The manner in which the chanvsof sign take place is as follows :— When x= —co , or even when it is numerically greater than any negative root, the criterion presents nothing but changes. Alterations of the criterion consist in : 1, Loss of one or more changes at the head of the criterion (showing real roots); 2, loss of changes in even numbers in the middle of the criterion (showing imaginary roots); 3, elevation of changes, or alteration of their place in a direction towards the head of the criterion. This last takes place only when an odd number of derived functions vanishes, the including functions (pre ceding and following) having different signs. So soon as a root has been passed, there is a permanency + + or — — at the head of the criterion • before another root is arrived at, this permanence must have a change, since a change there must then bo at the head to be lost in passing through the root. Hence it follows that between two roots of ox.= 0, there must lie a root of ¢'x=0; and this root is either single, triple, quintuple, etc., but not double, quadruple, Zto.
For example, let cox= xi + 2, 30x-10, ¢),x= 6x2-21x + 15, ¢,x= cx=1.
There are no negative roots, as is obvious from their being nothing but changes among the co-effieienta; if we construct the criteria for x=0, 1, 2, 3, and 4, we find the following results When x=0, the criterion shows four changes ; at x=1, it is inde finite, owing to 0,1=0. But immediately before x=1, ¢,x must, by the preliminary theorem, have the sign contrary to that of or the sign +; consequently, for x=1—h, however small h may be, the criterion must be + + + — +. Two changes of sign are therefore lost in passing from x=0 to x=1, and there are either two real roots between 0 and 1, or two imaginary roots. To try this further, let x=4, the criterion of which is — + + — + ; so that there is one root between 0 and 4, and another between 4 and 1. When x=1 + h, the criterion is + + --+, so that there is no root between 1 and 2. Lastly, thero is one root between 2 and 3, and one between 3 and 4.
The theorem of Fourier, though very convenient in practice, is defective in theory, as requiring an indefinite number of trials. If two roots were very nearly equal, it would require very minute subdivision of the interval in which they are first found to lie, to distinguish them from a pair of imaginary roots. This theorem was not published till 1831, in Fourier's posthumous work, but its author had made his methods known, and among others to the late N. Sturm, a young Oenevese, employed in the bureau of M. de Ferussac, editor of the bulletin which bore ,his name, afterwards a member of the Institute, who died a few years ago. Sturm applied himself to the detection of functions which should stand in the place of ox, sb,x, st),x, &c., in such manner that the criterion formed from them, in the same way as in Fourier'a theorem, should never lose a change of signs except in pass ing through a real root. In this he signally succeeded; and thus, though his theorem presents great practical prolixity of detail, he fur nished a complete solution of the difficulty which had occupied analysts since the time of Descartes. This theorem may be proved
as follows:— Let there be any number of functions v, v„ the last of which is a constant independent of x, and all hut ;the la at are functions of x. Let them be connected together by the equations— v =P,V, — V, V, V„ rave- V, Yr-1= Pr-IYr-1 - Yr.
P,r,, etc., being any functions of x, which do not become infinite when &e., vanish. From this it follows, first, that no two con secutive functions of the set v, v„ &c., can vanish together ; for if v, and v„ for instance, vanished together, the third equation shows that v, would also vanish, the fourth that v, would vanish, and so on ; consequently a given constant, also vanishes, which is absurd. Secondly, when any one after v vanishes, the preceding and following must have different signs ; for gives v= —v,, gives —v„ &e. Now call the signs of v, v„ &e., the criterion, and let v= 0 when x a there being only one root of that value, so that v changes sign lu passing from x=a—it to x =a + h. Since v, does not vanish with v, we have one of the four cases following :— x=a -a t=a z=a4a x=a-h z=a x=a+A V — — 4 V, + 4 +. + If be the derived function of v, only the second or third cases can happen, by the theorem so often used in the preceding part of this article; so that a change of sign will be lost at the head of the criterion for every single root of v=0. Nor will any change of sign ever be gained or lost in any other manner; for suppose gives for Instance, then v, and v, have different signs, and in passing from x sm —h to x=a + h, if each be so small that no root of v, or v, lies between a + h and a wo must have one of the following cases :— x=a-71 x=a x=a+h x=a x=a+h V. + 0 0 V. + In no one of these is any change of sign lost, or anything except a change and a permanence when x= a— h, and a change and a perma nence, in a different order perhaps, when x=a + h. Consequently, if in passing front x=a, the less, to x=b, the greater, it appear that no changes of sign are lost, it is certain that there must have been no real roots of v=0 between x=a and Now, V, being the derived function of v, it remains to find v„ v„ &c. Divide v by which is of one dimension lower, and we have a quotient, say T„ and a remainder Then v= P, v + n or Again, divide by v„ giving a quotient , and a remainder n, : we have then v, + n, or v,= and so on. It appears, then, that being the derived function of v, we must proceed as in finding the greatest common measure of v and v„ only changing the sign of every remainder as fast as it is obtained. Iu order that the last, may be a finite constant, it is requisite that there should be no equal roots. We must then suppose the equal roots to be separated before hand, as in the usual method. In fact, this very process of finding the greatest common measure, with o'r without change of sign in the remainders, will first detect the equal roots, if any. It is important to remark, that at any step multiplication by any positive quantity is allowable, the signs (the only things we have to do with) not being in any case altered by such multiplication.