216 The Secular -inequality Determinantive Equation generalized. 



ber of positive and the number of negative squares has for its limit 

 the number of real roots of in the Discriminant (otherwise called 

 the Determinant) of the given function. The theorem actually 

 demonstrated above teaches only this much, viz. that the maxi- 

 mum difference in number between the two species of squares 

 (which depends only on the value given to 6) cannot exceed the 

 number of real roots in the discriminant ; it admits, however, of 

 an easy proof that this maximum difference is equal to the num- 

 ber of real roots, so that the one number is, in the strict sense 

 of the word, an exact limit to the other. 



Art. 4. I was led to the theorem, as given in art. 1, by having 

 to consider the following curious and important question. 



" Given i linear functions of x, say X„ X 2 , . . . X y to find the 

 i— -1 positive quantities, say /i v fi if . . . //i„i, which shall give the 

 least value to the greatest root, or the greatest value to the least root 

 of the equation 



(V-M, 2 )(x/-(M 2+ ^) 2 )(x^-(^ + l) 2 )...(x i -(£) s ) = o = 



The theorem in art. 1 enables me easily to demonstrate, that 

 if we take X\, X' 2 , X' 3 , . . . X' f identical with 



^r.Xj, */r.x 2 ,... \/r.x f , 



the sign of the square root being selected in each case so that 

 the coefficients of x in X^, X' 2 , . . . X' { shall have all the same 

 sign, then the least value of the greatest root, and the greatest 

 value of the least root, of the given equation will be respectively 

 the greatest and least finite roots of the equation 



X' l l l -0*. 



Al ~xv=~T7"*x'. ; 



the two systems of values of fi lt fi 2 , . . . /&;_! required being the 

 two systems of values of 



Y' Y' Y' Yi 



A l> -^2 Y' > 8 ~" Y' \T ' ' ' A -f-i-"Y' Y~l •••yT> 



■^1 -^-2 — A l A »'-2 — -A-t-3 Aj 



corresponding respectively to these two values of x. 



And it is by means of this solution that the statement of the 

 rule for finding the superior and inferior limits to the real roots 

 of an algebraical equation made in the last August Number of 

 the Magazine, is capable of being converted into the statement 

 contained in the third observation on the same rule in the pre- 

 sent Number. 



7 New Square, Lincoln's Inn, 

 August 7> 1853. 



* The finite roots of this are the same as those of 



v. i L_ i _ 



