i.e. t=4. or (3t-2) (t3-6t°+10t-8)=0; 3t-2-0, or t3-61+101-8=0, (or t1-21+2=0, li.e. (t−1)2 =−1, which is impossible. Therefore, taking the first value of t, and substituting in (1) Taking the second value of t, other values of x and y remain to be determined. Ex. 58. {a (b + c)' + (b2 + c2) cz} × √ √ (b−c)*+(b+c°)'z' 缨 - {@b (b−c) - (b'+ c')cz} * √↓ = (b+c)*+ (b*+c')'z'; find z. Let b+c=m, and b−c=n, so that b2+c2='/(m2+n2), m*a2b2+2m2(m2+n2)abcx+(m2+n2)2c2x2 _ m*a3+ (m2+n2)2x2 = n*a2b2— 2n2(m2+n2)abcx+(m2+n3)2c2x2 ̄ n'a2+(m2+n2)2x2 ' n*a2b2— 2n2(m2+n2)abcx+(m2+n3)3c2x2 ̄ ̄ n*a2+(m2+n°)*x3 ? (m3—n3)n*a3b3+(m*—n*)(m2+n2)ab2x2+2(m2+n3)n*a*bcx+2(m2+n3)3bcx3 =(m2-n3)n*a3b2—2(m1—n1)n3a2bcx+(m1—n")(m2+n2)ac2x2, (m*— n1)ab3x+2n*a*bc+2(m2+n2)3bcx3— (m*—n1)ac3x—2(m3— n2)n3a2bc, 2(m3+n3)3bcx2+(m1—n1)a(b3—c3)x+2m3n3a2bc=0, Taking the upper sign, and substituting in (a), Taking the lower sign, and substituting in (a), 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. , and t; both of which forms are frequently required. whole distance = Thus, if A travels p miles in q days, then the distance (v) travelled in 1 day : 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. 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 x cubic feet of the second be added. Then, since 1 cubic foot of the first weighs a cubic feet of water, .. the whole 1+ a 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. c-b 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 1, 2, 3, • ., be the number of gallons of wine remaining in the vessel after 1, 2, 3,...n drawings off respectively. (1) Then x-a-b. (2) x-a-b-quantity of wine in b gallons of first mixture. Now, a gallons contain a-b of wine; .. 1 gallon contains and gallons contain b. .. x=a-b-b. a a a- b a -quantity of wine in b gallons of second mixture. And so on, for each succeeding mixture; so that, generally, (a - b)" PROB. 4. The advance of the hour-hand of a watch before the minutehand is measured by 153 of the minute divisions; and it is between 9 and 10 o'clock. Find the exact time indicated by the watch. Let x=number of minutes past 9 o'clock; then since the minute-hand goes 12 times as fast as the hour-hand, 45+ 12 ;=number of minute divisions the hour-hand is past 9, x=distance in minutes between hour-hand and minute-hand, Hence the time required is 28 minutes before 10 o'clock. |