EQUATION 505 of the letters, then the general form of such a function is P + QA ; this has only the two values P + Q P - QA . In the case of 4 letters there exist (as appears above) 3-valued functions : but in the case of o letters there does not exist any 3-valued or 4-valuecl function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the demonstration of the non-existence of a solution of a quin tic equation by radicals. The theory is an extensive and important one, depend ing on the notions of substitutions and of groups, 1 23. Returning to equations, we have the very import ant theorem that, given the value of any unsymmetrical function of the roots, e.y., in the case of a quartic equation, the function ab + cd, it is in general possible to determine rationally the value of any similar function, such as The a priori ground of this theorem may be illustrated by means of a numerical equation. Suppose that the roots of a quartic equation are 1,2, 3, 4, then if it is given that ab + cd=H, this in effect determines a, b to be 1, 2 and c,d to be 3,4 (viz. a = l,b = 2 or a = 2,b= 1, andc = 3, c? = 4 or c = 3, (I = 4) or else a, b to be 3, 4 and c, d to be 1,2; and it therefore in effect determines (a + b) 3 + (c + d) 3 to be = 370, and not any other value; that is, (a + b) 3 + (c + d) 3 , as having a single value, must be determinable rationally. And sve can in the same way account for cases of failure as regards particular equations ; thus, the roots being 1,2, 3, 4 as before, a 2 b = 2 determines a to be =1 and b to be -- 2 , but if the roots had been 1, 2, 4, 16 then a-b = 16 does not uniquely determine a,b but only makes them to be 1,16 or 2,4 respectively. As to the a posteriori proof, assume, fur instance. t } = ab + cd, y, = (a + If + (c + d)
then y } +y. 2 + y y ^yi + % + ^. /3> . t J + tf J-z + 3^3 will be respectively symmetrical functions of the roots of the quartic, and therefore rational and integral functions of the coefficients ; that is, they will be known. Suppose for a moment that t v t. 2 , t 3 are all known, then 1 A substitution is the operation by which we pass from the primi tive arrangement of n letters to any other arrangement of the same letters : for instance, the substitution , ) means that a is to be (abed changed into b, b into c, c into d, d into a. Substitutions may, of course, be reprasented by single letters a.,/3 , . . ; , -/,=!, is the aocd substitution which leaves the letters unaltered. Two or more substi tutions may be compounded together and give rise to a substitution; i.e., performing upon the primitive arrangement first the substitution /3 and then upon the result the substitution o, we have the substitution a)3. Substitutions are not commutative; thus, aB is not in general = #a; but they are associative, a/3.y = a.&y, so that afiy has a determinate meaning. A substitution may be compounded any number of times with itself, and we thus have the powers a 2 , a 3 . . &c. Since the num ber of substitutions is limited, some power a" must be =1, or as this may be expressed every substitution is a root of unity. A group of substitutions is a set such that each two of tl*em compounded together in either order gives a substitution belonging to the set; every group includes the substitution unity, so that we may in general speak of a group l,o, /3, . . (the number of terms is the order of the group). The whole system of the 1.2.3... n substitutions which can be performed upon the n letters is obviously a group: the order of every other group which can be formed out of these substitutions is a submultiple of this number ; but it is not conversely true that a group exists the order of which is any given submultiple of this number. In the case of a determinant the substitutions which give rise to the positive terms form a group the order of which is =il.2.3...w. For any function of the n-letters, the whole series of substitutions which leave the value of the functions unaltered form a group; and thence also the number of values of the function is =1. 2.3... divided by the order of the group. the equations being linear in y v y. 2 , ?/ 3 these can be expressed rationally in terms of the coefficients and of t v t 2 ,t s ; that is, yp Vv 2/3 w iM b e known. But observe further that y l [3 obtained as a function of t v t y / 3 symmetrical as regards t v t.^; it can therefore be expressed as a rational function of LI and of t., + L, t.jt 3 , and thence as a rational function of /t and of t l + t, + t 3 , Va + Va + -/si Ws > but tuese last are symmetrical functions of the roots, and as such they are expressible rationally in terms of the coefficients ; that is, ?/j will be expressed as a rational function of ^ and of the coefficients ; or t l (alone, not / 2 or ? 3 ) being known, //] will be rationally determined. 24. We now consider the question of the algebraical solution of equations, or, more accurately, that of the solution of equations by radicals. In the case of a quadric equation x- - px + q = 0, we can by the assistance of the sign N /( ) or ( )* find an ex pression for x as a two-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby identically satisfied ; it has been found that this expression is
- =J{^ ^^Tg],
and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. Or we may by writing x=-^p + z, reduce the equation to z ~ = l(l j2 ~ 4<?) viz., to an equation of the form z 2 = a ; and in virtue of its being thus reducible we say that the original equation is solvable by radicals. And the question for an equation of any higher order, say of the order n, is, can we by means of radicals (that is by aid of the sign ^/( ) or ( ) , using as many as we please of such signs and with any values of m) find an ^-valued function (or any function) of the coefficients which substituted for x in the equation shall satisfy it identically. It will bs observed that the coefficients p, q . . are not explicitly considered as numbers, but even if they do denote numbers, the question whether a numerical equa tion admits of solution by radicals is wholly unconnected with the before-mentioned theorem of the existence of the n roots of such an equation. It does not even follow that in the case of a numerical equation solvable by radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients. In the formula x = ^(p p lq), substituting for p,q their given numerical values, we obtain for x an expression of the form x = a + j3i /y + 8?, where a, /?, y, 8 are real numbers. This expression substituted for x in the quadric equation would satisfy it identically, and it is thus an algebraical solution ; but there is no obvious a priori reason why N /y + Si should have a value = c + di, where c and d are real numbers calcul able by the extraction of a root or roots of real numbers ; however the case is (what there was no a priori right to expect) that N /y-r 8^ has such a value calculable by means of the radical expressions ,J{ N /y- + S- y}: and hence the algebraical solution of a numerical quadric equation does in every case give the numerical solution. The case of a numerical cubic equation will be considered presently. 25. A cubic equation can be solved by radicals. Taking for greater simplicity the cubic in the reduced form a? + qx - r = 0. and assuming x = a + b, this will be a solu tion if only 3ab = qand u 3 + b 3 = r, equations which give (a 3 b 3 ) 2 = r 2 - T/^q 3 , a quadric equation solvable by radi cals, and giving a 3 - b 3 = Jr- - -^q 3 , a two-valued function of the coefficients : combining this with a 3 + b 3 = r, we have a 3 = (r+ f jr~--^q y ), a two-valued function: we then have a by means of a cube root, viz.,
VIII. 64