moting the work, either obftructed it, or undid it to a certain extent. P miles in q bours; the fecond x miles in s hours; let the interval between their Setting out be a; Then, by working as formerly, gra gr-ps If particular values be inferted for thefe letters, a particular folution will be obtainèd for that café. 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 alfo from the nature of the question, as the fecond courier muft 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 fecond as to that is, as ps to qr; q S and therefore qr must be greater than ps. SCHOLIUM. Sometimes when there are many known fim quantities in a general folution, it may plify the operation to express certain combinations of them by new letters, ftill to be confidered as known. С НА Р. СНАР. IV. Of Involution and Evolution. N order to refolve equations of the high IN er orders, it is neceffary to premise the rules of INVOLUTION and EVOLUTION. LE M M A. The reciprocals of the powers of a quantity may be expreffed by that quantity, exponents of the fame deThat is, the feries a, 1, with negative //nomination. a', a°, a—', a—2, a-3, a—, For For the rule for dividing the powers of the fame root was to fubtract the expo X nents; if then the index of the divifor be greater than that of the dividend, the index of the quotient must be negative. Cor. 1. Hence any quantity which multiplies either the numerator or denominator of a fraction, may be tranfpofed from the one to the other, by changing the fign of its index. xy—2. y yox 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. |