suit. At what rate did A. travel, and what was the rate that B. traveled at first? Ans. A.'s rate, 9 miles; B.'s, 81. 25. There are 32 gallons of wine in two casks. If from the first there be drawn into the second as much as there is in the second; then if there be drawn from the second into the first as much as remains in the first; and then if there be drawn from the first into the second as much as remains in the second, there will be 16 gallons in each cask. How many gallons were there originally in each? Ans. 22, and 10. + 2 26. A. cistern can be filled by 3 pipes. The first can fill it in 4 hours, the first and second together can fill it in 3 hours, and the third can fill it in 2 hours. How long will it take for them all to fill it together, and how long will it take the second alone to fill it? Ans. All in 1 h. 12 m.; the second in 12 hours. 27. A cistern has two discharge cocks: they both run together for two hours when the first one is closed; the second one then empties it in 2 hours and 48 minutes. Had the second one been closed at the end of two hours, the first one would have emptied it in 4 hours and 40 minutes. In what time could each empty it alone? Ans. The first in 10 hours; the second in 6 hours. III. EXPLANATION OF SYMBOLS AND DISCUSSION OF PROBLEMS. Explanations and Principles. 89. The symbol 0 is called zero; the symbol is called infinity; and the symbol is called the symbol of indetermination. To explain the meaning of these symbols, let us take the equation, d Which can be written under the form dxt = a = (2.) Any set of values of a, d, and t, that will satisfy equation (2) will, of necessity, satisfy equation (1). It is obvious that satisfy this equation. and (1), we have, 1o. If we suppose a to be equal to 0, and d to be finite, that is, to contain a limited number of units, equation (2) will become 0 x d = 0; and (1.) dx t = 0. 0 is the only value of t that will Making a = 0 and t = 0 in (2) Hence, we say that 0 multiplied, or divided, by a finite quantity is equal to 0. 2o. If we supposed to be equal to 0, and a to be finite, equation (2) becomes 0x t = a. It is obvious that no finite value of t can satisfy the last equation; this fact we express by saying that t is infinite, that is, that it is greater than any assignable quantity. Making d = 0 and to in equation (1), we have, Hence we say that a finite quantity divided by 0 is equal to infinity, and that a finite quantity divided by infinity is equal to 0. 3o. If both a and d are supposed equal to 0, equation (2) becomes t = 0 x t = 0. It is obvious that this equation will be satisfied for every finite value of t, (principle 1°). Making a = 0 and d = 0 in (2), we have Which is true for all finite values of t. In this case the equation does not determine the value of t, a fact that we express by saying that t is indeterminate. Hence, we say that an indeterminate quantity is one that has an infinite number of values. A fraction may reduce to the indeterminate form in consequence of a common factor in both terms, which factor becomes 0, for the particular hypothesis. Thus, the fraction 7(x + 1) (x − 1) 2(x-1) 0 reduces to for the particular value x = 1. If we strike out the 0 ― common factor x - 1 and then make x = 1, we find the true value of the fraction to be 7. Before deciding on the nature of the expression we must, therefore, determine whether it results from the existence of a common factor which reduces to 0 for the particular hypothesis; if not it is a true symbol of indetermination. Definitions. 90. The discussion of a problem consists in making every possible supposition on the arbitrary quantities which enter it, and interpreting the results. An arbitrary quantity is a quantity to which a value may be given at pleasure. The method of interpreting results is illustrated in the solution and discussion of the following problem : Problem of the Couriers. 91. Two couriers, A. and B., travel along the same line, R' R, in the same direction, R' towards R, and at uniform rates; the courier A. travels m miles per hour, and the courier B., n miles per hour. Now, supposing them to be separated by a distance a at any epoch, say 12 o'clock, when are they together? R' A B R Let the position of the rearmost courier, A., be taken as the origin of distances, and suppose all distances estimated towards B. to be positive. Denote the number of hours from the epoch to the time they are together by t. Denote the distance the courier B. travels in the time t, by x; then will the distance that the courier A. travels, in the same time, be denoted by a + x. Then, since the distance traveled is equal to the number of hours multiplied by the rate per hour, we have the equations: Discussion. 92. In discussing the value of t, found in the last article, it is to be observed that the distance between the couriers may be assumed at pleasure; hence, a is arbitrary: the rates of travel may also be assumed at pleasure; hence, m and n are arbitrary. From the conditions of the problem, a can never be negative; hence, the only suppositions that can be made on a are a > 0, and a = 0. The only suppositions that can be made on m and n are m>n, m<n, and m = n. By combining these hypotheses we obtain six suppositions, as follows: 1o. a > 0, and m>n. 3°. a= 0, and m>n. 5. a > 0, and m=n. ·2°. a > 0, and m <n. 4°. a= 0, and m < n. 6°. a= 0, and m=n. We shall consider these hypotheses in order: 1o. a > 0, and m>n; 2°. a > 0, and m <n. The first supposition makes the value of t, in article 91, essentially positive, and the second supposition makes the value of t essentially negative. |