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L. Observations on the Theory of Equations of the Fifth Degree. 

 By James Cockle, M.A., F.R.A.S., F.C.P.S., Barrister- 

 at-Laiv, of the Middle Temple^. 



THE " Reflections/' &c. of Mr. Jerrard (Phil. Mag. Suppl. 

 June 1845, pp. 545-574), starting from the solvable form 

 of De IMoivre, throw doubts upon the proposition -which it was the 

 design of the argument of Abel to establish. The following in- 

 vestigations are based upon a form more general than the qua- 

 drinomials of De Moivre or of Euler, which indeed are but par- 

 ticular cases of the quinomial here employed. In order to 

 facilitate the comparison of Mr. Jerrard's results with mine, I 

 have, as far as my own objects and formulae would permit^ 

 adopted his arrangement and notation. 



Introduction. 

 1 . The elimination of x between 



«^ -t- A,«'* + A^ + AgO^^ -f- A4a? -f As = 



x^-{-p^x+p^=y 



and 

 leads to 



where B are known functions of jo and A. 



2. The first two equations lead also to 



*' = ?4 + ?3y + W'^ + q\f + 9oJ^> 

 where q are known functions of;? and A. 



3. The elimination of p between 



^=pt + p^u + pH + p'^w 

 and 



p5_l=0 

 gives 



z'-^ + G'^z'^ + C'3^'2 -h C'^^' + C'5= 0, 



where C are functions of t, u, v and w. But the functions are 

 uusymmetric, and the determination of /, v, v and tv cannot be 

 made to flow from the solution of a biquadratic. 



4. If, however, one of the quantities t, u, v or w vanish, each 

 of the others is expressible in terms of C. And, denoting by^' 

 any value of p, the identical equalities 



?t +fn +fv = if ft +fu + ij^v 



= (f^'t + {ff>^ +f^- = if ft + i/fu + urv 



show that it is immaterial which one v.e suppose to be zero. Let 

 it be w that vanishes, and let its evanescence change s' into is and 

 C into C. 



* Communicated bv the Author. 



