respecting Equations of the Fifth Degree.- 191 



in which 6/', = ij" , = 1, we easily derive, from results obtained before, (by 

 merely changing b^'\„, V'.p *i"',oj *i"',i to •*"i> •*"2> -^3' •^4>) the following ex- 

 pressions for the four separate terms of this development : 



'o ,0 =^ "o ,0 = 5 (•*"! ~r "^2 "T •^'3 "1" •^4)> 



'1 ,0 = "1 ,0 ^1" = ¥ (•*"! "i" "*"> -^3 ■^4)' 



^0 ,1 — = "0 ,1 ^2 -—4 (-^l -^2 "T" -^3 "^'4 j» 



.r,, jTj, o-'j, x^ being some four unequal roots, and therefore all the four roots of 

 the proposed biquadratic equation. 



And when that equation has a root represented in this other way, which also 

 has been already indicated, and in which i/" = 1, 



X = ¥''= ^+ <"+ 0/"= v"+ &/" ar= tr^ tr, 



then each of the two terms of this last development may be separately expressed 

 as follows, 



tr=h:"=^{x,-\-x,), 



t,'''=brar=i(:v,-x,), 



Xy and x^ being some two unequal roots of the same biquadratic equation. 

 A still more simple example is supplied by the quadratic equation, 



x^ -\- ttiX -\- a^-= ] 



for when we represent a root x of this equation as follows, 



a; = 6' = -g-^ + < = C+ C 



we have the following well-known expressions for the two terms t^', t/, as rational 

 and linear functions of the roots x^, x^, 



t(= a; = |(a;, — »,). 



In these examples, the radicals of highest order, namely, 0/ in h\ a" in b", 

 VOL. xviii. 2 D 



