1905 - 6 .] Dr Muir on the Theory of Alternants. 
365 
/ (x) = (x - a\)(x - - a 3 ) . . . (x - a n ) 
f 1 (x) = 'Z(x-a 2 )(x-a 3 )(x-a 4 ) .... 
f 2 (x) = i:,(a 1 -a 2 ) 2 .(x-a 2 )(x-a i ) .... 
ffx) = 2 («! - « 2 ) 2 ( fl 2 - a 3) 2 ( a 3 - ai) 2 -(x -a A ) ... . 
P 2 
f m (x) —fix) . _ a^(oc - <x 2 ) . . . (x - af ) ’ 
where P stands for the difference-product of , a 2 , . . . , or 
for what Sylvester afterwards denoted by t}(a l , a 2 , ... , a m ) : 
and the problem is professedly to express f m (x) “ par les coefficients 
de/(a:),” but in reality to express it as a series arranged according 
to descending powers of x. 
This is accomplished by partitioning P j(x — a 1 )(x — a 2 ) . . . 
(x - a m ) into an aggregate of fractions having x - a 1 , x - a 2 , . . . 
for denominators,* namely, 
/ _ x m -T £H a i , «2 > • ' • > tt m) = » a 3 > • • • » a m) 
' ' {x- a 1 )(x — a 2 ) . . . (x - a m ) x — a Y 
t\a l , a s , . . . , a m ) + 
x- a 2 
so that the coefficient of x~ r is seen to be 
• £ 2 (^2 ’ ^3 5 ■ • • j ^m) ^2 £ 2 (^1 j a 3 3 • • * J ®m) P 
and therefore to be 
1 
«1 
a 2 . . 
a™ 2 
«; 1 
(-)”- 1 
1 
« 2 
a\ • • 
. . a™ -2 
«r l 
1 
< • • 
• • <~ 2 
c 1 
Multiplying both sides by P and performing the requisite sum- 
mation we find that the coefficient of x~ r in f m (x) — f(x) is 
s Q % ... s m _ 2 s r-l 
s 2 ... S m _ i s r 
or Y r _i say, 
1 • • • ^2m— 3 ^r+m— 2 3 
where is the sum of the q th powers of all the a’s ; in other 
words, that 
* It may be noted in this connection that 
C*(a 1 , <h 3 • • • , On) P </>'(«&) - ( - ) n_ *C 2 («i , «2 3 • • • > a K-l 3 3 • • • 3 On)- 
if cf){x) = (x- %) (x -a 2 ) . . . (x- a n ). 
