respecting Equations of the Fifth Degree. 251 



dratic, besides linear equations and De Moivre's solvible form ; and therefore 

 ultimately through the extraction of a finite number of radicals, namely, a 

 square-root, a cube-root, three square-roots, a cube-root, a square-root, a cube- 

 root, three square-roots, and a fifth-root. Accordingly, the fallacy of this pro- 

 cess of reduction has been pointed out by the writer of the present paper, in an 

 "Inquiry into the Validity of a Method recently proposed by George B. Jer- 

 RARD, Esq., for transforming and resolving Equations of Elevated Degrees :" 

 undertaken at the request of the British Association for the Advancement of 

 Science, and published in their Sixth Report. But the same Inquiry has con- 

 firmed the adequacy of Mr. Jerrard's method to accomplish an almost equally 

 curious and unexpected transformation, namely, the reduction of the general 

 equation of the fifth degreq to the trinomial form 



jr* -j- D.r -|- E = ; 



and therefore ultimately to this very simple form 



XT' ■\- X — e ; 



in which, however, it is essential to observe that e will in general be imaginary 

 even when the original coefficients are real. If then we make, in this last form, 



X — p(co%6 -^ \/ — Ismff), 

 and 



e =: r (cos u -|" V^ — Isinu), 



we can, by the help of Mr. Jerrard's method, reduce the general equation of the 

 fifth degree, with five arbitrary and imaginary coefficients, to the system of the 

 two following equations, which involve only real quantities : 



p^ cos 50 4" P cos = /• cosw ; 

 /)^ sin 5 -|- /> sin = r sinw ; 



in arriving at which system, the quantities r and v are determined, without ten- 

 tation, by a finite number of rational combinations, and of extractions of square- 

 roots and cube-roots of imaginaries, which can be performed by the help of the 

 usual logarithmic tables ; and p and 6 may afterwards be found from r and v, by 



