(44) 



Sir William Rowan Hamilton on Equations of the Fifth Degree. 345 



— 1-1-b4-c + d = Au)\ 



— 1+B — c— DZI 4w', 



— 1 — B-|-C — D = Ad?, 



— 1— B — C + D = Aw, 



while w, tt>^ w', w* are the four imaginary fifth roots of unity, we shall have, by 

 the theory of biquadratics already explained, the following identical equation : 



{{x-\- \f - (b^+ c^ + d*)}* - 8bcd («+ 1) — 4 (bV+ c^*+ dV) 



= {(a; + l)^+5r + 40(:c+l) + 180, (45) 



the second member being equivalent to 



«* + 4ar' + 4 V + 4'a; + 4^ 



we find,' therefore, that 



b2 4-c*-|-d2 = — 5; BCD = -5; bV+ cV + dV = — 45; (46) 



and, consequently, 



B*+C* + D*= 115. (47) 



Hence, by (37), the common value oi g and g-', considered as a function of a, 

 e, J/, £, is : 



g- = ^ = 125 (— 25a* + 50a?e — GOa'f} + 31 ae* - lOOai + 4>eri) ; (48) 



and if in this we substitute, for the quantities a, e, i], i, their values (16), or 

 otherwise eliminate those quantities by the relations (15), and attend to the de- 

 finitions (41) of the quantities Hj and H2, we find : . 



H, = ^ (25a;'* + 25^^=' + 25^0;'^ + 25rar' + pg) ; (49) 



and, as was said already, 



H, = 0. (50) 



It is therefore true, of these two quantities h, that they are independent of the 

 root a/ of the proposed equation of the fifth degree, or remain unchanged when 

 that root is changed to another, such as a:", which satisfies the same equation : 

 since it is possible to eliminate a/ from the expression (49) by means of the pro- 



VOL. XIX. 2 Y 



