respecting Equations of the Fifth Degree. 183 



Again, if we wish to isolate any term f of the development above 



assigned for the function of third order h'", which represents a root of the general 

 biquadratic equation, we may employ the formula 



t — .a. . a„ — -^^-^ . Zj .10 . p„ , p, 1 



~*U.o+(-l) 6..o + (-l) 6o.i+(-l) ^.,/' 



in which we have introduced the function 



(/3i"'ri"' /3;"y,"' /3,"' /8,"'v 

 .p2 -p-i .Cfi .Oa 1 



// y,'" /^ /// y/" » /// 7i"'+ys " " '" '" 



SO that, in particular, we have the four expressions 



^0,0 = Oo.o = t (Oo,o+ Oi,o+ Oo.i + 0i.i)» 



//I // (// /// /(/ /// ;// 



' ^1.0 = ^.0 «i = i (*o.o — *.,o +*o.. - ^1,0' 



(// /( /// III III III III 



*o.i=0(,,ia2 = i(Oo,o+*,,o — Oo,,— *i,i), 



/// // /// (/; (// ;// III III 



in which 



,// (// , ;/ III ^11 III III 



Ill II II III II III II III III 



«'l.0= *0,0 — *1.0 «1 + *0.1 «2 — Ol.l «1 «2 . 



//; // // III II III II III III 



Oo,\— Oo,o+0i,o«i — Oo,i«2 — Ol,l«l «2 > 



'" // /; /// /; /(/ /; /// iii - 



In these examples, the truth of the results is obvious; and the general 

 demonstration follows easily from the properties of the roots of unity. 



[5.] We have hitherto made no use of the assumed irreducibility of the 

 irrational function b^"'\ But taking now this property into account, we soon 



VOL. XVIII. 2 c 



