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VII On Algebraic Equations of the Fifth Degree. By 

 James Cockle, Esq., M.A., of Trinity College, Cambridge; 

 Barrister-at-Laxv, of the Middle Temple*. 

 I T FEEL it right to premise that the object of this paper 

 1 is not the finite algebraic solution of any of the higher 

 equations. Nevertheless it must be confessed that remarks 

 of Dr. Peacock and Mr. Jerrard leave the question of the pos- 

 sibility of such solution still open. Mr. Bronwui appears to 

 recrard the point as, at all events, doubtful. The advantages 

 of^such transformations as those discussed below are, that they 

 enable us to confine our attention to a comparatively smiple 

 case, instead of our having to reason on the general equation. 

 They are curious and interesting ; they serve to mark the 

 profTiess of theory and to indicate its direction. The first ot 

 them (the subject of the first article, and due originally to Mr. 

 Jerrard) is here shown to be connected with an important 

 department of analytical geometry. It is capable ot being 

 performed with extraordinary facility— at least ni one sense. 



Let . . » 



u^ + K^iC" + ^4^ + A3M- + k^u + A5=: 



be the general equation of the fifth degree. Also let 



and represent the transformed equation in v by 



v^ -f Bit;'* + Bg?^ + Bgt^^ ^ B,t; + 65= ; 

 then this last equation may be put under the form 



(„+ l;B^)VB',t>3+B'3r^^ + C, (t,+ Ib,) + C,=0, 



where 



B', = B.-4b^; B'3=B3-|b^; 



■^ " Ft J 



now, making 



5 

 1 



C,= B,-iB^; C,= B,-iB^-lBA; 



W; = W + — B,, 



we shall have transformed the given equation in u into the 



following: . . 



w^ + C4t£) + C5=0, (1.) 



provided that x, y, and -r satisfy the conditions 



B'2=^0, and B'3=0. 



* Communicated by the Author. 



