DETERMINE THE EORM OE THE ROOTS OE ALGEBRAIC EQUATIONS. 753 
Now substitute for the imaginary exponentials their trigonometrical value, and there 
results 
|sn ~ ^ — asV sin v m \'dv 
1 — 2asV cos - + # 2 V 2 
n 
■=i+?r 
Jo 
As x enters this expression only in combination with V, it is suggested to us to repre- 
sent xV by V. If we do this the final theorem will be 
Theorem. If y m represent the m th power of that real root of the equation 
y n —xy n ~ l — 1=0 
which reduces to 1 when x=0, then , supposing m to he between the limits 1 and — n+1, 
the value of y m will he 
_« / • /»*— 1 \ , 7 • mir\ dV 
if ( sm (~v n zrjh*’ 
1 — 2 Y cos - + V 2 
n 
(IV) 
in which 
V= 
1 +v n 
10. Hence too we have the value of a remarkable definite integral, viz. 
„ / . m— 1 
C ( sin 7 r — 
\ » 
Tr . mir\dV , 
-V sin — -t— v m dv 
n J dv 
1 — 2V cos- + Y 2 
n 
under the above conditions and with the above interpretations. 
It may be desirable to verify this result. 
Since 
we shall have 
V=Tj- n , 
l+v n 
dV (n-l)V _nV* 
dv v x 
so that the definite integral is resolvable into 
(n- 1 ) 
i 
17/ * I)?" if • 
V I sin Y sin 
mi r\ 
-*) 
V'dv 
1 — 2V cos - + V 2 
\ n n ) 
dv 
1 — 2 V cos- + V 2 
n 
• (V) 
— n 
