[ 342 ] 



LIII. Observations on the Theory of Equations of the Fifth 

 Der/ree. By James Cockle, M.A., F.R.A.S., F.C.P.S. ^c* 



[Continued from p. 54.] 



60. T ET the sextic in 6 be decomposed into 



where 



^^ = 0^ + 0^ ^2=O^ + 0fi, ^3 = 03 + 6s> 



8 J — 6^64, 8^ = 6cfiQ, S3 = ^3^5, 



and let 



y = Sj-f 82 + 83; 



then 7 is the root of a 15ic which I shall write 



yH3G,7"+..+G/=0. . . . (k) 



61. But 7 is a rational and symmetric function of three quan- 

 tities 8, each of which is of the form R(^a) ^b)j. that is to say, 

 each of which is a rational and symmetric function of the pair 

 6^ 6\,, Consequently 7 is a rational and integral function, say 

 r[x^, of x^, and is a root of the quintic 



{7-)-(a?5)}{7-?-(a?4)}..{7-)-(«,)}=0, 



the coefficients of which are symmetric functions of x. 



62. The equation (k) is therefore of the form 



(7^ + 0,7^+..+ 0^)3=0, 

 and y is determined by the quintic 



7^0,7'*+.. +65=0, (1) 



the roots of which, expressed in terms of 6, are 



eA+0A+dA=To=r{x,), 



dA+dA+0A=ys=r{^3)> 

 eA+OA+0A=y4=r{x,). 



63. The symbol 8, which occurs in the cubic 



may by known processes be expressed as a rational function of 7. 



64. If the sextic be an Abelian, is the quintic (1) an Abelian ? 

 Now in such case 7 is a rational function of any one root 0. Hence, 

 forming the equations 



* Coratuuuicatcd by the Author. In art. 4.3, line 2, interchange x and 

 «'. 6 and 6'. In art. 55, for (e,"-e,"')^ read (e,"+e,"')'. 



