Prob. 9. Find two numbers such that the product of their sum and difference may be a, and the product of the sum of their squares and the difference of their squares may be ma. Prob. 10. A laborer dug two trenches, whose united length was 26 yards, for 356 shillings, and the digging of each of them cost as many shillings per yard as there were yards in its length. What was the length of each? Ans. 10, or 16 yards. Prob. 11. What two numbers are those whose sum is 2a, and the sum of their squares is 26? Ans. a+√b-a2, and a-√b-a2. Prob. 12. A farmer bought a number of sheep for 80 dollars, and if he had bought four more for the same money, he would have paid one dollar less for each. How many did he buy? Let x represent the number of sheep. would be the price of each if he had bought four Prob. 13. A person bought a number of articles for a dol lars. If he had bought 26 more for the same money, he would have paid c dollars less for each. How many did he buy? Prob. 14. It is required to find three numbers such that the product of the first and second may be 15, the product of the first and third 21, and the sum of the squares of the second and Ans. 3, 5, and 7. third 74. Prob. 15. It is required to find three numbers such that the product of the first and second may be a, the product of the first and third b, and the sum of the squares of the second and third c. a2+b2 Ans. Prob. 16. The sum of two numbers is 16, and the sum of their cubes 1072. What are the numbers? Ans. 7 and 9. Prob. 17. The sum of two numbers is 2a, and the sum of their cubes is 26. What are the numbers? Prob. 18. Two magnets, whose powers of attraction are as 4 to 9, are placed at a distance of 20 inches from each other. It is required to find, on the line which joins their centres, the point where a needle would be equally attracted by both, admitting that the intensity of magnetic attraction varies inversely as the square of the distance. Ans. { 8 inches from the weakest magnet, or -40 inches from the weakest magnet. Prob. 19. Two magnets, whose powers are as m to n, are placed at a distance of a feet from each other. It is required to find, on the line which joins their centres, the point which is equally attracted by both. Ans. The distance from the magnet m is The distance from the magnet n is avm √m± √ ñ ±a√n √m± √nR Prob. 20. A set out from C toward D, and traveled 6 miles an hour. After he had gone 45 miles, B set out from D toward C, and went every hour of the entire distance; and after he had traveled as many hours as he went miles in one hour, he met A. Required the distance between the places C and D. Ans. Either 100 miles, or 180 miles. Prob. 21. A set out from C toward D, and traveled a miles per hour. After he had gone b miles, B set out from D toward C, and went every hour 4th of the entire distance; and after he had traveled as many hours as he went miles in one hour, he met A. Required the distance between the places C and D. Prob. 22. By selling my horse for 24 dollars, I lose as much per cent. as the horse cost me. What was the first cost of the horse? Ans. 40 or 60 dollars. Prob. 23. A fruit-dealer receives an order to buy 18 melons provided they can be bought at 18 cents a piece; but if they should be dearer or cheaper than 18 cents, he is to buy as many less or more than 18 as each costs more or less than 18 cents. He paid in all $3.15. How many melons did he buy? Ans. Either 15 or 21. Prob. 24. A line of given length (a) is bisected and produced; find the length of the produced part, so that the rectangle contained by half the line, and the line made up of the half and the produced part, may be equal to the square on the produced part. α Ans. (1+√5). Equations of the Second Degree containing Two Unknown Quantities. 266. An equation containing two unknown quantities is said to be of the second degree when the highest sum of the exponents of the unknown quantities in any term is two. Thus and x2+y2=13, are equations of the second degree. (1.) (2.) 267. The solution of two equations of the second degree containing two unknown quantities generally involves the solution of an equation of the fourth degree containing one un known quantity. Thus, from equation (2), we find Substituting this value for y in equation (1) and reducing, we have x2+2x3-11x2-48x=-108, an equation which can not be solved by the preceding methods. Yet there are particular cases in which simultaneous equations of a degree higher than the first may be solved by the rules for quadratic equations. The following are the principal cases of this kind: 268. 1st. When one of the equations is of the first degree and the other of the second.-We find an expression for the value of one of the unknown quantities in the former equation, and substitute this value for its equal in the other equation. Ex. 1. Given { { x2+3xy—y2 = 23 From the second equation we find x=7-2y. to find x and y. Substituting this value for x in the first equation, we have 49-28y+4y2+21y—6y2—y2—23, which may be solved in the usual manner. For equations of this class there are in general two sets of values of x and y. 269. 2d. When both of the equations are of the second degree, and homogeneous.-Substitute for one of the unknown quantities the product of the other by a third unknown quantity. v2+v v-2 From which we obtain v=8 or 3. Substituting either of these values in one of the preceding expressions for y2, we shall obtain the values of y; and since x=vy, we may easily obtain the values of x. 8 1 y=±1 or ± √6 y= ±3 or ± |