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



L = (/i + «•!/ + \v) (n + Okv + e^Xv) (fi 4- eVv + exv) ; 



that is, by (10), 



L = (/i — ev-\- ^\) (fji — ev + r/u) (fi — ev + h\) ; 



in which, by the same equations, and by (63) and (57), 



M-ev = (-5 + f)(5a'+|/3y8)+(l-^)a(/3^ + y + 8^); 



V 



(91) 

 (92) 



(93) 



(94) 

 (95) 



= (8 + 4^)a; f = x/-15. 

 Thus, L is seen to be composed of three factors, 



L = MjMjMa, 

 Ml := /x — ev + ^v, Mj = /i — ev -\- 7^1/, M3 = /it — 61/ + g'l/, 



of which each is a rational, integral, and homogeneous function, of the third di- 

 mension, of the four quantities a, /3, 7, 8, and, therefore, by (7), of the four 

 roots xf' ■, x'", x^^, x^, of the biquadratic equation (6); or finally, by (4), of the 

 five roots a:,, x^, x^, x^, x^, of the original equation (1) of the fifth degree : be- 

 cause we have 



Xf' = OTa — ^ (or, + X2 + Xj + Xi+ OTj), &c. ; (96) 



or because 



20a = x^ + X3 + x^+ X, — 4x^, 



4/3 = jCj + iBg — or^ — x^, 



4:y = x^—X3 + x^ — x^ 



TcO ^^ iJTrt ^^ "^s """" ^4 "T" 5* 



(97) 



And the first of these three factors of L may be expressed by the following equa- 

 tion: 



100m, = 5m', - f m", ; (98) 



in which, 



M', = 4.x,' - 3a:.* (or, + x, + x, + x,) - 2x, {x^^ + x^ -f x^ + x^) n 



— 1x, {x^3 -f x^x^ -\- 6a;, {x^ + 0^3) (or, + x^ > {m) 



+ 2{-«^2^3(^2+^3)+^4-«^5(^4+^6)} " 3 { J^2^3(-«^4+^5) + ^A(^2 + ^3)} ; J 



and 



