ME. CAYLEY’S NOTE ON THE EOEEGOINO PAPEE. 
323 
Let s have any one of the last-mentioned values, then the expression 
n.n — l.n — 2,.,n—x> 1 
n—s 
s.s—1...2.1. — l. — 2... — {x—s) 
which, as regards w, is a rational and integral function of the degree w (the factor w— s, 
which occurs in the numerator and in the denominator being of course omitted), vanishes 
for each of the values w=0, 1, except only for the value w=5, in which case it 
becomes equal to unity. The required formula is thus seen to be 
^\ n.n-l.n-2...n-x 1 1 
n-s s.s-1...2.1.-l.-2...-(a?-s) J’ 
where the summation extends to the several values 5=0, 1, or, what is the same 
thing, it is 
-n^_^{ n.n-\.n-2..n-x -pJ . 
n-s 1.2...s.l.2...(a?-s) J’ 
or changing the sign of it is 
^fn.n + l.n + 2.,n + x 
or, as this may be written. 
ra + s 
i l! U,*]. 
1.2...s.l.2...a?-s J’ 
B- 
[a? + n] vJ (-) 
« l^ + ^i vj 
-[n-1] 
B 
} 
[s] [«— s](w + s) 
or substituting for 5 the values 0, 1, 2, ..oe, the formula is 
R-”— J T?o_ pij ?: P 2 __ 1 
continued up to the term involving E*, which is the theorem in question. 
