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XXXIII. The Algebraical Theory of the Secular-inequality Deter- 

 minantive Equation generalized. By J. J. Sylvester, F.R.S.* 



Art. 1. T ET 



v v rax + a; bx + fi~\ v 



X x -=ax + * Xo= ' ^ X 3 = 



1 ^ Ux + P; cx + ryJ 8 



~ax + a; bx + /3; dx + 8~ 

 bx4-/3; cx + y; ex + e &c, 

 _dx-\-8; ex + e ; fa + <f> _ 



and let the first coefficients of X„ X 2 , X 3 , &c. have all the same 

 sign ; then I say that the roots of any such function as X f will 

 be all real, and will lie respectively in the intervals comprised 

 between + x the successive descending roots of X,_! and — x . 

 [When«=l, c=l,/=l, &c,and&=0, d=0, e=0, &c, X i= =0 

 becomes the well-known secular-inequality equation.] 



Demonstration. For greater simplicity, let all the first coeffi- 

 cients be taken positive, and suppose the theorem proved up to i, 

 it will be true for i + 1. For by a well-known property of sym- 

 metrical determinants, when X £ =0, X*_ u and X, +1 , will have con- 

 trary signs. Let the roots of X*~i be 



h x h 2 . . . A f _!, 

 and the roots of X t , k x k 2 . . . k t _ x k. 



When x=k v which is greater than h v the greatest root of X t _! 

 will be positive; when x=k 21 which lies between the first and 

 second root of X t _j, X**.| will be negative; and so on, X,-_i 

 alternately becoming positive and negative as we pass from root 

 to root of X f . 



Hence X, +1 , which is positive when #=x , becomes negative 

 when x = h v positive again when x = h 2 , and so alternately ; being 

 finally, when x= h., positive or negative, and when x=— x, 

 negative or positive, according as i is even or odd. Hence 

 X t+1 , which changes sign i-\-\ times between -f x and — x, 

 must have all its roots real, and lying severally in the intervals 

 included between + x , the successive roots of X f and — x . 

 Hence if the theorem be true for i— 1 and «, it is true for all 

 numbers above i ; but if we take 



ax+a and \ ax + a ** +/3 l 

 Lbx + f3 or + y, -J 



the latter is (ax + a)(cx + y) — (&r-f/3) 2 , which is positive for 

 x — x , negative for ax + a = 0, and positive for x = — x . Hence 

 the theorem is true for X! and X 2 , and therefore universally. 



In the above demonstration it was supposed that the leading 

 coefficients are all positive ; but the demonstration will be pre- 

 cisely the same, mutatis mutandis, if they are all negative. 

 * Communicated b} r the Author. 



