DETERMINE THE EORM OE THE ROOTS OE ALGEBRAIC EQUATIONS. 743 
6. In determining C 0 the following theorem will be of use, viz. : — 
If r be a positive integer , and & positive and less than r, then 
( 8 ) 
This may be proved as follows : — 
Let i be the greatest integer in a, and let a — i= a'. Then 
r(a)T(r-a)=[a-l]T(a') x [r-a-lJ-^Til-a'). 
But a' being a positive proper fraction, 
and 
Hence 
But 
T(a')T(l-a')=-r-^, 
K J ' sin (a'ir) 
[a— lj=(a— l)(a— 2) . . ( a—i ) 
= (_l)<(i_0)(i_ a _l) . . (1 - a ), 
[r—a—l] , '~ i ~ 1 =(r—a—l)(r—a—2 ) . . (i—a-j- 1) ; 
\r— a— l] r-i-1 [a— 1J=(— l)‘(r— a— 1) . . (i— a-\-l)x(i— a) . . (1 —a) 
r(a)r(r-«)=(-l)‘[r- a -l]'-^. 
sin (a'x) = sin (ax — «V) = ( — 1 )' sin (< 2 w), 
v ' v ' sin («7r) 
as was to be proved. 
Now in the instance before us we have by (5') 
( , m\ „ fn—m\ 
r(n+l)<p(ra) 
where 
C„=— A„ 
<p(n)=- 
Hence, since r(w+l)=[w]“, 
M" 
C n = A, 
r n— 1 + 
?) r (‘-£) 
n + 2 x — 
L n J n 
wherefore 1 — - being a positive quantity, and n a positive integer, we have, by the 
