240 
[262 
262. 
ON THE EQUATION OF DIFFERENCES FOE AN EQUATION OF 
ANY ORDER, AND IN PAETICULAE FOE THE EQUATIONS 
OF THE OEDEES TWO, THEEE, FOUE, AND FIVE. 
[From the Philosophical Transactions of the Royal Society of London, vol. cl. (for the 
year 1860), pp. 93—112. Received March 2,—Read March 29, I860.] 
The term, equation of differences, denotes the equation for the squared differences 
of the roots of a given equation ; the equation of differences afforded a means of 
determining the number of real roots, and also limits for the real roots, of a given 
numerical equation, and was upon this account long ago sought for by geometers. 
In the Philosophical Transactions for 1763, Waring gives, but without demonstration 
or indication of the mode of obtaining it, the equation of differences for an equation 
of the fifth order wanting the second term : the result was probably obtained by the 
method of symmetric functions. This method is employed in the Meditationes A lye- 
hraicce (1782), where the equation of differences is given for the equations of the 
third and fourth orders wanting the second terms ; and in p. 85 the before-mentioned 
result for the equation of the fifth order wanting the second term, is reproduced. 
The formulae for obtaining by this method the equation of differences, are fully 
developed by Lagrange in the Traité des Equations Numériques (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 coefficients of the equation of differences, 
qua functions of the differences of the roots of the given equation, are leading