362 Sir William Rowan Hamilton on Equations of the Fifth Degree. 



in wliich, Mj and Mg are formed from M„ as in the twenty-sixth article ; be- 

 cause the second factor (119) may be formed from the first, by interchanging x., 

 and x^, and multiplying by — 6"-; and the third factor may be formed from the 

 second, by interchanging x^ and x-^, and multiplying again by — 0'. If then we 

 multiply the three expressions (135) (143) (144) for the three factors (119) 

 together, and divide by three, we find : 



18\/h, = — 5^(0 — e^)7^M,M3M3; (145) 



-sr denoting here the product (116) of the six differences of the four roots x^ . . . 

 x^. The expression (101) for H4 itself is therefore reproduced under the form : 



H,= - 2-^3-'5'«w^m,^m/m3^; (146) 



and the conclusions of former articles are thus confirmed anew, by a method 

 which is entirely different, in its conception and in its processes of calculation, 

 from those which were employed before. 



33. It may not, however, be useless to calculate, for some particular equa- 

 tion of the fifth degree, the numerical values of some of the most important 

 quantities above considered, and so to illustrate and exemplify some of the chief 

 formulae already established. Consider therefore the equation : 



x'' - Hx'' -\- Ax = Q ; (147) 



of which the roots may be arranged in the order : 



X, = 2, or, = 1, X3 = 0, x,=z — 1, x^ = —2; (148) 



and may (because their sum is zero) be also written thus : 



x' = 2, x" = 1, x'" = 0, x'" =-h x" =-2. (149) 



Employing the notation (32), in combination with (22) or with (105), we have 

 now : 



T,^, = (2 + «,*-ft,^-2«,)''; 1 



T3,,,=r(2 + «.3-2«,*-«.>^; ^ ^j5Q^ 



T,,,3 = (2-«.''-2«.^ + «,7'; 

 T3,3, - (2 - 2«.^ -u?^ u>)\ 

 But ftt* 1= 1 ; therefore. 



