234 Sir William R. Hamilton on the Argument of Abel, 



the coefficients of this expression being functions of the second or of lower 

 orders. If we suppress entirely one of the two last radicals, such as aj", without 

 introducing any higher radical a/'', we shall indeed obtain a simplified ex- 

 pression, but cannot thereby represent any root, such as ^ , of the proposed 



biquadratic equation ; for if we could do this, we should then have a system of 

 two expressions for two different roots, x , a; , of the forms 



which would give 



but this last rational function, although six-valued, cannot be put under the 

 form (II. II.), and therefore cannot be equal to any function of the second order, 

 such as &„". Retaining therefore both the radicals, a/", a.^", we have next to 



observe, that if the function ^"' can coincide with the sought function b "' , 

 so as to represent some one root x of the proposed biquadratic equation, it must 



give a system of expressions for all the four roots x , x , s , x^, in some ar- 

 rangement or other, by merely changing the signs of those two radicals of the 

 third order ; namely the following system, 



,// (// ,// /// ,// /// /// 



// // /;/ ;/ /// ;/ //; //( 



iTy = Oo,, — 6,,,a, +Oo,i«2 — Oi.iOi «2 , 



,// ((/ // III II III III 



which four expressions for the four roots conduct to the four following relations, 



