REPORT ON CERTAIN BRANCHES OF ANALYSIS. 331 



In order to render the preceding propositions more easily in- 

 telligible, we will apply them to two examples. 



Let X = x'* — 4<r^ — 3x+ 23 = 0, and underneath X'", 

 X'", X", X', X, let us write down the signs of the results of the 

 substitution of 0, 1, 2, 3, 10, in the place of x, in conformity 

 with the following scheme : 



X', X, 



- + 



- + 



+ + 



For X = 0, there is a result placed between two similar 

 signs ; there is therefore a pair of imaginary roots correspond- 

 ing to it. Every value of x less than will give results alter- 

 nately + and — , and there is therefore no real negative root. 



For X = I, there is a result placed between two dissimilar 

 signs : there is therefore no pair of imaginary roots corre- 

 sponding ; and since there is no loss of changes of sign in pass- 

 ing from to 1, there is no real root between those values. 



For X = 2, there is a result placed between two dissimilar 

 signs ; there is therefore no pair of imaginary roots correspond- 

 ing, and there is no root between 1 and 2. 



For X = 3, there is a loss of one change of sign, and there is 

 therefore one real root between 2 and 3. 



For jr = 10, there is a loss of one change of signs and all the 

 resulting signs are positive ; there is therefore one real root 

 between 3 and 10. 



The limits of the real roots are thus completely determined, 

 and the substitution of the successive whole numbers, from 3 

 upwards, will show the nearest whole numbers 3 and 4, between 

 which the greatest root is situated. 



Let X = ^6 _ 12 > + eOx'^ + 123 .r^ + 4567 x - 89012 = 



All tlie real roots of the equation are included between the 

 extreme values — 10 and 10. 



