166 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
determine the p unknown quantities, v, v,, ...... At) 
the simple equations, 
ie by means of 
v= V, 
2 =V,, 
@, V2 =V., 
Then equation (15) gives us 
|b 
2 
@ + v,U- &e.) (BR +R,U,+ &e.) = 
Al 
(tees Ue Be (RE Ree Be.) 
Hence, by Prop. X., one of the surds in F, (2), viz. T, is of the form 
of Y in (1) Prop. XII.: which (by hypothesis) is impossible. Hence 
U cannot be a chief subordinate of T; and therefore / (p) involves 
no surds except Y and its subordinate T; the latter being of the 
form T= ./C. Consequently f (p) is of the form (1). 
We may notice a particular form in which f (p) admits of being 
expressed. By means of equation (5), we can reduce (1) to the 
following : 
f@)=A + (A, +B, JC) 4D, JOY 
+ (A, 28/0) Ore Dey” 
a BOM yi 
2 
is AS 
+ (A,—B, /C)(D—D, /C) . 
But, as thus exhibited, /(p) is not in a simple form. 
Cor. 1.—The exact resolution, in algebraical functions, of an equa- 
tion of the fifth degree, is only possible when X admits of being 
broken into rational factors, or when the roots can be reduced to the 
form (1). Hence, in the most general case, the exact resolution, in 
algebraical functions, of an equation of the fifth degree, cannot in the 
nature of things be effected. A fortiori, the exact resolution, in 
algebraical functions, of equations of degrees above the fifth, cannot, 
in the most general case, be effected. 
