DETERMINE THE FORM OF THE ROOTS OF ALGEBRAIC EQUATIONS. 751 
So too 
sin jit 
=0 
for all values of j taken from the series 1, 2, . . n except the value n, for which its true 
value is 
n cos nit 
~^r = ± n 
as n is odd or even. 
Hence whenj stands for any of the integers 1, 2, . . n— 2, we have 
When j=n — 1, we have 
,in(^)* r =°. 
X sin 
( m — r \ 
■2\/-V 
the upper or lower sign being taken according as n is odd or even. To the second 
member we may give the form 
— 2^-1 en ^ 1 ( cos ( w + 1 ) ,r — V / — 1 sin(w+l)T)=2^^ye» V_1 
since sin (w +1)^=0, cos + as n is odd or even. 
Thus whenji'=% — 1, we have 
v . fm—r \ _ r n ™5 v _i 
2 ' sm (,~ *)“< = 27=i e ■ 
In the same way when^=?i, we find 
fm—r \ _ r n 
—2V-l en • 
X sin 
It results therefore that, according as ji is less than n— 1, equal to n— 1, or equal to 
n, we shall have 
C,=0, or * or 
3 nit 2V-1 nit 2 \f — \ 
In the general integral (II), art. 7, we shall therefore have 
2C,=- 
=V-i _=v- : 
1 nit\ 2 V — } 
-j* dv v n ~ l log(l — a„_jiry) — ^«’” _1 log(l-a^Y)|, (5) 
where V= 
l + w 
