of the Fifth Degree, 357 



is the biquadratic, in p, 



+ { 6{x'') - V^ . r{x) . xi^) }/ 

 + 5 . t{x^) . x{a:^) -P + ^(*^) =0, 

 in which t and x ^^^ defined by 



r[u) = UiU^ + MgMg + Mg?/,j + W^Mg + u^u^, 

 X[u) =2t(m) — SWiMg- 



37. But, if 



we are led to 



Fi(m) =2m2 + if + i3)2MiM2+ \/5 . t(m), 



F2(m) = 2w2 + (i + i4)2MjM2 -Vo, t(m), 



38. Consequently, if 6 can be determined generally, F, t, j^, 

 /?, y, and x will be known. 



39. The algebraic roots, if any exist, of the above trinomial 

 are all included in the expression 



^'"-^V + ^ i^-Xa + i'«x y 



as those of the general quintic are in the corresponding formula 



40. Hence, instead of proceeding by the above transforma- 

 tion, we may, when 6[x) is known, seekX by means of the rela- 

 tions given in art. 36 of Euler's paper, to which I have already* 

 referred. 



40. In a paper which will be printed in the forthcoming 

 (fifteenth) volume of the Manchester Memoirs, I have given the 

 coefficients of the equation in d{x) when the given quintic is of 

 the above trinomial form. Mr. liarley has verified them all, 

 and has devised a striking process of calculation, of which I 

 have availed myself in obtaining some of the above results. For 

 the expression t{x) in the general case I am indebted to Mr. 

 Ilarley, who brought it under my notice in a correspondence to 

 which my paper gave rise. At first I had employed, in parti- 

 cular instances only, an operation equivalent to F, 



4 Pump Court, Temple, E.G., 

 April 26, 1859. 



* Vol. XV. p]). 389-90, where the S which occurs in the symmetric pro- 

 duct equals P*+R. 



