69 



January 10, 1861. 



Major-General SABINE, R.A., Treasurer and Vice-President, 

 in the Chair. 



The Right Hon. Sir William Erie was admitted into the Society. 

 The following communications were read ; 



I. "On the Equation for the Product of the Differences of all 

 but one of the Roots of a given Equation." By ARTHUR 

 CAYLEY, Esq., F.R.S. Received November 30, 1860. 



(Abstract.) 



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 0v=(a, . . .^f, l) n =a(v a)(v /3). . ., 

 then putting for the moment a= 1, and disregarding numerical factors, 

 \/ Q , the square root of the discriminant, is equal to the product of 

 the differences of the roots, and 0'a is equal to (a /3)(a 7). . ., 

 consequently the product of the differences of the roots, all but a 



is equal to V D -r<p'a, and the expression -j- is the root of an equa- 



a 



tion 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, four, 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," Phil. Trans, t. cl. p. 112 (1860), 

 viz. the last coefficient of the given equation is put equal to zero, so that 

 the given equation breaks up into 0=0 and into an equation of the 

 order n 1 called the reduced equation; and this being so, the re- 

 quired equation breaks up into an equation of the order n I (which, 

 however, is not, as for the equation of differences, that which 

 corresponds to the reduced equation) and into a linear equation ; the 

 VOL. xi. G 



