EXISTENCE OF A RELATION, &c. 69 



n + 1 functions of x. We substitute in them the limits a 

 and b, and the number of changes of sign lost between these 

 limits is the number of real roots of w, which are to be found 

 in this interval. Consequently we have only to write plus and 

 minus infinity in Sturm's functions, to get the whole number of 

 real roots belonging to the equation. 



The signs of each function, when - is put for x, will of 



course be that of the first term, supposing each function to be 

 arranged in a series of decreasing powers of x. And if the first 

 term of each be positive, the series of signs at the superior limit 

 will be all permanences, and at the inferior all alternations ; that 

 is, all the roots of the equation will be real. 



Hence the reality of all the roots depends on the signs of 

 n+ I terms. But of these, the sign of/(#) is determined at the 



limits + - ; so is that of ~?~ , which is Sturm's first function. 

 Consequently there remain but n 1 terms on the sign of which 



A> A> -r^ 1 



the reality of roots depends. Instead, therefore, of ' 



conditions, there are in reality but n 1 . 



Thus, in the equation of the third degree we find two condi- 

 tions, in that of the fourth, three, and so on, agreeing with what 

 Lagrange found in these cases, and suspected in that of the fifth 

 degree. 



It is not very difficult to see why some of the coefficients of 

 the equations of differences must be so connected with the rest, 

 as not to give any independent condition. 



In order to get an idea of this connection, let us imagine 

 n 1 independent conditions, that is, n 1 functions of the 

 coefficients off(x) 0, which have a definite sign when all the 

 roots are real. These functions are coefficients of the equation 

 of the squares of the differences. Let them all become equal to 

 zero, then we have n 1 relations among the n roots, which 

 will give every one of these roots, except one, in terms of that 

 root. Now the relations b = a, c = a, . . . k = a, will fulfil our 

 n I equations, because these evidently make all the coefficients 

 of the equation of the squares of the differences vanish. Hence 

 these are the relations implied in the n 1 equations we have 

 assumed; and these would, it has just been said, make all 



