eliminating f, g, and putting y for ei, we obtain yö + 2a,y+ (as - 4 a)y-a=D. This equation may be solved by either of the preceding methods, if two of its roots are impossible, and the possible value of y, or e obtained; the values of w are $ а. + --;+(--:+ i +(---) (W. 832—3.) The same reducing cubic equation may be obtained by substituting y + x for a, and in the resulting equation putting 4xy+(4x3 + 20,8 + a)y=0. (F. 551-2.) [2] Waring's Method. Let the equation be 2* + a, 3 + 0,2? + a,a + a=0, у* adding +y y.d + to both sides of this equation, 4 or the second side becomes a perfect square, if yö - azy2 + (aq .az - 4a)y- a(a — 4a,) – ai=0. By substituting the possible value of y obtained from this equation, the preceding is reduced to two quadratics, whence the roots may be found. (W.334, 5.) This and the preceding method are applicable only when the proposed equation has two possible, and two impossible, roots. [3] Euler's Method. Let the equation be reduced to the form * + apa? +a,+ a=0. H Assume x=VB.+ VB, + VB3, (1) y' + b2y2 + b2y +b=0. Raising (1) to the 4th power, we have ** +2b,« –8(-6)*x+63 – 6,=0; equating the coefficients of x we obtain If a, is positive, the values of x are x= + VB. + VB, VB, x=-VB-VB-VB3. x = + VB. + VB, +1Bz, x=-VB-VB+ VBz. (E. 773—83.) The same result may be obtained by assuming x=u+y+%, and 8uyx +a=0, (G. A. 58; Bour. 412, 3.) Solution of a biquadratic equation by the method of divisors. (E. 759.) Solution by the theory of symmetrical functions. (Bour. 416, 7; G. A. 53, 5.) THE EQUATION Y" +p=0. a (48.) Properties of the equation y" + p=0. This equation may be reduced to " +1=0, by assuming y=p* x. que m-1-1=0 has only one possible root, 1. If is a root of the equation con-1=0, a is also a root, qe being any integer. And a=a"-n. If a is a root of the equation 2n+1=0, a?r – 1 is also a root. Sum=0, unless mcon, in which case Sim If non?.nps.n,3 ... &c. ni, ng, &c. being prime numbers, the solution of the equation 2" – 1=0 may be made to depend on the solution of the equations 2011 – 1=0, 20-1=0; &c. (Bour. 393—401; L. C. 15—8; G. A. 46, 7.) (49.) Theorem of Fermat. If p is a prime number, and a prime to p, then ap-1-1 p. If a is such a number that no power of a < ap-160p and a, a’, a’,... a®-1 be respectively divided by p, the remainders will all be unequal. En. The values of a for all values of p from 3 to 37 inclusive will be found in the annexed Table. 13 17 19 23 2 6 7 11 14 29 31 37 (G. A. 48, 49; Mem. Acad. Berlin, Ann. 1771.) (50.) Algebraical solution of the equation XP-1=0, p being prime number. a The roots of this equation, a, a, a, a', &c. ad - 1 may a, ao, ao, ao, &c. ar, in which a may have any value corresponding to the value of P in the preceding Table. If qa be substituted for a, the roots become a“, aa, aa, aa, &c. &c. qapa? where can saj -1 cop In this case the roots are evidently the same as before, but årranged in a different order. Let B, BE, B9, BP-e, &c. be the roots of the equation, DUB-1-1=0; assume e=a+Baza +B* &** + ... + B2-%2CP-?; then CP-1=(a + Baca + B2 cup + ... + BP – *)? –, = A + AB+ 4żße + ... + Ap-2BP-2: observing that all indices of a > al-1, and of ß >p-1 may be respectively divided by those quantities, and the quotients neglected. Each of the quantities A, A1, A2, &c. may be reduced to the form B+cla + a® + az? + ... + aan%), in which B and c are known quantities independent of a; hence the values of A, A1, &c. are not altered by substituting successively a“, &c. in the place of a. Let cp-, -1, &c. Cp-2 P-1, be the values of P = 1, when B*, ßi, &c. BP-1, are substituted for B, then |