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III, On the Equation for the Product of the Differences of all hut one of the Roots of a 
given Equation. By Aethue Cayley, Esq., F.R.S. 
Eeceived November 30, 1860, — Bead January 10, 1861. 
It is easy to see that for an equation of the order n, the product of the differences of 
all but one of the roots will be determined by an equation of the order n, the coefficients 
of which are alternately rational functions of the coefficients of the original equation, 
and rational functions multiplied by the square root of the discriminant. In fact, if the 
equation be (pv=(a, . . .'fv, lY=a(v — ci)(v — (5).. ., then putting for the moment a~\, and 
disregarding numerical factors, □ , the square root of the discriminant, is equal to 
the product of the differences of the roots, and fa is equal to (a — /3)(a— y),.., con- 
sequently the product of the differences of the roots, all but a, is equal to \/ □ 
and the expression ^ is the root of an equation of the order n, the coefficients of which 
are rational functions of the coefficients of the original equation. I propose in the 
present memoir to determine the equation in question for equations of the orders three, 
fom’, and five : the process employed is similar to that in my memoir “ On the Equation 
of Differences for an Equation of any Order, and in particular for Equations of the 
Orders Two, Three, Four, and Five*,” \iz. the last coefficient of the given equation is 
put equal to zero, so that the given equation breaks up into 'y=0 and into an equation 
of the order n — 1 called the reduced equation ; and this being so, the required equation 
breaks up into an equation of the order n—1 (which however is not, as for the equa- 
tion of differences, that which corresponds to the reduced equation) and into a linear 
equation; the equation of the order n—\ is calculated by the method of symmetric 
functions; and combining it with the linear equation, which is known, we have the 
required equation, except as regards the terms involving the last coefficient, which terms 
are found by the consideration that the coefficients of the required equation are semin- 
variants. The solution leads immediately to that of a more general question; for if the 
product of the differences of all the roots except a, of the given equation 
l)’'=a(v — a){v—j3) . . . =0 
(which product is a function of the degree n — 2 in regard to each of the roots /3, y, S . .), 
is multiplied by (oc—uyY~^, the function so obtained will be the root of an equation of 
the order n, the coefficients of which are covariants of the quantic (ffcc, yf, and these 
coefficients can be at once obtained by writing, in the place of the seminvariants of the 
former result, the covariants to which they respectively belong. In the case of the 
* Philosophical Transactions, vol. cl. p. 112 (1860). 
H 
MDCCCLXI. 
