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VI. On the Equation of Differences for an Equatmi of any Order, and in ^particular for 
the Equations of the Ordei's Tivo, Three, Four, and Five. By Aethue Cayley, 
Esq., F.R.S. 
Eeceived March 2, — Bead March 29, 1860. 
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 equa- 
tion, and was upon this account long ago sought for by geometers. In the Philosophical 
Transactions for 1763, Waeiyg gives, but without demonstration or indication of the 
mode of obtaining it, the equation of differences for an equation of the fifth order want- 
ing the second term : the result was probably obtained by the method of symmetric 
functions. This method is employed in the ‘ Meditationes Algebraicse ’ (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 formulse for obtaining by 
this method the equation of differences, are fully developed by Lageange 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 Waeing, 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 coefficients of covariants, or (to use a shorter 
expression) they are “ Seminvariants*,” that is, each of them is a function of the coeffi- 
cients which is reduced to zero by one of the two operators which reduce an invariant 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 (*X^? 1)"=0 ; then, when the last coefficient or 
constant term vanishes, the equation breaks up into '^=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 equa- 
tion, 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, 5, &c., those 
of the equation of differences are (a— (3)^, (a~yf, [a — lf, &c.. ((3 — yf, ((3 — ^)^, (y~^)% 
&c. ; but in putting the constant term equal to zero, we in effect put one of the roots, say a, 
* The term “ Seminvariant ” seems to me preferable to M. Beioschi’s term “ Peninvariaut.” 
MDCCCLX. 0 
