DETERMINE THE EORM OF THE ROOTS OF ALGEBRAIC EQUATIONS. 749 
But by the theorem of art. 6, 
Hence 
( 2 ) 
Giving, in this equation, to r any particular system of n values, we shall obtain a system 
of n linear equations for the determination of the n constants C 1? C 2 , .. C n . We shall 
form this system by giving to r the values 1, 2, . . n. 
Now ^representing any particular root selected from the series a„ a 2 , . . a n , multiply 
the above typical equation by a,j~ r , and then, giving to r the successive values 1, 2 . . n, 
form the sum of the equations thus arising. The result may be expressed in the form 
( 3 ) 
the summations with respect to i and r being both from 1 to n inclusive. 
But 
% r u r i otj~ r — 1 -j-a” 
= 1 + . . +«r) 
a? — «r* 
— 05j- — —• 
Uj- CCi 
Now when a ; is not equal to this expression vanishes^ since af=a”= — 1. When, 
however, the fraction ^ — — becomes indeterminate, and its true limiting value is 
seen to be nctf— — n. Hence (3) becomes 
We have thus solved the linear system of equations. We have still to reduce this solu- 
tion by effecting the summation in the second member. 
3/+V A-! 
Now to ccj we may give the form e n , which will represent all the nth. roots of —-l 
in succession if we give to j the series of values 1, 2, .. n. Hence substituting for a, 
5 G 2 
