METHOD OF TSCHIRNHAUSEN FOR SOLVING EQUATIONS. 370. As another application of the theory of elimination, we shall briefly illustrate the principle upon which Tschirnhausen proposed to accomplish the general solution of equations, but which, as observed at Art. 277, was soon found to be of but very limited application, not extending beyond equations of the fourth degree; and, even within this extent, too laborious for general use. The principle consists in connecting with the proposed an auxiliary equation of inferior degree with undetermined coefficients, and of as simple a form as possible consistently with the office it is to perform, but involving, besides the unknown quantity x, a second unknown y. The unknown, common to both equations, is then eliminated according to the method at Art. 315, and a final equation in y thus obtained, of which the coefficients are functions of the undetermined coefficients in the auxiliary equation. The arbitrary quantities, thus entering the coefficients of the final equation in y, are then determined so as to cause certain of these coefficients to vanish; by which means the equation is ultimately reduced to a prescribed form, supposed to be solvable by known methods. 371. As an example, let it be required to reduce the cubic equation and eliminate x from (1) and (2) in the usual way. The remainder arising from dividing the first member of (1) by the first member of (2) is (a'aa'+b-b'—y)x+(a'—a)(b'+y)+c, which, equated to zero, gives (a-a')(b'+y)-c x= ́a'2—aa'+b—b' —y' and this value of x, substituted in the proposed equation, transforms it, after reduction, into the form k=b'3-ab'a'+bb'a'-ca"+(a2—2b)b'2+ (3c-ab)a'b'+aca'2+(b2-2ac)b'—bca'+c2. Hence, in order to reduce (3) to the prescribed form, we must determine the arbitrary quantities a', b' conformably to the conditions h=0, i=0; that is, these quantities must satisfy the equations 3b' — aa'+a2 —26=0 3b'2 —2b' (aa' —a2+2b)+ab+ of which the first is of the first degree with respect to a' and b', and the other of the second degree, so that their values may be determined by a quadratic equation. And these values, or, rather, the expression for them in terms of the given coefficients, being substituted in the preceding expression for k, ren der that symbol known; and thus the required form 372. In a similar manner may the general equation of the fourth degree +ar+bx2+cx+d=0 be transformed into one of the form y'+hy+k=0, which is virtually a quadratic, by eliminating x from the pair of equations x+ax+bx2+cx+d=0, x2+ax+b' +y=0, which elimination will conduct to a final equation in y of the form y+gy3+hy+iy+k=0, from which the second and fourth terms will vanish by the equations of condition g=0, i=0, the first of which will be of the first degree as regards the arbitrary quantities a', b', and the second of the third; both quantities are, therefore, determinable by means of an equation of the third degree, and thence the quantities h, k, which are known functions of them. All this is very laborious, but it really does effect the object proposed thus far; that is, it reduces the solution of equations of the third and fourth degrees to those of inferior degrees; but beyond this point the method fails, as the conditional equations resolve themselves ultimately into a final equation that exceeds in degree that which they are intended to simplify. On this subject we may add that Mr. Jerrard has greatly extended the principle of Tschirnhausen, and has succeeded in reducing the general equation of the fifth degree x+A421+A ̧x3+A ̧x2+Ax+N=0 to the remarkably simple forms 25+ax+b=0 2+ax+b=0 x+ax2+b=0 x+ax+b=0; so that the solution of the general equation of the fifth degree might be considered as accomplished if either of the above forms could be solved in general terms. For a very masterly analysis of Mr. Jerrard's researches, the reader is referred to the paper of Sir W. R. Hamilton in the Report of the sixth meeting of the British Association. METHOD OF LAGRANGE FOR SOLVING EQUATIONS. 373. A remarkable application of the theory of symmetrical functions is that made by Lagrange to the general solution of equations; by that means he solves the general equations of the first four degrees by a uniform process, and one which includes all others that have been proposed for that purpose, the common relation of which to one another is thus made apparent. It consists in employing an auxiliary equation, called a reducing equation, whose root is of the form denoting by x1, X2, r the n roots of the proposed equation, and by a one of the nth roots of unity; and the principle on which it is based is as follows: Let y be the unknown quantity in the reducing equation, and let a1, a2,... a, denoting certain constant quantities; then, if n-1 values of y, and suitable values of the constants a1, ar an, can be found, so that we may have n-1 simple equations, these, together with the equation Now, supposing the constants in the value of y to preserve an invariable order, a1, a, &c., since the number of ways in which the n roots may be combined with them to form the expression a11+at+, &c., is the same as the number of permutations of n things taken all together; therefore, the expres sion for y will have n(n−1)...3.2.1 values, and the equation for determining y will rise to the same number of dimensions, or will be of a degree higher than that of the proposed equation; hence the method will be of no use, unless such values can be assumed for the constants a, a, ... as shall make the solution of the equation in y depend upon that of an equation, at most, of n-1 dimensions. Now this may be done (at least when n does not exceed 4) by taking the nth roots of unity ao, a, a2, a3, ... a-1 for a1, α, y=a°x1+ax2+...