49x2+49 x 150. 15. 846 +47 æ3 +47 2-46x+8= 0. PROBLEMS IN PHYSICS 19. The Simple Pendulum. If a simple pendulum swings through an arc the extremities of which are A and B, the motion of the pendulum from A to B or from B to A is called an oscillation or a vibration. 20. The time of vibration is the time required for the pendulum. to make a single vibration, that is, to move from A to B, or from B to A. 21. The distance from the lowest point of the arc AB to either extremity, A or B, that is, one-half of the arc AB, is called the amplitude of vibration. 22. When the amplitude of vibration is very small, an approximate value of the time of vibration, t, expressed in seconds, of a pendulum of length 7, expressed in feet, is found by the formula t = T In the following examples the approximate value 22/7 may be taken for the numerical constant . EXERCISE XXIV. 6 Solve each of the following problems relating to the simple pendulum : 1. Find the length in feet of a pendulum which vibrates once in a second at a place at which g = 32.16. 2. Find the value of g at a certain place if a pendulum which is 10 feet in length makes 20 vibrations in 35 seconds. 3. If a certain pendulum vibrates once in a second, find the time required for a pendulum which is twice as long to vibrate once. 4. Find the length of a pendulum which makes 80 vibrations per minute at a place at which the value of g is 32.16. 5. Find the length of a pendulum which vibrates once per second at a place at which the value of g is 32.19. 6. If at a certain place a pendulum 39 inches in length vibrates once in a second, find the length of a pendulum which at the same place will make one vibration in one minute. 7. If a ball suspended by a fine wire makes 88 vibrations in 15 minutes, find the length of the wire. 8. If a pendulum which is 39.1 inches in length vibrates once in a second at a certain place, find the length of a pendulum which will vibrate once in 5 seconds. EXERCISE XXIV. 7. Review 1. If a = 1, b = 3, and c = 2, find the value of 5. Find the prime factors of (a a2)3 + (a2 − 1)3 + (1 − a)3. 15. Express ( a − b) ÷ (— a-1-6) with the minimum number of minus signs. 17. Show that (a2 — b2)2 = a3 — ab + b3, if a + b = 1. Simplify each of the following expressions : 18. (3√2)(2 — √— 3). 19. ab + √ab + (a − √b)(√a − b). 21. (√√5 − √7 + 2)(√5 + √/7 − 2). 23. (√II — √6 + 5)(√11 − √6 − 5). CHAPTER XXV SYSTEMS OF SIMULTANEOUS EQUATIONS INVOLVING SYSTEMS OF TWO EQUATIONS CONTAINING TWO UNKNOWNS 1. THE most general form for an equation of the second degree containing two unknowns, a and y, is ax2 + 2 hxy + by2 + 2 gx + 2fy + c = 0, in which a, b, c, f, g, and h all represent real known numbers. If one or more of these letters be given the value zero, the terms of which they are the coefficients disappear, and we have special types of quadratic equations containing two unknowns, x and y, such as the following: If f, g, be zero, we have ax2 + 2 hxy + f, g, h, b, f, g, a, f, g, b, g, h, a, f, h, etc. by2 + c = 0. (i.) 2. In this chapter we shall obtain the solutions of certain systems of simultaneous equations, containing one or more equations of the second degree, of the general types shown above. 3. In Chapter XVII, we found that there exists a definite set of values of the unknowns which satisfies all of the equations of a given set simultaneously, provided that the given equations are independent and consistent, and that the number of equations is equal to the number of different unknowns appearing in the system. We shall find, whenever one at least of the given equations composing a system is of the second or higher degree, that there can be found more than one set of values which satisfies all of the equations at the same time. 4. There are many systems consisting of two equations of the second or higher degrees with reference to two unknowns which cannot be solved by means of quadratic or linear equations. 5. Whenever a system consists of one equation of the second degree and another of the first degree with reference to two unknowns, say x and y, the equation of the second degree having the form either of the general equation ax2+2hxy + by2 + 2 gx + 2 fy + c = 0, or of one of the special forms ax2 + by2 + c = 0 (see § 1), we may always make the solution of the system depend upon the solution of equations of either the second or of the first degree. 6. From the equation of the first degree, we may express the value of one of the unknowns, say y, in terms of the other, x; on substituting this expressed value for the same unknown, y, wherever it appears in the equation of the second degree, we shall derive an equation of the second degree containing but one unknown, x, called the x-eliminant of the system; this x-eliminant may be solved by the methods already shown for the solution of quadratic equations containing one unknown. From the given system is thus derived an equivalent system consisting of the given equation of the first degree with reference to both of the unknowns, and, as the case may be, either the x-eliminant or the y-eliminant of the system, which contains but one of the given unknowns. The number of solutions of the particular eliminant employed depends upon its degree with reference to the unknown contained in it, and by substituting the solutions of the eliminant separately in the remaining original equation of the first degree, we shall, for each value of the unknown substituted, say x, obtain a corresponding value for the remaining unknown, y. The number of sets of values thus obtained is the same as the degree of the "eliminant" equation. If the eliminant be of the second degree, the number of solutions of the given system for finite values of the unknowns is two; if it be of the third degree, three; etc, |