of Algebraic Equations of the Fifth Degree. 5Q5: 



Now from the preceding number we perceive that 



(pO' = /«' + «' A 



in which «' and a" are the roots of the equation u^— 1=0, and 

 Hence 



Again 



(pO X {fu) = p5 ^^2 _ , ^ _ J^ 



,4„_.,_ A?^ (pjr 



5 (/^ + a V v) 

 and, as might be foreseen, 



The equation (q.) may therefore, without altering the root 

 a; be resolved into the two following equations, 



:r^r\+r',{pt)+r',ipt)^+r',{ptr+r^,{pt)% 1 



X = 7\ + r^'3 (p4 u) + 7-"2 (p4 uf + 7'\ {p^ uf + t\ {p' u)^ ; J ^ "^ 

 ^V ^■'s) • • ^■'o heing rational with respect to q^^ q^, . . q^, /u,, 

 v V, A^2 ; and / becoming r" when ex! is changed into a". 

 33. If now in the equation 



.r = r^ + rgV + rgt;^ + ?'i t^ + Vq-c^, 



which may represent either of the equations (r.), we write i^v 

 instead of?;; taking ^ successively equal to 0, 1, 2, 3, 4, we 

 shall, if 



arrive at the system of equations 



^« = ©0 + e^ + 02 + 63 + 64, 1 



j;^ = 00 + iGj + .202 + '^©3 + '^04» 



^y=0O+ «2ei + ,402 + l03 + ,3e4, V. . (fl.) 



^3 = 00 + «^0, + » ©2 + ''^©a + *''04> 



J7, = 60 + '"^ 01 + 1^ 02 + ''-^ ©3 + * ©4; J 



from which there will result 



