312. To determine the imaginary roots of the equation if—2i— 5=0. Computing the equation of the squares of the differences from the general formula for the third degree at Art. 279, viz., Z3+ 6pz*+9p^z + 4p>-f27 q*=0, in whichpss—2 and q——5, we have z3—12z5+36z + 643=0. In order to determine the negative roots of this equation, change the alternate signs, or put z = —to, and then change all the signs, converting the equation into is>+12toJ+36to—64 3=0, and seek the positive root, which is found by trial to lie between 5 and 6. Adopting Lagrange's development. Art. 297, this root proves to be 6+, &c, from which we get the converging fractions (see Continued Fractions) Knowing thus an approximate value of to, we know /?= ——. In order now to get the equations (1), p. 385, substitute a-\-j3 V —1 for x in the proposed equation, and form two equations, one with the real terms of the result, the other with the imaginary terms; we shall thus have the equations (1) referred to, viz., <r>—(3/y+2)a—5=0 in which /? is known. Seeking now the greatest common measure of the first members of these equations, stopping the operation at the remainder of the first degree in a, and equating that remainder to zero, we have _ 15 and thus both a and (3 are determined in approximate numbers. 313. There is another method of proceeding for the determination of imaginary roots, somewhat different from the preceding, being independent of the equation of the squares of the differences. It is suggested from the following considerations: Since the quadratic, involving a pair of imaginary conjugate roots, is always of the form i*_2az+o»+/3J=0, every equation into which such roots enter must always be accurately divisible by a quadratic divisor of this form; that is, the proper values of a and ft are such that the remainder of the first degree in x, resulting from the division, must be zero. This furnishes a condition from which those proper values of a and /? may be determined; the condition, namely, that the remainder spoken of. Ax—B, must be equal to zero, independent of particular values of x; and this implies the twofold condition A=0, B = 0, from which a and /?, of which A and B are functions, may be determined. As an example, let the equation proposed be x<+4.r,+6r«+4x+5=0. Dividing the first member by i2 —2ax+a»+0«, we have for quotient at + (4 + 2a)x+ 6+8a+3a» —/3», and for the remainder of the first degree in x (4 + 12a+ 12a*+4a3—4a/3«—4^)r— (a'+^)(6+8a+3a'-/J»)+5, which, being equal to zero whatever be the value of x, furnishes the two equations 4 + 12a+12a»+4a3—iap—4/3*=0 (a*+/3i)(6+8a+3a«—/?»)+5=0. From the first of theje we get /3»=(l+a)« nud this, substituted in the second, gives 4a«+16a»+24as+lGa=:0, two roots of which are 0 and —2; the other two are imaginary, and must, consequently, be rejected as contrary to the hypothesis as to the form of the indeterminate quadratic divisor. The two real values of a, substituted in the expression above for /3s, give fora= 0, /J>=1» .-./3= + l a=_2,/3a=(-l)'.-. 0=-l and, consequently, the component factors of the original quadratic divisor, viz., the factors x—a—(i V—1, x—a+0 V—1. furnish these two pairs of imaginary roots, viz., *=•/—i, i=— V—i, and x=— 2— •v/^1, x=—2+ v^-i. This method, like that before given, is impracticable beyond very narrow limits, because of the high degree to which the final equation in a usually rises. And it is further to be observed of both, and, indeed, of all methods for determining imaginary roots by aid of the real roots of certain numerical equations, that whenever, as is usual, these real roots are obtained only approximately, our results may, under peculiar circumstances, be erroneous. For instance, in the two methods just explained we have two equations, f(a)=0, F(/3)=0, where the coefficients of a in the first are functions of (3, and the coefficients of (3 in the second functions of a; hence, whichever of these symbols be computed approximately, in order to furnish determinate values for the coefficients of the other, these coefficients must vary slightly from the true coefficients; and, consequently, under this slight variation of the coefficients, real roots may become converted into imaginary, and imaginary into real. The terms imaginary and impossible have been thought objectionable when applied to the roots of equations, inasmuch as definite algebraic expressions are always possible for these roots. A specimen of a strictly impossible equation would be the following: 2x—5+ y/x-—7—0, when plus before the sign y/ implies the positive root yj£' — 7. No expression, either real or imaginary, can satisfy the condition or represent a root of this irrational equation. The terms imaginary and impossible, when used, should be understood rather as applying to the solutions of the problem from which the equation is derived than to the expressions for the roots. The number of solutions which the problem admits will ordinarily be expressed by the degree of the equation, but certain suppositions affecting the values or signs of the coefficients may cause some of these solutions to become absurd or impossible, and these will be indicated by the form a-\-b V — 1 for the roots, in which b is not zero. THEORY OF VANISHING FRACTIONS. 