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



(^ _ a!f — 3»/ {z — a!) — 2e' = ; 

 in which 



a' z= 4a^ + 2rj, 



vi = j3'y = V- + iGa'-*/ — Sae, 



e' = 1 (^'3 + y'3) — 2e2 - t;3 — 12aej7 + 48a^ 17^ - 64a^e. 

 Substituting for a, 7], e, their values, as functions of x', p, q, we find 



V = i (— I2x'* — 12j9a;'^ - 12^0;' +/) ; 

 €' = 315: (72jo:c'* + 72pV* + 72iJ(?a/ + 27?^ + 2p^) ; 

 and eliminating x', by the condition 



«''' -|- px'^ -\- qx' := — r, 



we obtain 



V=i(12r + ;>^); 



e' = 3^(-72p/- + 27?^ + 2/). 



The auxiliary cubic in z becomes therefore 



(^ + ^Py - i (12r +/) (2r + §;>)+ 2V (72;>r - 27?^ - 2f) = ; 



that is 



;23 + 2j9 0^ + (/ — 4r) 5? — 9-^ = ; 



and if its three roots be denoted by z', z", z"\ in an order such that we may 

 write 



z' = \{:d^x" -x'" -x'^y-d-^^^i, 



Z" = l {x' - x" + x'" - x'y = a' + e^ + e'y', 



s'"= 1 (x' - x" - x'" + x'^f =: a' + 0^-p'+ ey, 



we may express the four roots of the biquadratic equation under known forms, 

 by means of the square roots of z', z", z'", as follows : 



x' =+^/^ + iV^" + ^V^", 

 y = + |V^' - 1/^" - iV^", 

 a/" = - ^/^' + \^z" - \V2f", 



x'^=-i^z'-^v'z"-\-iv^z"'. 



