4, ibx=9x —7 , Now Descartes's theorem tells us that there may be one negative and two positive roots, and we see that in passing from x= — os to x=0, or through the whole range of negative quantity, there is one change of signs lost ; while in passing from x=0 to x= + a, , or through the whole range of positive quantity, two changes of sign are lost. Fourier's theorem would suggest itself as highly probable to any one who put Descartes's theorem in the preceding form : it is as follows :— When let the signs of ipa, (ha, site. be ascertained, and let this be called the criterion series, or simply the criterion. Then in passing from x=a, the less, to x=b, the greater (greater and less being understood in the algebraical sense), the criterion never acquires changes of sign, though it may lose them. When in changes of sign are lost to the criterion in passing from x= a, the less, to x=b, the greater, it follows that there are either in real roots of the equation lying between a and b, or some number, p, of pairs of imaginary roots, and m-2p real roots lying between a and b. If in be odd, there must be at least one real root lying between a and b. And if no changes of sign be lost in passing from a to b, there is certainly no root lying between a and b. For example, examine the preceding function and its derivatives when x= — 1 and x= +1. In the former case the criterion is — + —+ (three changes), and in the latter + + + + (no changes). Three changes then are lost to the criterion in passing from —1 to +1 : so that there are either three real roots, all lying between —1 and + 1; or one such real root and two imaginary roots. Again, in passing from-1 to 0, one change is lost : there is certainly then one negative root between-1 and 0. The remaining roots are then either both Imaginary, or positive and lying between 0 and 1 : the least consideration of the equation will show that the former is the case.
Fourier's theorem is proved as follows :—changes of sign take place only when quantities become nothing or infinite ; those before us cannot become infinite, and therefore the criterion can never be disturbed except when one or more of the set ipx, &c. vanish. Now when any function 4x, vanishes, say at x=a, its previous sign must have been the contrary of that of its derived function, and its subsequent sign the some; that is, in passing from a—h to a + h, II being very small, 4/x x 4.1* must pass from negative to positive. An algebraical proof may be given of this, but none which in brevity comes near to the following. The function 4/x x tpx is the derived function or differential coefficient of a positive quantity. Now if tpa= 0, must diminish (being positive) from x= a— h to x= a, and increase from x=a to x=a +h. But a differential coefficient is negative when its function diminishes with an increase of the variable, and positive when its function increases with an increase of the variable. Consequently ifix x tpx is negative from x=a—h to x=a, and positive from to x= a +h; as asserted. We now proceed to the proof of the theorem.
1. When x= , the criterion is +—+— &c. or — + — + &c.;
and when x= +so , it is + + + &c. or — — — &c. This follows im mediately from the nature of the functions ipx, &c., in which, when x is numerically great enough, the sign is always governed by that of its highest term. Thus, in some place or places, so many changes are certainly lost as there are units in the dimension of qsx, neither more nor fewer, unless changes be gained and afterwards lost.
2. When x passes through a root of ipx, as many changes are lost as there are roots of ˘x equal to that root. Let there be only one root equal to a, so that ip,a does not vanish. We have then one or other of the following :— The signs in parentheses are those which follow from the theorem above proved. ip,x cannot change its sign in the process, for by hypo thesis it does not vanish when x=a, and we take It so small that there shall be no root of cp,x between a—h and a+ h. At x= a —h, we must have x cpx negative, and at x=a + h we must have it positive, by the theorem ; which gives the signs in parentheses as marked.
Now let there be, say five roots equal to a, or let Oa, Ca, ip,a, 4i,a all vanish, not vanishing. We must have then one or other of the following:— All the signs except those in the lowest line are dictated by the preliminary theorem. Thus thx, in the first case, is negative by hypothesis ; now ep,x is 0%x+ 5, so that 41,x x ip,x must be negative before vanishes, and positive afterwards. Hence continuing negative, Cx must change from positive to negative. Again, ipx x makes a similar change. The least consideration will show that, the signs in the lowest line being given, those in_ all the upper ones must be as written.
3. When intermediate functions vanish, changes of sign arc never gained, but only lost ; and are never lost but in even numbers. Suppose, for instance, that ip,a vanishes, but not (pa nor•tha. We have then one of the four following :— The signs in the middle Hoes are dictated by the preliminary theorem. Next let Ca, Ca, ip,a, ip,a vanish, but not ib,a nor Ca. We have then, by the preliminary theorem, one or other of the four following :— Tho same conclusions will be found from other cases, and we have now examined every way in which the criterion can undergo an altera tion in the order of the signs of which it is composed. And since, the function being of * dimensions, there are altogether s changes, and n only, to lose, it follows that every pair of signs lost by the vanishing of any of the derived functions, in any internal part of the criterion, shows that there inust*be two imaginary roots : for there must be roots only, every mad root must be accompanied by a change lost at the head of the criterion, and every lose of changes which takes place anywhere else diminishes the number which can take place at the bead. Again, since losses other than at the head of the criterion must tako place in oven numbers, it follows that of any odd number of losses, one must have been effected at the head, or must have risen from a real root ; or If not one, some other odd number.