+a2-1x,+ a2 I1+1+... -1 ... aa, so that For, in the first place, with this assumption, the reducing equation will contain only powers of y which are multiples of n; for, since a=1, which is the same result as if we had interchanged r, and r1+1, 2 and + &c., so that if y be a root of the reducing equation, a-ty is also a root; therefore, the reducing equation, since it remains unaltered when aty is written for y, contains only powers of y which are multiples of n; if, therefore, we make y"=z, we shall have a reducing equation in z of only 1.2.3... (n−1) dimensions, whose roots will be the different values of z which result from the permutations of the n-1 roots x2, x3,... x among themselves. We shall now have, expanding and reducing, z=y"=uo+u1a+μ‚a2+ ... ... in which uo, U1, U2, u are determinate functions of the roots, which will be invariable for the simultaneous changes of r1 into x1, x2 into +2, &c., since z=(a'y)"; and when their values are known in terms of the coefficients of the proposed equation, we shall immediately know the values of the roots. For let zo, Z1, Z2, ... Zn-1 be the different values of z, when 1, a, ß, y, ... . 2, the roots of y"-1=0, are substituted for a; then, since y=V, we have therefore, adding, and taking account of the properties of the sums of the powers of 1, a, b, y, &c., (Art. 357, [2]), we get Again, multiplying the above system of equations respectively by 1, a"-1, 3-1,... 2n-1, we get ... 0-19 and so on for the rest. Hence, since -p1o, and (—P1)"=Z1=Uo +ni+ the problem is reduced to finding the values of u1, U2, • • • Un—1• 374. When n is a composite number, the above general method admits of simplifications. For let n have a divisor m, so that n=mp, and let a be a root of y-1=0; then, since am=1, am+1=a, am+2=a2, &c., a2m=1, a2m+1=a, &c., we have where uo, u1, &c., are known functions of X1, X2, &c.; and when they are found in terms of the coefficients of the proposed equation, we shall be able to determine immediately the values of X1, X2, &c., as before. To deduce the values of the primitive roots x1, X2, X3, , we must regard separately those which compose each of the quantities X1, X2, &c., as the roots of an equation of p dimensions. Thus, let the roots whose sum is X1 be those of the equation ... XP-X1xP1+LxMx-3+... =0, where L, M, &c., are unknown; then the first member of this equation is a divisor of the first member of the proposed, since all its roots belong to the latter. Hence, effecting the division and equating to zero the coefficients of x31, xp—2, &c., in the remainder, we shall have p equations in X1, L, M, &c., of which the first p-1 will give the values of L, M, &c., in terms of X, by linear equations. It will then remain to solve the equation so formed of p dimensions. Similarly, substituting the value of X, in place of that of X1, we shall have an equation giving the next group of roots T2, Tm+2, &c.; and so on. EXAMPLE I. r-pr2+qr-r=0. Let the roots be a, b, c, and let y=a+ab+ac; ••• z=y3=a3+b3+c3+6abc+3(a2b+b2c+c2a)a+3(a2c+b2a+c2b)a2, Hence, denoting by z1, zg, the values of z when a and a2 are respectively written for a, we have a+b+c=p a+ab+a2c=Vz1 a+ab+ac=√/22; from which we obtain the values of a, b, and c, viz., Since 4 2.2, let a be a root of y2-1=0, so that a2=1; then if where u。=X2+X2, u1=2X1X2, and u。+u1=zo=p2. Hence u1=2(r1+x3)(x2+x,), by interchanging the roots among themselves, will admit the two other values 2(x1+x2)(X3+x1), and 2(x1+x1)(x2+13), and will, therefore, be a root of an equation of the form 3 the coefficients being symmetrical functions of X1, X2, X3, 14, and, consequently, assignable in terms of p, q, r, s. It is easily seen that if we make u1=2q—2u, we shall have an equation in u whose roots are X1X3+X2X4, X1X2+X3X49 X1X4+Loł3 ; and the transformed equation is (Art. 362) u3-qu2+(pr—4s) u — (p2—4q)s—r2=0. Let u' be a root of this equation, then u1=2q-2u'; hence, making a=—1, z1=U。-u1 = p2-2u1 = p2-4q+4u'; Hence 1, r, may be regarded as roots of a quadratic ra-X1x+L=0; dividing the proposed by this, and putting the first term of the remainder equal to zero, we find therefore, 1, 3 are known; and x, x will result from the same formulæ by interchanging X, and X2, or by changing the sign of the radical √ If r be one of the roots, and a be a primitive root of the prime number ʼn (that is, a number whose several powers from 1 to n−1, when divided by n, leave different remainders), it will be proved hereafter that all the roots of this equation may be represented by Let r, ra, ra2, ra3, ... |