YojJi^iG— Product of the Squares of the Differences, etc. 749 



reduced to unity by the terms of the equation being each of them 

 divided by Ao. 



Take, for instance, the equation following : — 



P=2x^ - 3^- - 7^ + 5 = 0, 



•one of the roots of which is found to be 2' 5 [Analysis and Solution of 

 Ctibic and Bipiadratic Equations, p. 179). To obtain the remaining 

 roots we proceed thus : — 



2-3-7 + 5(2-0 

 5 5-5 



2-2 d" .-. 2^-- + 2.y - 2 = 0, or a;- + :r - 1 = 0, 

 "which equation gives, for the other two roots of the equation P = 0, 



11/;: 11/- 



^ = -2 + 2^5, -^—^-s^^J 

 so that the product of the squares of the differences is 



Again: Q'-ZPF = Ax^ +Bx+ C =blar-Q'dx-\-M] therefore, 

 £^-4AC 4761-19176 -14415 



AJ' 16 16 



(2) 



•and, changing the sign of this, (1) x 3 = (2). 



The square of the dilference of the two roots Vi, n, of the 

 equation 



Ax-+Bx+ C=0 

 is of course 



^ ^-4^(7_ 69--204x94 _-14415 



{■n-r,y - -j^— - ~ — 5p-' 



A'' 51- . - 14415 



the 



which, multiplied by — -, that is, by — - , gives 



^3 16 16 



same result as that marked (2) above; and which, by the expres- 

 sion (1), is three times the product of the squares of the differences 

 of the roots of the equation P^O, when the sign of this product is 

 changed. 



It may not be superfluous to remark here, that the relation 

 established in this Paper between the product of the squares of the 

 three roots i?i, jRz, i?3, of a cubic equation, 



P=^3^ + ./2r- + .-ii.r + A-0, (1) 



