DETERMINE THE EORM OE THE ROOTS OF ALGEBRAIC EQUATIONS. 741 
Hence if we $rite 
r(i+l)C,_ A 
we shall have 
\ n n J \ n J 
r('i3_ 1 D+”'\r('5_“') 
\ n n J \n n J 
T(D + 1) 
<n — 1 ^ m\ „ /D 
Ai(e i0 - e (n+i < d +e (2n * &c. ) 
and therefore 
D + -W---) 
_ \ n nj \n n) A^ 9 
F(D + 1) 1+e” 
iY2=Id+=W5_“') 
S,C,{l+f)-V=S, > ” r(U-/i) " ” 
e™ 
1+1” 
the summation extending from i = 1 to i—n—1. 
Consider next the case in which i— 0. We have, when p is not less than 1, 
g p C 0 =f~' s C Q 
=rw c „ 
=C n <p(n)p p - l e ne . 
But changing in (4)p into^? — 1, and i into n, 
T> 
r(^D+™W5_™\ 
-P-: i V n n / \ n n J 
S _______ 
r(»+i) 
Hence, if we write 
r( n-l+ 
c 0 <pO)- 
r(n+ 1) 
we have for all positive integral values of^>, 
;) r (”) 
= —A n , . 
ppn9 
ec,= 
r /» = l D + »\ r /D_m\ 
A V n n) \n n] ^ 
T(D+1) ’ 
and therefore 
(1— gAg 2 — £ 3 + &c.)C 0 — C 0 + 
(1+f) 'C 0 — C 0 +. 
?UiJ.d + ^\ r — — 
n n \ n n 
( 5 ) 
(S') 
(D + l) 
_+-V, 
n n / \n n ) A n e 
(A n e n0 — A„e 2 ” 9 +A„e 3 ” 0 — &c.) ; 
n ~ 1 D + -\]V — — m 
r(D + i) 
5 f 2 
l+e” 
