moting the work, either obstructed it, or undid it to a certain extent. y In question stb, let the firft courier travel 9 bours ; the fecond r miles in s bours; let the interval between their setting out be a; p miles in If particular values be inserted for thefe letters, a particular solution will be obtainèd for that cafe. Let them denote, the numbers in Example 5. Here it is obvious, that qr must be greater than ps, else the problem is impoffible; for then the value of x would either be infinite or negative. This limitation appears also from the nature of the question, as the the second courier must travel at a greater rate than the firft, in order to overtake him. For the rate of the firit courier is to the rate of the second as to, that is, as ps to qr; and therefore qr must be greater than ps. 9 Sometimes when there are many known quantities in a general solution, it may sim, plify the operation to express certain combinations of them by new letters, still to be considered as known, CH A P. N order to resolve equations of the high er orders, it is necessary to premise the rules of INVOLUTION and EVOLUTION. INO L E M M A. The reciprocals of the powers of a quantity may be expressed by that quantity, with negative exponents of the same denomination. That is, the series a, 1, , &c. may be expressed by a2 a', ao, a 'a a For I I I I a a am 1 a a For the rule for dividing the powers of the same root was to fubtract the exponents ; if then the index of the divisor be greater than that of the dividend, the index of the quotient must be negative. Thus, a' a? a? m و الموت في وربيون يورو on of others. a? Cor. 1. Hence any quantity which multiplies either the numerator or denominator of a fraction, may be transposed from the one to the other, by changing the fign of its index. 9 b's sit 2017 Ls 97199429 Thus, * =*—. And ax yo Mitto 90 bin oli - Cor. 2. From this notation, it is evident that these negative powers, as they are called, are multiplied by adding, and divided by fubtracting their exponents. 20445 Thus, 3 |