or by changing the signs in the two members, We see that the distance passed over by the courier from the a. point B, is c d am, or what remains of B C after A B and AR are subtracted, that is CR, and that AC=cdThis is just what would take place if the second courier had started immediately from the point C, where he is at the departure of the first; but as they travel in opposite directions, they must necessarily meet between A and C. Thus, this case is similar to the first of those of art. 74, where it is sufficient to change a cd into cda, in order to obtain the value, which m has according to the above equation.* 76. The problem of art. 56, taken in its most enlarged sense, may be enunciated as follows; A labourer having passed a number a of days in a family, and having with him his wife and son during a number b of days, received a sum c; he lived afterward in the same family a number d of days; he had with him this time his wife and son during a number e of days, and he received a sum f; we inquire what he earned per day, and what was allowed per day to his wife and son. Let a represent constantly the daily wages of the labourer, y that of his wife and son; for the number a of days he has ax, and for the number b of days his wife and son have by, so that, and ax + by = c; for the number d of days, he has da, and for the number e of days his wife and son have ey, thus, dx+ey=f. These are the general equations of the question. We deduce from the first x = c―by. α multiplying this value by d, in order to substitute it in the place * See note at the end of the Elements of Algebra. cd-bdy+aey=af, aey-bdy-af-cd, y=af-cd Having the value of y, if we substitute it instead of y in the expression for x, this last will be known To simplify this expression, we should, in the first place, perforin the multiplication indicated upon the quantities and then reduce c to a fraction having the same denominator as the fraction which accompanies it, and perform the subtraction of this fraction (53); and it becomes *There might be some doubt as to the meaning of this expression; but it is obviated by attending to the bar denoting division, which is placed in the middle of the line. Thus, in the expression A x= A represents the dividend, whether integral or fractional, and B' B the divisor, which may also be a whole number or a fraction. So also Suppressing the factor a, common to the numerator and denom are applied in the same manner as those, which we before found for literal equations, with only one unknown quantity; we substitute in the place of the letters, the particular numbers in the example selected. We shall obtain the results in art. 56, by making 77. The values of x and y are adapted not only to the proposed question; they extend also to all those, which lead to two equations of the first degree with two unknown quantities, since it is evident, that these equations are necessarily comprehended in the formulas, signifies, that x is equal to the quotient of the B A A fraction divided by B, and the expression x = で indicates for x B A B the quotient arising from A divided by the fraction; and lastly, we denote by the expression x = the quotient resulting from the di It will be perceived by these remarks, that it is necessary to place the bars according to the result, which we propose to express. The points A and B being coincident, we have on this supposition a = 0, and constantly b= c; it follows then, that In order to interpret these values, that indicate a division, in which the dividend and divisor are each nothing, I go back to the equations of the question. The first becoming x-y=0 gives x=y; and substituting this value in the second equation, which is The last equation having its two members identical, that is to say, composed of the same terms with the same sign is verified, whatever value is assigned to y, and this unknown quantity can never be determined. Besides, it is evident that the equation x=14/ becomes x=Y, and consequently can express nothing more than the first.* The only result both from the one, and from the other is, that the two couriers are always together, since the distances x and y from the point ♬ are equal, their value in other respects remains indeterminate. The expression then is here a symbol of an indeterminate quantity. I say here, for there are cases where it is not; but the expression has not then the same origin as the preceding. 70. To give an example, let there be This quantity becomes in its present form when a = b; but if we reduce it first to its most simple expression, by suppressing the factor a b, common to the numerator and denominator, we find *For the sake of conciseness, analysts apply to the same equations the epithet, identical. y b = y b is an identical equation, 5-3x 5-3x is another, and when two equations express only the same thing, we say that these equations also are identical. It is not the same with the values of x and y, found in the preceding article, for they are not susceptible of being reduced to a more simple expression. It follows, from what I have just said, that when we meet with an expression which becomes, it is proper before pronouncing upon its value, to see if the numerator and denominator have not a common factor, which becoming nothing, renders the two terms at the same time equal to zero, and which being suppressed, the true value of the proposed expression is obtained. There are, notwithstanding, some cases which elude this method, but the limits of this work will only allow me to note the analytical fact. It belongs properly to the differential calculus to give the general processes for finding the true value of quantities, which become g. 71. It is very evident, from what has been said, that algebraic solutions either answer perfectly to the conditions of a problem, when it is possible, or they indicate a modification to be made in the enunciation, when the things given imply contradictions that cannot be reconciled; or lastly, they make known an absolute impossibility, when there is no method of resolving with the same things given, a question analogous in a particular sense to the one proposed. 72. It may be remarked, that in the different solutions of the preceding question, the changing of the signs of the unknown quanties x and y corresponds to a change in the direction of the journeys represented by the unknown quantities. When the unknown quantity y was counted from B towards A, it had in the equation x+y=a, the sign+, and it takes the sign for the second case, when the motion is in the opposite direction from B towards C, art. 65, since we had for the first equation |