533] 
FACTORIALS, AND DERIVATIONS. 
467 
where the symbols ( ) 0 denote sums of products of the differences of 
and a lower letter, 6 factors in each product; and the several sums 
follows : 
For 
a, ¡3, y, 8, e 
a, b, c, d, e)i 
write a 
/3 
7 
8 
a 
b 
c 
d 
e; 
the expression is 
For 
(a — a) 4- (/3 — b) + (y — c) + (8 — d) + (e — e). 
a, /3, y, 8 
m, b, c, d, e, 
a, a 
a, 8 
a, /3 
a, c 
a, y 
a, o? 
/3, /3 
c 
a, 8 
a, e 
/3, y 
6, d 
/3, 8 
b, e 
7> 7 
c, d 
7» $ 
c, e 
8, 8 
d, e 
viz. the expression is 
(a — a) (a — b) + (a — a) (/3 — c) + ... + (8 — d) (8 — e). 
For 
write a, 
a , a 
a, 
b, 
c 
a, 
a, /3 
a, 
b, 
d 
a, 
a, y 
a, 
b, 
e 
a, 
/8, /8 
a, 
c, 
d 
a, 
/3, y 
a, 
C, 
e 
/3, 
/3, /3 
b, 
C, 
d 
a, 
7. 7 
a, 
d, 
e 
/3, 
/3, 7 
b, 
c, 
e 
A 
7> 7 
b, 
d, 
e 
7> 
7’ 7 
c, 
d, 
e 
viz. the expression is 
(a — a) (a - b) (a — c) + ... + (y — c) (y — d) (y — e). 
an upper letter 
being formed as