1 for X, substitute be + and repeat the same process with the XV 2 resulting equation in x, : and so on, as far as may be thought necessary. Then 1 1 1 @=b+ 6, +63 +64 + &c. [2] Newton's Method. Determine by experiment a number C, such that one of the roots is > C1, but > C1 +0,1; for æ substitute C1+y, and from the resulting equation in y, obtain an approximate value of y, by neglecting ya and all superior powers: thus an approximate value of x is obtained. If greater accuracy is required, the same process may be repeated. (Bour. 341-9; Lagr. Equ. Num. Chap. 3.) Case 2nd. Suppose the difference between two possible roots to be <l. 1 Determine the inferior limit 7 of the positive roots of the equation of differences, and let k be the least integer > VI: transform the given equation into one whose roots are k times as great, in which equation if the series of natural numbers be successively substituted for X, one root and only one will lie between any two numbers m, and m+ 1, and therefore the k corresponding root of the given equation will lie between ka and A nearer approximation may be made to this root, m+1° by either of the preceding methods. The equation must be cleared of equal roots, before this method can be applied. (Bour. 350-7; L 217—20.) QUADRATIC FACTORS. (43.) Every equation of n dimensions has at least one possible quadratic factor, if every equation of In(n-1) dimensions has one possible factor either of the first or second degree. Hence every equation of 2m dimensions may be decomposed into m quadratic factors. (L. C. 38, 9.) To find the quadratic factors of Q(x)=0 divide Q() by ** + px +q, and continue the division, until the remainder contains only the first power of x, and may be represented by fi(p, q). +f2(p, q)=0, then fi(p, q)=0, and fx (p, q)=0; from which equations q may be eliminated, and the corresponding possible values of p and q determined as above (Art. 34.) will give the quadratic factors required. (Bour. 338, 9.) Further investigation of the commensurable divisors of literal and numerical equations. (Clairaut, Elém. d'Alg. P. 3.) IMPOSSIBLE ROOTS. (44.) If a+B-1 is a root of an equation of which the coefficients are possible quantities, a and ß being possible, a-BJ 1 is also a root. Impossible quantities of all degrees may be reduced to the form A + BV -1, A and B being possible quantities. To find the impossible roots : substitute y +x_ - 1 for x in the given equation, and another equation will be obtained +A+B1=0. Determine all the corresponding values of y and x that satisfy the equations A=0, B=0; let these be az, ag, &c. Bu, B., &c. then az +B-1, a, +.B. / -1, &c. are the required roots. The series of quantities, – 4Bi, – 4B, &c. will all be found amongst the negative roots of the equation of differences : to determine whether – C, a negative root of that equation, is one of these quantities, substitute IV e for x in A and B, and if it be such, the resulting polynomials in y will have a common divisor. If the equation of differences has other negative roots besides the above series, it must also have equal roots, which may be determined by preceding methods. If any series of quantities be successively substituted for x, there can be only as many changes of sign in the results, as the equation has possible roots. The equation p'(x)=0, has at least as many possible roots as Q(x)=0, wanting one. (W. 355–62; G. A. Ch. 2; Bour. 385-92; F. 533—5.) Newton's rule for discovering impossible roots. (W. 363.) APPLICATION OF THE THEORY OF EQUATIONS TO SURDS. ; (45.) To extract the cube root of a + Vb: assume a + vb=x(x + Vy)}, then we=y=+. (a— b)1* ? x must be so assumed that (a? – b)x=c, c being an integer. Eliminating y we obtain which equation must have at least one possible root, in order that we may have a + V bitx (x + Vy)”. To extract the nth root of a + Vb: assume a + Vb=z(x + y)"; ; x must be so assumed that (ao — b)x=?=c", c being an integer. Eliminating y between the equations y=x? — C, and n(n-1) a=% (+ 30"-'y + &c.) 2 we obtain a final equation in a, which must have at least one possible root in order that we may have a + V6+*(x + Vy)". (Bour. 403—5; L. C. 47-9; G. A. Chap. 15.) CUBIC EQUATIONS. (46.) Cardan's Method. Let the equation be reduced to the form 2013 + ax +a=0. 1 Assume y +8=X, and 3o1 > 0, y and > are possible, and a result may be + obtained. In this case the equation has one possible, and two impossible roots. a i and - 2 a 27 be obtained. In this case all the roots are possible. Let the equation be x? - a,&- a=0, which has at least one possible root; let this be a, the other two roots are - ( a + 4a, -34?))) from which their values may be obtained, if the value of determined by approximation. (W. 325—31; E. 734-49; Bour. 406—11; G. A. 58.) Solution of a cubic equation by the method of divisors. (E. 719_-33.) Solution by the theory of symmetrical functions. (Bour. 415; G. A. 52.) a be BIQUADRATIC EQUATIONS. (47.) [1] Descartes': Method. Let the equation be reduced to the form 20* + a222 + aya+a=0; the first side of this equation may be supposed to be the product of two quadratic factors 202 +ex+f, and x? - ex+g. Multiplying these factors together, and equating the coefficients of , g+f-e=an, fg=a; |