ON THE EXISTENCE OF A RELATION AMONG 

 THE COEFFICIENTS OF THE EQUATION OF 

 THE SQUARES OF THE DIFFERENCES OF 

 THE ROOTS OF AN EQUATION*. 



THE equation of the squares of the differences of the roots 

 gives the means of ascertaining whether any assigned equation 

 has all its roots real ; for if they be so, all the roots of the equa- 

 tion of differences must be real and positive, and consequently, 

 by Descartes' rule of signs, all its coefficients must be alternately 

 positive and negative. Accordingly, Waring applied it to this 

 purpose, and in the Philosophical Transactions for 1763 gave the 

 conditions of the reality of all the roots in equations of the 

 fourth and fifth degree. 



There will be as many conditions as there are coefficients 

 that is, as there are units in the degree of the equation of the 

 squares of the differences ; and therefore, for an equation of the 

 w th order the equation of the squares of the differences will of 



course be of the - th order. Thus, in the third order 



there would be three conditions, in the fourth, six, and so on. 



Lagrange remarked, however, that the number of conditions 

 in these two cases reduced itself to two and three respectively ; 

 and he suggested, that a similar simplification might be possible 

 in the ten conditions of the fifth order. 



Sturm's theorem, which, however different in form, is still in 

 substance intimately connected with the theory of the equation 

 of the squares of the differences, enables us to ascertain the true 

 number of independent conditions. 



By this theorem, we deduce from a given equation f(x) = 

 a series of n functions. These, with the original f(x), make 



* Cambridge Mathematical Journal, No. VI. Vol. I. p. 256, May, 1839. 



