752 Proceedings of the Royal Irish Academy. 



and, therefore, the product of the squares of the differences is 



that is, 



or, 



that is, 



21 x^^ + 54^071* + Tlp^x^- + 4// = - IP', 

 - IT' - 4^/ 





27 

 - L^' - 4^/ 



27 ' 



But by the equation (1), {x^ + pxi)- = q- ; therefore, 

 I- = - (27^2 ^ 4^3)_ 



From the foregoing results we may deduce the equation of which 

 the roots are the squares of the differences of the roots of the equa- 

 tion (1), with remarkable facility, thus : — Let 



z^ + az^ + hz + c = 



represent the equation of which the three roots are the squares of 

 the three expressions (2), in which expressions, Xi is either one, 

 indifferently, of the three roots of the equation (1). Then the 

 co-efficient a will denote the sum of the squares of the three ex- 

 pressions (2), when the sign of each square is changed ; the co- 

 efficient h will denote the sum of the products, taken two and two, of 

 these same squares, whether the signs of them be changed or not, 

 since the resulting products are the same ; and c will denote the 

 product, with changed sign, of all three of the squares. 



Now, each of these co-efficients has but a single definite value ; so 

 that no quantity iuTolviug Xi (which has a threefold value) can enter 

 any of them, except, indeed, the quantity be of the form 



VI {xi^ +_^j^i)" = ui (- q)", 



where 7i is a whole number, and m a numerical factor ; because only 

 then, and when Xi is entirely absent (in consequence of the terms 

 involving Xi neutralising one another), can the co-efficients a, b, c, 

 have, each of them, single unambiguous values. 



