respecting Equations of the Fifth Degree. • 187 



(m— 1) (m) 3 



f, +G, =0, 



must be either identically true, in which case we may substitute for the radical 

 a . , in the proposed function of the m'* order, the expression 



(m) 3y (m) 



a. =z — V 1 . Gj ; 



or at least it must be true as an equation of condition between the remaining 

 radicals, and liable as such to a similar treatment, conducting to an analogous 

 result. 



A more simple and specific example is supplied by the following function of 

 the second order, 



or = — 3^ + t/(c, + /Ci' — c/) + t/ (Ci— /c,'— c/), 



which is not uncommonly proposed as an expression for a root x of the general 

 cubic equation 



x^ -\- a^ x^ -\-a^x-\-a^z=.0, 



c, and Cj being certain rational functions of a,, a^, a^, which were assigned in a 

 former article, and which are such that the cubic equation may be thus written : 



{x-\- -J— 3 c, {x-\- 1-) — 2 c, = 0. 



Putting this function of the second order under the form 



in which the radicals are defined as follows, 



<" = c,4-<, a^'^z=c, — a(, al^ = c^—c.^, 



we easily perceive that it is permitted by these definitions to suppose that the 

 radicals a,", a^' are connected so as to satisfy the following equation of con- 

 dition, 



diy ci.2 ^^ Cg ; 



and even that this supposition must be made, in order to render the proposed 



