this, let us suppose that the difference between m and n without being absolutely nothing is very small; in this case, it is evident, that the values of x and y will be very large, Let, for example, m= =3, m―n= = 0. 01, we shall then have n=2. 99, whence Again let mn=.0001, m being equal to 3, n will then am whence am m-n an an =30000 a, m-n =29999 a In a word, so long as there is any difference, however small, between m and n, the couriers will meet in one direction or the other; but the distance of the point, in which they meet, from the points A and B will be greater in proportion as the difference between m and n is less. If then the difference between m and n is less than any assignable quantity, the distances will be greater than any assignable quantity or infinite. Since then 0 is less than any assignable quantity, we may employ this character to represent the ultimate state of a quantity which may be decreased at pleasure; and since the value of a fractional quantity is greater, in proportion as its denominator is less, the expression and in general, any m-n m -n a m 0 quantity with zero for a denominator may be considered as the symbol of an infinite quantity, that is, a quantity greater than any, which can be assigned. am To show how the notion indicated by the expression does away the absurdity of the equations x - y = a, x— y = 0, 0 'from the second of these equations, we deduce the value of y and substitute it in the first, we then have x ing both sides of this last by x, we have divid Here, as we put for x values greater and greater, the fraction will differ less and less from 0, and the equation will apIf then ≈ be greater proach nearer and nearer to being exact. a than any assignable quantity, will be less than any assigna ble quantity or zero. Xx 4. Let us suppose next m=n, and at the same time a= = 0, we shall then have 2= y= But how shall we interpret this new result? Returning to the enunciation, we perceive, that if the couriers set out each from the same point and travel equally fast, there is no particular point in which they can be said to meet, since in this case, they will be together through the whole extent of their route. Indeed, on this hypothesis the equations of the problem become equations which are identical; the problem is therefore indeterminate, since we have in fact but one equation with two unknown quantities. The expression is therefore a sign of indetermination in the enunciation of the problem. The preceding hypotheses are the only ones, which lead to remarkable results. They are sufficient to show the manner in which algebra corresponds to all the circumstances in the enunciation of a problem. GENERAL FORMULAS FOR EQUATIONS OF THE FIRST DEGREE WITH ONE OR TWO UNKNOWN QUANTITIES. 87. Every equation of the first degree with one unknown quantity may, by collecting all the terms, which involve x into one member and the known quantities into the other, be reduced to an equation of the form ax= b, a and b denoting any quantities whatever, positive or negative. Comparing this equation with the general formula, we have a=27, b=216. Again let there be the equation Freeing from denominators, transposing and uniting terms, we have (mn) x = n (p − q). Comparing this equation with the general formula, we have m · n = a, n (p − q) = b. 88. Resolving the equation a x — = b, we have x = is a general solution for equations of the first degree with one unknown quantity. Discussion. 1. Let it be supposed, that in consequence of a particular hypothesis made upon the known quantities, we have a=0, the value of a will then be b But the equation a x= b on 0 this hypothesis becomes 0 x xb, an equation which it is evident, cannot be satisfied by any determinate value for x. The equation 0 x x=b may, however, be put under the 0. Here, if we consider x greater than any as b signable quantity, the fraction will be less than any assign x able quantity or zero. On this account we say that infinity in this case satisfies the equation. It is evident, at least, that the equation cannot be satisfied by any finite value for x. 2. Let us suppose next a = = 0, and at the same time b — 0. Ο the value of x will then take the form 0. = In this case the equation becomes 0 x x 0, an equation which may be satisfied by any finite quantity whatever, positive or negative. Thus the equation, or the problem, of which it is the algebraic translation, is indeterminate. It should be observed, however, that the symbol does not always indicate that the problem is indeterminate. Let for example the value of x derived from the solution of a problem be x= a3 -b3 If we put ab in this formula, it will, under its present form, be reduced to o; but this value for a may be put under (a — b) (a2 + ab + b2) (a—b) (a + b) If then before making the hypothesis ab, we suppress the factor ab, the value of x becomes a2+ab+b2 from which we obtain x = on the hypothesis ab. We conclude therefore that the symbol is sometimes in alge bra the sign of the existence of a factor common to the two terms of a fraction, which in consequence of a particular hypothesis becomes 0, and reduces the fraction to this form. Before deciding then, that the result is a sign that the problem is indeterminate, we must examine whether the expressions for the unknown quantities, which in consequence of a particular hypothesis are reduced to this form, are in their lowest terms, if not, they must be reduced to this state; the particular hypothesis being then made anew, the result o shows that the problem is really indeterminate. 89. Every equation of the first degree with two unknown quantities may be reduced to an equation of the form a, b and c denoting any quantities whatever, positive or negative. It is evident, that all equations of the first degree with two unknown quantities may be reduced to this state, 1°. by freeing the equation from denominators; 2°. by collecting into one member all the terms, which involve x and y, and the known quantities into the other; 3°. by uniting the terms, which contain x into one term, and those, which contain y into another. This is a general solution for all equations of the second degree with two unknown quantities. To show the use, which may be made of these formulas in the solution of equations, let there be the two equations, 5x+3y=19, 4x + 7y=29 Comparing these with the general equations, we have a=5, b=3, c = 19, a' = 4, b' = 7, c' = 29, In the above formulas for x and y let a b' -ba': = = 0, c b' -b c' and a c-c a' being each different from zero, we |