If in equation (1) we put the value of A', we shall have This last equation will apply to the following problems : 5. The annual rent of a freehold estate is p pounds or dollars, to continue forever. What is the present value of the estate, money being worth 5 per cent., compound interest? p Here, as n is infinite, the term, becomes 0, and equation An 20p; that is, the present value of the estate is worth 20 years' rent. 6. The rent of an estate is $3000 a year; what sum could purchase such an estate, money being worth 3 per cent., compound interest? Ans. $100000. 7. What is the present value of an annuity of $350, assigned for 8 years, at 4 per cent.? Ans. $2356.46. 8. A debt due at this time, amounting to $1200, is to be discharged in seven annual and equal payments; what is the amount of these payments, if interest be computed at 4 per cent.? Ans. $200, nearly. 9. The rent of a farm is $250 per year, with a perpetual lease. How much ready money will purchase said farm, money being worth 7 per cent. per annum? Ans. $35713. 10. An annuity of $50 was suffered to remain unpaid for 20 years, and then amounted to $1413.98; what was the rate per cent., at compound interest? N. B. This question is the converse of problem 3, and, of course, the answer must be 3 per cent. But the general equa tion gives us 1413.98 50[(1+r)20—1], r an equation from which it is practically impossible to obtain r, except by successive approximations. GENERAL THEORY OF EQUATIONS. (Art. 156.) In (Art. 101.) we have shown that a quadratic equation, or an equation of the second degree, may be conceived to have arisen from the product of two equations of the first degree. Thus, if x=a, in one equation, and x=b in another equation, we then have x-a=0, and x-b=0; By multiplication, x2-(a+b)x+ab=0. This product presents a quadratic equation, and its two roots. are a and b. If one of the roots be negative, as x-a, and a=b, the resulting quadratic is x2+(ab)x-ab=0. If both roots be negative, then we shall have x2+(a+b)x+ab=0. Now let the pupil observe that the exponent of the highest power of the unknown quantity is 2; and there are two roots. The coefficient of the first power of the unknown quantity is the algebraic sum of the two roots, with their signs changed; and the absolute term, independent of the unknown quantity, is the product of the roots (the sign conforming to the rules of multiplication). When the coefficients and absolute term of a quadratic are not large, and not fractional, we may determine its roots by inspection, by a careful application of these principles. EXAMPLES. Given 2-20x+96=0, to find x. The roots must be 12 and 8, for no other numbers will make -20, signs changed, and product 96. Given y2—6y—55=0 to find y. Given y2+12y-589=0 to find y. Roots 11 and -5. Roots 10 and —4. Roots 7 and —13. Roots 6 and -1. Here it is not to be supposed that we can decide the values of the roots by inspection; the absolute term is too large; but, nevertheless, the equation has two roots. Let the roots be represented by P and Q. Hence P=19 or -31, and Q=-31 or +19, the true roots of the primitive equation; and thus we have another method of resolving quadratics. (Art. 157.) In the same manner we can show that the product of three simple equations produce a cubic equation, or an equation of the third degree. Conversely, then, an equation of the third degree has three roots. The three simple equations, x=a, x=b, x=c,* may be put * Of course, x cannot equal different quantities at one and the same time; and these equations must not be thus understood. in the form of x-a=0, x-b=0, and x-c=0, and the product of these three give (x—a)(x—b)(x—c)=0 ; and by actual multiplication, we have x3—(a+b+c)x2+(ab+ac+bc)x−abc=0. If one of the simple equations be negative, as x=-c, or x+c=0, the product or resulting cubic will be x3—(a+b—c)x2+(ab—ac—bc)x+abc=0. If two of them be negative, as x=-b and x=-c, the resulting cubic will be x3+(b+c—a)x2+(bc—ab—ac)x—abc=0. If all the roots be negative, the resulting cubic will be x2+(a+b+c) x2+(ab+ac+be)x+abc=0. Every cubic equation may be reduced to this form, and conceived to be formed by such a combination of the unknown term and its roots. By inspecting the above equations, we may observe 1st. The first term is the third power of the unknown quantity. 28. The second term is the second power of the unknown quantity, with a coefficient equal to the algebraic sum of the roots, with the contrary sign. 3d. The third term is the first power of the unknown quantity, with a coefficient equal to the sum of all the products which can be made, by taking the roots two by two. 4th. The fourth term is the continued product of all the roots, with the contrary sign. It is easy, then, to form a cubic equation which shall have any three given numbers for its roots. Assuming for the unknown quantity, what will the equation. be which shall have 1, 2 and 3 for its roots? Ans. (1+2+3)x2+(2+3+6)x-6=0; x-6x2+11x=6. Find the equation which shall have 2, 3, and 4 for its roots. Ans. x3-x2-14x+24=0 W Find the equation which shall have -3, -4, and +7 for its x0x2-37x-84=0. roots. Ans. Or x3-37x-84-0. These four general cases of cubic equations may all be represented by the general form. (Art. 158.) When the algebraic sum of three roots is equal to zero, equation (1) takes the form of Equation (1) is a regular cubic, and is not susceptible of a direct solution, by Cardan's rule, until it is transformed into another wanting the second term, thus making it take the form of equation (2). To make this transformation, conceive one of the roots, or x, in equation (1), represented by u+v. By addition, and uniting the second member according to the powers of u, we shall have u3+(3v+p)u2+(3v2+2pv+q)u+(v3+pv2+qv+r)=0, for the transformed equation. But the object was to make such a transformation that the resulting equation should be deprived of its second power; and to effect this, it is obvious that we must make the coefficient of u2 equal zero, or 3v+p=0. Hence, we perceive that if x, in the general equation (1), be put equal to u, there will result an equation in the form of 3' u3+qu+r=0, or the form of equation (2). p As x-u- and if a, b, and c represent the roots of equa3' tion (1), or the values of x, the roots of (2), or values of u will be a+šp, b+ip, |