518 
Proceedings of Royal Society of Edinburgh. 
and the corresponding values of 
being 
{a ; s, I) ^(a; s,l) 
\n h h/ 'w h h/ 
( 11 11\ /n 2 2\ 
a; a, a) Via ] a ^ a) 
n h hJ h h/ 
( 11 11 \ 
a] s , a\ 
n h hJ 
( 11 n 11 \ 
a] a ^ a] 
n h hJ 
it is required to show that the solution of the set of ?^^-l equations 
n-\-\n+l nn IcJc 11 
a X ax ax +...+ ax = s 
1 1 1 11 
n+ln+l nn Tele 11 
a X + ax +...+ ax ax = s 
2 2 2 2 2 
n+l?i+l 
k k 
1 1 
IS 
I 
x = 
a X 
-}- ax + . . 
. + a X 
. . . -f a X = s 
'/i-t-l 
n+l 
?i+i 
%+i ?i+i 
1\ 
/n+l n-fl\ 
p( a j 
s 
, a 
?i+i 
( a ; s , a j 
Vn+l 
h 
hJ 
\?^^-l h h J 
. . , 
1 
1 , . . . . 
/?i+l »+l w4-l\ * ' 
p( a ; 
a 
,a) 
1 
( a ; « , a.) 
\?«+i 
h 
iJ 
\?j+l Ji h / 
(xiii. 4.) 
Before proceeding, the notation 
,n 
’(«; 
\n 
s, a 
h h 
requires attention. It is meant to be an epitome of Cramer’s rules ; 
the first half of the group of symbols, viz. P( a implying permuta- 
12 3 n 
tion of the under-indices of the product aaa ... a and aggregation 
12 3 n 
of the different products thus obtained, each taken with its proper 
sign : and the second half implying that in every term of this 
aggregate s is to be substituted for a. A modern writer would 
h h 
denote the same thing by 
2 
3 
11 
S 
a 
a . 
. . a 
1 
1 
1 
1 
2 
3 
n 
a 
a . 
. . a 
2 
2 
2 
2 
2 
3 
n 
S 
a 
a . . 
... a 
n 
11 
11 
11 
