". The common denominator of the values of x, y, z, is R=ab'c"—ac'b"-\-ca'b"—ba'c" -\-bc'a"—cb'a", and it may be written in three ways: R=a{b'c" —c'b")+a'(cb" — bc") + a"(bc'— cb'), Place b'c"=c'b", cb" = bc". From these equations we deduce bc'=cb', and, consequently, R becomes zero. Then the numerator of x, which is formed from R by changing a, a', a" into k, k', k", becomes zero also. But as the numerator of y is formed by placing k, k', k" in R instead of b, b', b", there is no reason why this numerator should become zero, unless we make some new hypothesis. The same may be said of that of z. Thus the value of x can take the indeterminate form g, where the values of y and z are infinite. But with regard to this indeterminate form, another error still is to be avoided, because it may be that the indetermination is only apparent (see Art. 155). In order to judge better of it, we shall have regard only to the single relation c'b" b'&'^b", .: c"=-jr Substituting this value of c" in the general value of x, it will be seen that be'—cb' becomes a common factor of both numerator and denominator. But by hypothesis this factor is zero; it is its presence, then, which produces tho appearance of indetermination. Suppressing it, we have the true value of x, which appears no longor indeterminate, unless some new hypothesis be joined to those already mad*.* * An important observation should be made before quitting the subject of indetermination. When the two terms of a fraction decrease so as to become less than any assignable quantity, if the suppositions which cause one of them to decrease indefinitely are entirely independent of those which cause the other to do so, the vnlues of these terms may be taken as near zero as we please, and such that their relation, which is the value of the fraction, may be equal to any quantity whatever; consequently, the symbol -, at which we arrivo when the two terms shall have attained the limit of their decrease, will express ramplete indetermination. But it may happen that the two terms of the fraction are connected together in such a way, that to a very small value of one there corresponds always 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. PROBLEM. Two couriers set off at the same time from two points, A and B, in the same —^7 j g ^— straight line, and travel in the same direction, A C. 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 A C and B C, expressed in miles. We have manifestly for the first equation x—y—a (1) Since m and n denote the number of miles traveled by each in an hour, that is, the respective velocities of the two couriers, it follows that the time rear *j quired to traverse the two spaces, x and y, must be designated by —, —; these two periods, moreover, are equal; hence we have for our second equation x y -=- (2) The values of x and y, derived from equations (1) and (2), are x= , y= . m—n J m—n 1°. So long as we suppose m~>n, or m—» 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. 2°. Let us now suppose m<n, or m—n negative, the values of x and y are both negative, and we have am an x n—m' * n—m The solution, therefore, in this case, points out that some absurdity must exist in the conditions of the problem. In fact, if we suppose m<n, 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 -J-x, and —y for +y, in equations (1) and (2); when modified in this manner, they become a very small value of the other; and that, when they converge toward zero, their relation converges toward a determinate limit, which it does not attain till the moment that tho two terms vanish, and the fraction presents itself under the form A particular example of this last case is the vanishing of a common factor of the numerator and denominator The same remark is applicable to the symbol —. 00 * * TLLa principle is fully exemplified in the d y—x=a i equations which, when resolved, give am an n—m J n—m in which the values of r 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<Cji, but of the following, which is the translation of the changed equations. Two couriers set out at the same time from the points A and B, and travel in the direction B C, ice. (the rest as beforo); the values of x and y mark the distances A C, B C, of the point C, where tho couriers are together, from the points of departure A and B. From this problem, as well as that of the father and son above, may be deduced tho following rule, when the value of the unknown quantity is found to be negative: Change the sign of the unknown quantity in the first equation, or the one derived immediately from the problem; this changed equation, translated into common language, will furnish the problem which will give a positive solution. If the problem be at first enunciated in a general manner, then negative values of the unknown quantity may be regarded as furnishing a true solution, but are to be interpreted in a contrary sense. Thus, if positive values represent distance to the right, negative •will represent distance to the left; if positive express distance upward, negative distance downward; if the former indicate time future, the latter must indicate time past; if Oie one gain, tlic other loss; if the one a rate of increase, the other a rate of decrease, &fc* 3°. Let us next suppose m=n; tho values of x and y in this case become am an x=t< y=~o' or r=<x>, y = <»; 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 samo, and if the couriers travel forever, they can nover be together, f * Applications of this use of positive and negative quantities constantly occur in trigonometry and analytical geometry. t Since m=n, equation (2) gives x=y, and equation (1), in consequence, a=0. To understand this, we must recur to the principle stated in (Art. 154). Wo may here extend a little the statement there made. All zeros are equnl when compared with finite quantities, but not when compared with one another. Thus, 2x is twice as great as x, thouirh x be 0; but 2x^-a=-x-\-a=.a, if x=0. In the first of these cases one zero, 2x, is compared with another, and then they are not equal; in the second, both zeros, 2x and x, are compared with the finite quantity, a, and then aro equal. A^ain, x-\-a=x-\-\0a=x~\-0=x, if :r=ac; but 10a is ten times as great as a, when unconnected with infinity. Finite quantities are, therefore, all equal to one another, and all equal to zero when compared with infinite ones, but not when simply compared with one another. It is rare that algebra can be employed to demonstrate moral or religious tmth; |