314. From the principles established in (Art. 253), we readily derive the following consequences, viz.: Since f(x)=(x—al)(x—a,)(x—a,)(x—at) and _/!(*)«(*—« ){x—ai)(x—a3) \-(x—a,)(x—a^x—a4) 1-, dec., it follows that t filx) 1111 iii—'= .... _i_ 4. _i .... (1) f(x) x—a, 1 x—a3' x—a.3 x—ai' In like manner, for any other equation F(x) = 0, we have F,(.r) 1 1 1 1 ■ 'F(x) = " ' ' ' x—bt+x—b^x—b^x—ft, • • • • (2) Suppose the two equations f(x)=0, F(*)=0, have a root in common, viz., a, = 6i, then, dividing (1) by (2), we have 1111 L 1 |_ f,(x) F(.t) • • • • x—at~x—a, x—a, ' x—a, x—b4 x—63 x—6j x—6| Hence, multiplying numerator and denominator of the second member by x—di, and then substituting for x its value x=aI, we have /(«.) F(a,) from which we learn, that if any two equations have a common root a, and their derived equations be taken, the ratio of the original polynomials, when n is put for x, will be equal to the ratio of the derived polynomials when a is put for x. This property furnishes us with a ready method of determining the value /(l) of a fraction, such as ^—, when both numerator and denominator vanish for E (x) a particular value of x, as, for instance, for x=a. For we shall merely have to replace the polynomials in numerator and denominator by their derived polynomials, and then make the substitution of a for x. If, however, the terms of the new fraction should also vanish for this value of .r, we must treat it as we did the original, and so on, till we arrive at a fraction of which the terms do not vanish for the proposed value of x. The following examples will sufficiently illustrate this method: (1) Required the value of x*—a* when x=a. Ma) 2a (2) Required the value of nx"+1 — (n+l)x"-\-l (1-*)'' when z=l. fi(x) n(n+l)x"—n(n+' 0 This still becomes - for z=l, fi(x) n'(n+l)x°-'—n(n+l)(n—l)x°-« /,(!) «(n+l) the value sought. (3) Required the value of when r=l. (4) Required the value of ELIMINATION. RESOLUTION OF EQUATIONS CONTAINING TWO OR MORE UNKNOWN QUANTITIES OP ANY DEGREE WHATEVER. 315. We have already indicated, at p. 157, the possibility of eliminating one of two unknown quantities from two equations by the method of the common divisor. The general theory of equations which has since been unfolded will afford the means of giving a more full development to this subject. The two given equations may be thus expressed: F(x,y)=0,/(z,y)=:0 (1) They are said to be compatible if they have common values of x and y. This is the case with two equations derived from the same problem, the conditions of which, for the determination of the required quantities, are expressed by the two given equations. Suppose now that one of the common values of y were known, and substituted for y in the two equations (1), the first members of both would become functions of x, and known quantities; the common value of x, corresponding to this value of y, must have the property of every root of an equation pointed out at, Prop. II. of Art. 238; that is to say, if o denote this value of z, each of the equations (1) must be divisible by (x—a); in other words, they must have a common divisor containing r. If, therefore, without kuowing and substituting the value of y, we proceed with the two given equations (1), according to the method for finding the greatest common divisor, until we arrive at a divisor of the first degree with respect to z, and to a remainder independent of x, or containing only y, as this remainder would have been zero if the value of y had occupied its place during the process, the value of y ought to be such as to reduce this remainder to zero. The values of y which will do this are found by putting this last remainder equal to zero, and thus forming what is called the final equation in y only. The values of y which satisfy the final equation are the only compatible values of this unknown in the two given equations (1). The corresponding values of z are found by substituting these values of y successively in the last divisor, which will ordinarily be of the first |