so that 

 and 



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



Let us now endeavour to apply similar methods of expression to a system of five 

 arbitrary quantities, or to an equation of the fifth degree. 



13. Let, therefore, x^, x.-^, x^, x^, x^, be the five roots of the equation 



X^ — AX* + BX^ — CX'^ -}- T>X — E = 0, (1) 



and let .r', x", x"', x^^, x^, be the five roots of the same equation when deprived 

 of its second term, or put under the form 



x" + px'^ + ya-'2 + rx' + * = 0, (2) 



a/ + or" + 3f" + x'"" + a;'' = 0, (3) 



^ x,zzx' + ^^, x^ = x"+^h, &c. (4) 



Dividing the equation of the fifth degree 



x"' -af^^p {x"^ - x") + q {af" - x'^) + r {x" -3f)zzO, (5) 



by the linear factor x" — a;', we obtain the biquadratic 



x"* + x'x"^+ {xf^ + p) 3f^+ {a/^ + px' + q)x"-\-x"-irp3/''-\-q3^ + r = 0, (6) 



of which the four roots are x", x'", x^^, x ^. Hence, by the theory of biqua- 

 dratic equations, we may employ the expressions : 



provided that a, j3, 7, 8 are such as to satisfy, independently of x", the condi- 

 tion : 



{oo"~«.y-2{^^f+l-^){x"-c.f-S^l{x"-c.) + ^+y*^i* 



-2(^Y-f 7^8^ + g'^p^) 

 = ar"" + x'x"' + {x"" + p) x'"" + {of' + px' + q) x" -f x'* + px'^ 



-\-qx' -\-r; 

 which decomposes Itself into the four following : 

 — 4a = a;' ; 



+ 6«^-2(^^.f 7^ + 8^) = x'^-j-j9; 



-4«'+4«(/3^ + 7^+8^)-8/378 = a/'+j9a;' + y; [ (9) 



+«^-2a^(|3^+ 7'^+g^).f 8a^7g-f (^* + 7^ + 8^)'^-4(py+7'8^+ 8^/3^) 



(8) 



