429 



obtaining by this method the equation of differences, are fully deve- 

 loped by Lagrange in the ' Traite des Equations Numeriques ' (1808) ; 

 and he finds by means of them the equation of differences for the 

 equations of the orders two and three, and for the equation of the 

 fourth order wanting the second term ; and in Note III. he gives, after 

 Waring, the result for the equation of the fifth order wanting the 

 second term. It occurred to me that the equation of differences 

 could be most easily calculated by the following method. The co- 

 efficients of the equation of differences, quti functions of the differences 

 of the roots of the given equation, are leading coefficients of covari- 

 ants, or (to use a shorter expression) they are "Seminvariants"*, 

 that is, each of them is a function of the coefficients which is reduced 

 to zero by one of the two operators which reduce a covariant to zero. 

 In virtue of this property they can be calculated, when their values 

 are known, for the particular case in which one of the coefficients of 

 the given equation is zero. To fix the ideas, let the given equation 

 be (*^0, l) w =0 ; then, when the last coefficient or constant term 

 vanishes, the equation breaks up into v=0 and into an equation of 

 the degree (n 1), which I call the reduced equation; the equation 

 of differences will break up into two equations, one of which is the 

 equation of differences for the reduced equation, the other is the 

 equation for the squares of the roots of the same reduced equation. 

 This hardly requires a proof ; let the roots of the given equation be 

 a, /3, y, S, &c., those of the equation of differences are (a 13) 2 , 

 (a-y) 2 , (-S) 2 , &c., (/3-y) 2 , (/3-S) 2 , (y-3) 2 , &c. ; but inputting 

 the constant term equal to zero, we in effect put one of the roots, 

 say a, equal to zero ; the roots of the equation of differences thus 

 become /3 2 , y 2 , S 2 , &c., (/3-y) 2 , (/3-) 2 , (y-) 2 , &c. The equation 

 for the squares of the roots can be found without the slightest diffi- 

 culty ; hence if the equation of differences for the reduced equation 

 of the order (n I) is known, we can, by combining it with the equation 

 for the squares of the roots, form the equation of differences for the 

 given equation with the constant term put equal to zero, and thence 

 by the above-mentioned property of the Seminvariancy of the co- 

 efficients, find the equation of differences for the given equation. The 

 present memoir shows the application of the process to equations of 

 the orders two, three, four, and five : part of the calculation for the 



* The term " Seminvariant " seems to me preferable to M. Brioschi'sterm Pen- 

 invariant. 



