70 EXISTENCE OF A RELATION, &c. 



the coefficients = 0. Consequently, n 1 independent relations 

 are all we can have, and it follows, that if n 1 independent 

 coefficients of the equation of differences become = 0, all will 

 be so. 



We may take the matter somewhat differently, still using 

 the case of equal roots in f(x) = to show the relations 

 among the coefficients of the equation of the squares of the 

 differences. 



= has m equal roots, there will be ~ - roots 



a 



of the equation of the squares of the differences, or A = 0, equal 

 to zero. Let/j(a?) = get another root equal to these m roots by 



any change in its coefficients, then there will be - - - roots 



in A = equal to zero ; the difference is m. Thus, a single 

 fresh relation among the coefficients of f(x] = makes m coeffi- 

 cients of A = vanish ; for obviously the last coefficients of this 

 equation disappear whenever it gets roots equal to zero. 



We may easily see, too, that the constant function (the last 

 in Sturm's process) is the same as the term independent of u in 

 A = 0. 



The equation A = may, theoretically, be got by eliminating 

 x between the two equations 



(Lagrange, p. 7) ; A will be the term independent of x : put, then, 

 u 0, A reduces itself to its last term, and the process becomes 

 simply that of finding the common measure of f(x) and/' (#), 

 which, abstracting the changes of sign, is exactly Sturm's pro- 

 cess ; hence his term independent of a?, will be " aux signes 

 pres" the constant term in A = 0. 



The development of this idea would undoubtedly lead to the 

 general theory of Sturm's method, and would make it more than 

 a happy artifice, by showing its intimate connection with the 

 equation of the squares of the differences. As is generally the 

 case, the different ways in which the subject may be viewed, 

 ultimately coalesce. 



