94 
ME. A. CAYLEY 02s THE EQrATI02s’' OE DIITEEEYCES 
equal to zero ; the roots of the equation of differences thus become /3", /, See., 
equation for the squares of the roots can be found without 
the slightest difficulty; hence if the equation of differences for the reduced equation of 
the order (w— 1) 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 coefficients, 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 equation of the fifth order was kindly peiffonned 
for me by the Eev. E. Haeley. It is to be noticed that the best conrse is to apply the 
method in the first instance to the forms (a, h, ..^v, 1)"=0, without numerical coeffi- 
cients (or, as they may be termed, the deniimer ate forms), and to pass from the results 
so obtained to those which belong to the forms {ci, h, ..y^v, 1)"=0, or standard forms. 
The equation of differences, for (a— (if, Sec., the coefficients of which are seminvariants. 
naturally leads to the consideration of a more general equation having for its roots 
{o(—^y{x—yyf{x—lyf. Sec., the coefficients of which are covariants; and m fact, 
when, as for equations of the orders two, three, and four, all the covariants are known, 
such covariant equation can be at once formed from the equation of differences ; for 
equations of the fifth order, however, where the covariants are not calculated beyond a 
certain degree, only a few of the coefficients of the co variant equation can be thus at 
once formed. At the conclusion of the memoir, I show how the equation of differences 
for an equation of the order n can be obtained by the elimination of a single quantity 
from two equations each of the order w-— 1; and by appljfing to these two equations the 
simplification which I have made in Bezout’s abridged method of elimination, I exhibit 
the equation of differences for the given equation of the order n, in a compendious form 
by means of a determinant; the first-mentioned method is, however, that which is 
best adapted for the actual development of the equation of differences for the equation 
of a given order. 
The equations successively considered are 
{a, h, c Jy, 1)"=0, 
{a, h, c, d yy, 1)^=0, 
{a, b, c, d, e 1)^=0, 
{a, h, c, d, e,fyy, 1)^=0. 
The equation of differences for the quadric, and that for the squares of the roots, are 
considered to be known, and the other results are derived from them : it will be con- 
venient to write down in the first instance the results for the quadric, the cubic, and the 
quartic equations, and then explain the process of obtaining them. 
