The Secular-inequality Determinantive Equation generalized. 215 



Art 2. And much more generally it may be shown, in like 

 manner, that if the successions of signs in the series consisting 

 of the sign + followed by the signs of the principal coefficients 

 in X„ X 2 , . . . X m+n consist of m variations and n continua- 

 tions, the number of real roots of the equation X m+n =0 will 

 be at least as great as the positive value of the difference between 

 m and n. This theorem, moreover, remains true if X x , X 2 , X 8 , &c. 

 be formed from a symmetrical matrix, in which the terms, instead 

 of being linear functions of x, are any odd- degreed rational inte- 

 gral functions of x, or fractional functions of which the numera- 

 tors (when rendered prime to their denominators) are odd-degreed 

 functions of x. My friend M. Borchardt, who has so beautifully 

 effected the decomposition of my formulae for the Sturmian cri- 

 teria of reality into the sums of squares for the secular-inequality 

 form of the equation, may now, if he pleases, tax his ingenuity 

 to effect a similar decomposition for the general case supposed in 

 art. 1 *. 



Art. 3. It is obvious that, in applying the theorem contained 

 in arts. 1 and 2, it is indifferent whether we look to the signs of 



the successive determinants a: ' : &c, or to those of a: 1 ; 



b, c' -ft? 



&c. ; or, more generally, to those ofa + ctO-, , -J ~ ; &c, 



6 being any arbitrary but real quantity. Conversely we obtain 

 the remarkable theorem, that when any homogeneous quadratic 

 function, whose coefficients are linear functions of 6 } is linearly 

 converted by real substitutions into a sum of positive and negative 

 squares, the greatest difference for any value of 6 between the num- 



* So, too, my own more simple method for proving the omni-reality of 

 the roots of the secular-inequality equation given in a previous Number of 

 this Magazine, August 1852, ought to be capable of being extended to the 

 general form in art. 1, i. e. we ought to be able to prove that the equation 

 whose roots are the squares of the roots of X^=0 will have all its coeffi- 

 cients alternately negative and positive. If we take ex. gr. i=2, the equa- 

 tion to the squares of the roots becomes 



(ac-tf) 2 x*-{ay+ccc-2bpy+2(b 2 -ac)(ccy-P 2 )}x+(»y-l3' 2 ¥=z0-, 



and we have to prove that the coefficient of — x in this equation is essen- 

 tially positive when ac — b 2 is positive : this may be shown by various modes 

 of decomposition ; amongst others, by writing the coefficient in question 

 under the form 



—{ (c 2 *+yb 2 —2bc!3) 2 +y 2 (ac-b 2 ) 2 +2(b r ~ c/3) 2 (ac-6 2 ) } • 



In general, if L is essentially positive when L 1? L 2 , ... L; are positive, 

 then, discarding all artifices of calculation, this must be capable of being 

 proved by virtue of an identity of the form 



L=2wr+2w 2 i . Ll+Sto 8 , . L 2 + ... +2m 2 . L^ 



