506 EQUATION a six-valued function of the coefficients ; but then, writing q = } we have, as may be shown, a + b a three-valued function of the coefficients ; and x = a + b is the required solution by radicals. Tt would have been wrong to com plete the solution by writing for then a + b would have been given as a 9-valued function having only 3 of its values roots, and the other G values being- irrelevant. Observe that in this last process we make no use of the equation 3ab = q, in its original form, but use only the derived equation 27 3 /> 3 = 2 3 , implied in, but not implying, the original form. An interesting variation of the solution is to write x = ab(a + b), giving 3 6 3 (a 3 + b*) = r and 3a 3 & 3 = q, or say a i + tf = ~, <1 = l/i ; and consequently i.e., here a 3 , /> 3 are each of them a two valued function, but as the only effect of altering the sign of the quadric radical is to interchange a 3 , 6 3 , they may be regarded as each of them one-valued; a and b are each of them 3-valued (for observe that here only a 3 6 3 , not ab, is given) ; and ab (a + b) thus is in appearance a 9-valued function ; but it can easily be shown that it is (as it ought to be) only 3-valued. In the case of a numerical cubic, even when the co efficients are real, substituting their values in the expres this may depend on an expression of the form where y and 8 are real numbers (it will do so if r" - -~ is a negative number), and then we cannot by the extrac tion of any root or roots of real positive numbers reduce to the form c + d< , c and d real numbers ; hence here the algebraical solution does not give the numerical solution, and we have here the so-called "irreducible case" of a cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all leal ; if the roots arc two imaginary, one real, then contrariwise the quantity under the cube root is real ; and the algebraical solution gives the numerical one. The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals : it consists in effect in reducing the given numerical cubic (not to a cubic of the form z 3 = a, solvable by the extraction of a cube root, but) to a cubic of the form 4z 3 -3x = a, cor responding to the equation 4cos 3 - 3cos0 = cos3$ which serves to determine cos0 when cos 30 is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4.C 3 - 32 = or, is solvable by " trisection" then the general cubic equation is solvable by trisection. 26. A quartic equation is solvable by radicals: and it is to be remarked that the existence of such a solution depends on the existence of 3 valued functions such as ab + cd of the four roots (a, b, c, d) : by what precedes ab + cd is the root of a cubic equation, which equation is solvable by radicals: hence ab + cd can be found by radicals ; and since abed is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a + b, is obtain able rationally ; and then from the values of a + b and ab we may by radicals obtain the value of a or b, that is, an expression for the root of the given quartic equation : the expression ultimately obtained is 4-valued, corresponding to the different values of the several radicals which enter therein, and we have thus the expression by radicals of each of the four roots of the quartic equation. But when the quartic is numerical the same thing happens as in the cubic, and the algebraical solution does not in every case give the numerical one. Tt will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or non-existence of &- valued functions of the five roots (a, b, c,d,e) ; the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters : and by reasoning depending in part upon this theorem, Abel (1824) showed that a general quintic equation is not solvable by radicals; and a fortiori the general equation of any order higher than 5 is not solvable by radicals. 27. The general theory of the solvability of an equa tion by radicals depends fundamentally on Vandermonde s remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expres sion of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the re sulting expression must reduce itself to any one at pleasure of the roots a, b, c . , ; thus in the case of the quadric equa tion, in the expression x = (p+ >/p 2 - 4r?), substituting for p and q their values, and observing that (a + b)" 2 - 4ab = (a-Vf, this becomes x=a + b + J(a- A) 2 }, the value being a or b according as the radical is taken to be + (a- b) or - (a - b). So in the cubic equation .r 3 -px- + qx- r = 0, if the roots are a,b,c, and if o> is used to denote an imaginary cube root of unity, w 2 + w+ 1 = 0, then writing for shortness p = a 4- b + c, L = a + wb + arc, M = a + orb + wc, it is at once seen that LM, L 3 + M 3 , and therefore also (L 3 - M 3 ) 2 are symmetrical functions of the roots, and consequently rational functions of the coefficients : hence is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, = L J or M 3 : taking it = L 3 , the cube root of the expres sion has the three values L,wL,w 2 L ; and LM divided by the same cube root has therefore the values M,u> 2 M,u>M ; whence finally the expression . -j-LM has the three values + M ), that is. these are = a, b, c respectively. If the value M 3 had been taken instead of L 3 , then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x 3 + qx - r = 0, it will readily be seen that the two solutions are identical, and that the function r 1 - if-g 5 under the radical sign must (by aid of the rela tion p = which subsists in this case) reduce itself to (L 3 - M 3 ) 2 ; it is only by each radical being equal to a rational function of the roots that the final expression can become equal to the roots a, b, c respectively. 28. The formula} for the cubic were obtained by La- grange (1770-71) from a different point of view. Upon
examining and comparing the principal known methods for