46 THE ROYAL SOCIETY OF CANADA 



Tartaglia's solution of the cubic Q(3) = 0, as published by Cardan, 

 is simply the reduction of Q(3) to C(3). In like manner Euler's 

 solution of the quartic Q(4) =0 rests on the explicit determination of 

 3'l^ 3'2S ys^ in terms of §2, Qs, Qi given Q(4) = C(4). 



Abel gave the radical /orw5 which appear in yi, y^, ys, yi in the 

 solution of the quintic but did not express the quantities under the 

 radical signs in terms of g2, Ss, g,i, Çs- He added that he had determined 

 similar forms for the equations of the 7th, 11th, 13th, etc., degrees. 

 (Crelle, V, 336;' Noiiv. An., IV, 536; Memorial Volume, "Corre- 

 spondence," 21-2.) 



The present writer has, in the case of the quintic, expressed 

 3'i. 3'2, 3'3, 3'4 explicitly in terms of go, Çs, §4, Çs and the rational root 

 of the dioristic sextic {Am. Jour, of Math., XIII, 49-56). He has 



also determined yi, yi, yz in terms of q^, qz,....q-; and the common 



root of the triple diacrinic system for the isodyadic septimics. These 

 form the widest class of septimics that hitherto have been solved, 

 including, as they do, the Gaussian cyclotomic septimics as special 

 cases. 



