202 Mr. Jerrard on solving Equations of the Fifth Degree. 



" of old like a pool of water," Calneh, Babylon, &c. " in the 

 low lands of the Tigris and Euphrates"; they resorted to the 

 valley of the Nile, fearless of the "flood of Egypt"; and they 

 peopled the vale of Siddim in the plains of the Jordan, " which 

 overflowed all its banks in the time of harvest". 

 [To be continued.] 



XXIV. On certain Transformations connected with the finite 

 Solution of Equations of the Fifth Degree. By G. B. 

 Jerrard, A.B. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 

 COON after I had discovered, by the method of investiga- 

 ^ tion contained in Part ii. of my " Mathematical Re- 

 searches", that the second and third terms of the transformed 

 equation in y might be made equal to zero, independently 

 of one of the indeterminate quantities, Q, involved in the ex- 

 pression for y, and that consequently the general equation of 

 the fifth degree might be reduced to any one of the trinomial 

 equations 



x 5 + D.r + E = 0, 



a* + C^ 2 + E = 0, 



x 5 + ftx 3 + E = 0, 



a.' 6 + A x* + E = 0, 

 I perceived, on comparing these forms with the solvable ones 

 which were known to mathematicians, that had it been pro- 

 posed to arrive by a direct process at the third of these equa- 

 tions, 



x 5 + Bx 3 + E = 0, 



and at De Moivre's solvable form, 



a*> + Bx> + \Wx + E = 0, 



the same analytical difficulty arising from the dimensions of 

 the equations of condition must in both instances have been 

 overcome ; since in the former case it would have been ne- 

 cessary to make 



A = 0, C = 0, D = 0, 



and in the latter, 



A = 0, C = 0, D - \ B 9 = 0. 



This led me to suspect, notwithstanding the almostoverwhelm- 

 ing weight of authority which pressed against the supposition, 

 that there must be some defectiveness in the train of reasoning 

 by which MM. Ruffini and Abel (in following out the views 



