HI 



STURM'S THEOREM. 



STURM'S THEOREM. 



858 



theorem arising from the possible existence of imaginary roots, but 

 j>iM|>"sed to divide those last roots themselves into two classes 

 corresponding to the true and false (or positive and negative) of the 

 real roots. The next step was made by De Gua (1741), who showed 

 that the roots of an algebraical equation <J> x= are never all real, 

 unless the roots of the derived equations <t> z=0, <f>" x0, &c. be also 

 all real ; <)>' x, <p* x, &c. being the derived functions, or differential 

 coefficients, of x. He also showed how to determine the conditions 

 of the reality of all the roots. (Lagrange, ' Kes. des Equ. Numer.', 

 note viiii. ; Peacock, ' Report,' ic., p. 327.) 



artes's theorem would be perfect if the roots of equations were 

 always real. For example, take .<' 13x + 40 = 0. If the roots be real, 

 they are both positive ; write x + 6 for x [INVOLUTION AND EVOLU- 

 TION], and we have x" x 2 = 0, of which the roots are less by 6 than 

 those of the former equation. But in the second equation, one root is 

 negative and one positive ; consequently the roots of the first equation 

 are one greater and one less than 6. In the same manner a more com- 

 plicated case might be treated. 



The theorem of Descartes, and the notion derived from it, that the 

 order of signs of coefficients regulates the signs of the roots ; with the 

 step made by De Gua, and the notion derived from it, namely, that 

 tin- derived functions must be consulted upon the question whether 

 the roots of an equation be real or not ; and the common theory of 

 equal roots, namely, that when ^ x=0 has m equal roots, m 1 of its 

 derived functions (neither more nor fewer) vanish at the same time, or 

 with the same root, were the hints on which FOURIER [Bioo. Dry.] 

 was able to make an advance upon his predecessors. The coefficients 

 of the equation are themselves nothing but the dii~id(d derived functions 

 on the supposition that z=0. Thus, if <^c = 3x* - 7-z* + Hz + 4, we 

 have 



'0 *"0 <t>'"0 



,0=4, E2_ Uj j.-7 f i.r.a=*- 



Let <fac, ,,ar, $jc, Ac. be the function in question, and its divided 

 derived functions. If we make x great enough and negative (say 

 infinite and negative), the signs of these functions arc all alternate, 

 that is, the series yields nothing but changes of sign in passing 

 from term to term. But if we make x great enough and positive (say 

 infinite and positive), the series yields nothing but permanences of 

 sign. Thus, in the preceding expression we have 



+ 

 + 



nothing two 



but changes. changes. 

 c=3jr J -7* 5 + ll*+4, <t>x=9x' 



no 



changes. 



Now Descartes's theorem tells us that there may be one negative and 

 loo positive roots, and we see that in passing from x= oo to j:=0, 

 nr through the whole range of negative quantity, there is one change 

 of signs lost ; while in passing from x=0 to .z= + , or through the 

 whole range of positive quantity, tiro 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 a;=a,lct the signs of <f>a, <f t a, fya, Ac. be ascertained, and let 

 this be called the criterion series, or simply the criterion. Then in 

 passing from x=a, the less, to a: =6, the greater (greater and less being 

 understood in the algebraical sense), the criterion never acquires 

 changes of sign, though it may lose them. When m changes of sign 

 are lost to the criterion in passing from xa, the less, to a; =6, the 

 greater, it follows that there are either m real roots of the equation 

 lying between a and 6, or some number, p, of pairs of imaginary roots, 

 and m2p real roots lying between a and 6. If m be odd, there must 

 be at least one real root lying between a and 6. And if no changes of 

 sign be lost in passing from a to ft, there is certainly no root lying 

 between a and b. For example, examine the preceding function and 

 its derivatives whenz= 1 and *=-rl. In the former case the 

 criterion is -I + (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 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 

 rnmmt become infinite, and therefore the criterion can never be 

 disturbed except when one or more of the set qac, <p L x, &c. vanish. 

 Now when any function i(sr, vanishes, say at x = a, its previous sign 

 must have been the contrary of that of its derived function, and its 

 subsequent sign the same; that is, in passing from a A to a + k, 

 A being very small, ty'x x ifr must pass from negative to positive. An 

 algebraical proof may be given of this, but none which in brevity 

 conies near to the following. The function tyx x ^x is the derived 



function or differential coefficient of \(tyx)-, a positive quantity. Now 

 if i^a=0, ^x)- 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 $'xx<fix is negative from x = ah to x=a, 

 and positive from x aio x=a + h; as asserted. We now proceed to 

 the proof of the theorem. 



1. When .r= w,the criterion is + H &c. or -I h &c. ; 



and when x= + oo , it is + + + &c. or &c. This follows im- 

 mediately from the nature of the functions <p;r, <p,a:, &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 <(>x, 

 neither more nor fewer, \inless changes be gained and afterwards lost. 



2. When x passes through a root of $>x, as many changes are lost as 

 there are roots of <px equal to that root. Let there be only one root 

 equal to a, so that Q^a does not vanish. We have then one or other of 

 the following : 



a h x = a 



<t>x 



One change lost. 



x = a h x=a x 



<& (-) 

 Pi* + + 



One change lost. 



The signs in parentheses are those which follow from the theorem 

 above proved. <t> t x cannot change its sign in the process, for by hypo- 

 thesis it does not vanish when x=a, and we take h so small that there 

 shall be no root of <t> t x between a A and a + h. At x= a h, we must 

 have <f> t x x ^.r negative, and at .r = a 4 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 <a, <[, <t>. 2 n, tp-fl, 

 <t> t a all vanish, <f> 5 a not vanishing. We must have then one or other of 

 the following : 



<t>x 



f.x 



x?=a h x = a 

 

 

 











xa^h 



Five changes lost. 



x=a h x a x = a 



+ 



+ + 



+ 



+ + 



+ 



+ . + + 



Five changes lost. 



All the signs except those in the lowest line are dictated by the 

 preliminary theorem. Thus <f>..r, in the first case, is negative by 

 hypothesis ; now Q^x is <(>',^-H 5, so that $> S .T x <f> t c must be negative 

 before <t> t x vanishes, and positive afterwards. Hence tp^x continuing 

 negative, tp t x must change from positive to negative. Again, <f> t x x 

 4>.,z 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 are never 

 gained, but only lost ; and are never lost but in even numbers. 

 Suppose, for instance, that 0,a vanishes, but not <f>a, nor </> 2 a. We 

 have then one of the four following : 



3T rt-f-A x^ n h ;r = rt x tt-i-h 



f.x + 



<t>. r c 



Ttoo changes lost. 

 <fx + + + 



^x - + 



M + + + 



Two changes lost. 



<t>,x 







No change lost.* 



No change lost* 



The signs in the middle lines are dictated by the preliminary 

 theorem. Next let $>,, 4> <J, <f> 8 o, <t>,a vanish, but not if>,a nor <,a. 

 We have then, by the preliminary theorem, one or other of the four 

 following : 



= a h x = a 



<f,x 



- 



+ - 



- 



+ 



- 



Four changes lost. 



<t>,x 



<t>,x + 



v - 











Four changes lost. 



r = a-h x = a ar=rt-J 



- 



+ 







+ 



Four changes lost. 



+ + 



- + 



+ + + 



Four changes lost. 



* Observe thnt In these caces a change is removed to a higher place in the 

 series, nearer to the head of the criterion. 



