(m* — n1) a2b3 + 2 (m2 + no)2 abc x = m1a2 + (m2 + n2)2x2 (m1 — n1) a2 (m2 - n3) ab2 + 2(m2 + n2) bcx = (m3 — n2) a n*a2b2 — 2 n2 (m2 + n2) abcx + (m2 + no)°c2x2 ̄ n*a2 + (m2 + n2)2x2' (m2 — n2) n1a3b2 + (m* — n1) (m2 + n3) ab2x2 + 2 (m2 + n2) n‘a2bcx + 2 (m2 + n2)3bcx3 = = (m2 — n2) n*a3b2 — 2 (m* — n1) n3a2bcx + (m* − n1) (m3 + n2) ac2x2, (m* — n1) ab3x + 2n*a2bc + 2(m2 + n2)2bcx2 = (m1 — n1) a c2x − 2 (m2 — n2)n2a2bc, 2(m2 + no)2bcx2 + (m1 − n1) a (b3 − c3) x + 2 m2 n2a2bc = 0, (m3 + n3)2x2 + 2 (m2 + n3) m na x + m2n3a2 = 0, ··· mn = b2 — c2 m2 + n2 4 Taking the upper sign, and substituting in (a), 4u2u-u — 2u + 1 = 1, (B). Taking the lower sign, and substituting in (a), from (ii), x = ± 1/1√33; and y-√5/33. PROBLEMS. IN reducing a Problem to an Equation, the course to be pursued is stated in Art. 199; but much depends here, as in the solution of equations, upon a practical acquaintance with particular artifices, by which the most convenient unknown quantities are assumed, and the problem most easily translated into algebraical language. The general question always is, having certain known quantities, represented by given symbols, and one or more other unknown quantities, represented by one or more of the letters x, y, z, &c., to connect the known and unknown symbols together by the conditions of the problem, so as to produce as many independent equations as there are unknown quantities. There is also one general property of a large class of such problems,. viz., that the increase or decrease, the selling or buying, &c., is after a uniform rate. Thus, if A is said to perform a piece of work in a days, he is supposed to work equally every day. If A is said to travel p miles in q days, he is supposed to travel one uniform distance each day. And so on, unless the contrary be expressed. So that the following Rule is of constant application, seeing that uniform increase or decrease of every sort may be represented by uniform motion :— RULE. If v represent the space described by a body moving uniformly in 1 unit of time, (whether it be 1 second, 1 hour, or any other known unit), and s the space described by the same body in t such units of time, then s= = tv. 1 day Also v = ==, and t ==; both of which forms are frequently required. S t' Thus, if A travels Р = whole distance (s) number of days (t) miles in q days, then the distance (v) travelled in whole distance distance per day The following Problems are added here as differing in some material respect from those in the text: PROB. 1. In the year 1830 A's age was 50 and B's 35. Required the year in which A is twice as old as B. .. the year required is 1810. PROB. 2. In what proportions must substances of "specific gravities" a and b be mixed, so that the "specific gravity" of the mixture may be c? DEF. By the "specific gravity" of a body is meant the number of times which its weight is of the weight of an equal bulk of water. To 1 cubic foot of the first substance let a cubic feet of the second be added. Then, since 1 cubic foot of the first weighs a cubic feet of water, and x ....... feet ...... ... the whole 1 + x cubic feet of mixture weighs a + bx cubic feet of water. But since c is the specific gravity of the mixture, the weight of 1 + x cubic feet is c(1+x) cubic feet of water, that is, for every cubic foot of the substance whose specific gravity is a a-c there must be cubic feet of the substance whose specific gravity is b. с PROB. 3. From a vessel of wine containing a gallons b gallons are drawn off, and the vessel is filled up with water. Find the quantity of wine remaining in the vessel, when this has been repeated n times. Let x1, x2, x3, x be the number of gallons of wine remaining in the vessel after 1, 2, 3,... n drawings off respectively. (2) x2 = a-b- quantity of wine in b gallons of first mixture. = a quantity of wine in b gallons of second mixture. And so on, for each succeeding mixture; so that, generally, PROB. 4. The advance of the hour-hand of a watch before the minute-hand is measured by 15% of the minute divisions; and it is between 9 and 10 o'clock. Find the exact time indicated by the watch. Let x = goes number of minutes past 9 o'clock; then since the minute12 times as fast as the hour-hand, hand number of minute divisions the hour-hand is past 9, x= distance in minutes between hour-hand and minute-hand. 11x 12 .. x = 32. 88 Hence the time required is 28 minutes before 10 o'clock. PROB. 5. In comparing the rates of a watch and a clock, it was observed on one morning, when it was 12h. by the clock, that the watch was at 11.59m. 49; and two mornings after, when it was 9h. by the clock, the watch was at 8. 59. 58. The clock is known to gain 0·1® in 24 hours, find the gaining rate of the watch. On the 1st morning the watch is behind the clock 11'; and after 45 hours it is only 2 behind; .. the watch gains upon the clock to the 1 amount of 9 in 45 hours, or of a second per hour. 0.1 Let x be the gaining rate of the watch per hour; then since 24 the gaining rate of the clock, a – 0.1 24 is the gain of the watch upon the Hence the gaining rate of the watch is 4.9 in 24 hours. PROB. 6. If A and B together can perform a piece of work in a days, A and C together the same in 6 days, and B and C together in c days; find the time in which each can perform the work separately. Let w represent the work, and x, y, z, the times in which A, B, C, can separately do it. |