moting the work, either obstructed it, or undid it to a certain extent. In question stb, let the firj courier travel р miles in 9 bours; the fecond r miles in s hours ; let the interval between their setting out be a; Then, by working as formerly, gra If particular values be inserted for these letters, à particular solution will be obtained 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 first courier is to the rate of the second as to that is, as ps to gr; P 9 and therefore gr must be greater than ps. Sometimes when there are many known quantities in a general solution, it may fima plify the operation to express certain combinations of them by new letters, still to be considered as known, N order to resolve equations of the high er orders, it is necessary to premise the rules of INVOLUTION and EVOLUTION. IN LE M M A. of a quan The reciprocals of the powers tity may be expressed by that quantity, with negative exponents of the same denomination. That is, the series a, 1, &c. be expressed by a a', ao, a-, a», a-?, amm, &c. For I I I: ! 1 a. a am 3 2 For the rule for dividing the powers of the same root was to fubtract the expo heim nents ; if then the index of the divisor be greater than that of the dividend, the index of the quotient must be negative. 1 a 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. Tug bs sit: 21 I 2, }', TS10A LE WITH a2x Thus,=xy—. And у sitius 90 bis can D1301 - Cor. 2. From this notation, it is evident that these negative powers, as they are called, are multiplied by adding, and divided by subtracting their exponents. con uns I Thus, 3 |