THE 

 LONDON, EDINBURGH and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



SUPPLEMENT to VOL. XXVI. THIRD SERIES. 



LXXX. Reflections on the Resolution of Algebraic Equations 

 of the Fifth Degree. By G. B. Jerraiid, Examiner in 

 Mathematics and Natural Philosophy at the University of 

 London*. 



COME years ago, while reflecting on the possibility of dis- 

 covering an expression which, consisting of a finite com- 

 bination of radicals and rational functions, would satisfy the 

 general equation of the fifth degree, 



x^ + Ai^''* + A^^ + A3<2?2 + A^x + A5 = 0, 

 I was led to a result which seems to indicate the possibility of 

 assigning, in a subsidiary equation of the third degree with 

 respect to x, 



such finite expressions to p^, p^, p^, that the equation for y 

 may take the known form 



If Pi, P2, Pq admit of having the requisite expressions as- 

 signed to them, it must, in opposition to what has been ad- 

 vanced by Abelfj be possible to discover five finite algebraic 

 expressions capable of satisfying the equation 



x^ + Ai«^ + A^x^ + Aqx'^ + A4X + A5 = 0. 

 In effect, we might obtain, as we shall see in another place, 



^ = ^4 + 73^ + <i^y^ + 9iy^ + %y\ 



or rather the system of equations 



^1 = 5'4 + §'32^1 + .• + 9'o2'A 

 ^2 = fi'4 + isV^ + . . + qQy^\ 



^5 = S'4 + (73S'5+ •• +%y6\ 



* Communicated by the Author. 



f A complete exposition of the argument of Abel is given by Sir Wil- 

 liam R. Hamilton in vol. xviii. part 2 of the Transactions of the Royal 

 Irish Academy. 



PhiU Mag. S. 3. No. 1 76. Suppl. Vol. 26. 2 O 



