On the Theory of Equations of the Fifth Degree. 355 



5. How, under these circumstances, to express t, u, v and z 

 in terms of C I liav^e elsewhere shown (Phil. Mag. August 1856, 

 p. 124; Diary for 1857, pp. 77, 78). My present object is to 

 inquire whether p^ and p^ can be so determined as to render the 

 equation in y identical with 



^ + Ca^ + Cgr V C40 + C5 = 0, 



and, consequently, solvable. Three questions are here involved : 

 (1) the degree of the final equation, (2) its susceptibility of de- 

 pression, and (3) of solution. The first two are the subjects of 

 this paper. 



Sbction I. 



6. The identity of y and z is expressed by the system 



!>Oa-VPlX«+p,i = t + U-^V, 



"'^ +i' 1^^ +^^2 = *^ + ^^^^ + * V 



x^^->rp^x^^p^-iH^-i^u^i''v, )>.... (a) 

 a!s^+PiXi+pci=i^t + i% + iv, \ 

 oo^-^p^Xt+p^=i'^t + iu +i^v,-J 



where i, i^, ? and ?'* are the unreal fifth roots of unity, and 

 a, /3, y, 8, e represent in an undetermined or arbitrary order the 

 five suffixes 1, 2, 3, 4, 5, by which the values of w may be distin- 

 guished. This system is evolved from 



x^ +p^x+pci^t + i'^u + i^v 

 by substituting i^, i^, i^ and i^ (or 1) for i. The right of the 

 last equation is derived from the third form of art. 4 by changing 

 j^ into i, t into v, and v into t. All these changes are permissible, 

 for j may be any fifth root of unity, and t, v and u are as yet 

 wholly undetermined. 



7. By combining any four of the equations (a) we can ehmi- 

 nate ^, m and f . Replacing x^^ +PiXy +P^ by y^, we thus arrive at 



where fx, are functions of i which have no common factor dififer- 

 ent from 1, and are such that one of them must admit of being 

 equated to zero. 



8. These functions admit of five diflferently derived sets of 

 values, deducible from equations which may be represented by 



V«y« + V^ yp+ ■+ V. 2/. = o> 



and all of which belong to the same system. 

 2B2 



