1905-6.] Dr Muir on the Theory of Alternants. 
385 
s o 
S 1 
• • S n-1 
S 1 
#2 
• V 
s n-l 
• • S 2n-2 
or Z say, 
he substitutes for it a determinant of the (2 n - 2) th order 
1 
0 
0 . . 
. . 0 
0 
0 
.... 0 
0 
1 
0 . . 
. . 0 
0 
0 
.... 0 
0 
0 
1 . . 
. . 0 
0 
0 
.... 0 
0 
0 
0 . . 
. . £ 0 
S 2 
.... s n-1 
0 
0 
0 . . 
• • *i 
S 2 
S 3 
.... s n 
0 
*0 
% • ; 
• • S n - 3 
«n- 
-2 S n-1 
.... S 2n _ 4 
*0 
s i 
s 2 . . 
• • S n _ 2 
«»- 
-1 S n 
• • • . $2n- 3 
S 1 
S 2 
s 3 . . 
• • S n- 1 
^ n 
S n+ 1 
• • • • S 2n-2 
the first 
n - 2 
row’s do 
not 
contain 
an s. and the 
following contain all the s’s in descending order from right to left, 
beginning with g n _j in the last place of the (n - l) 111 row, with s n 
in the last place of the n th row, and so on. He then multiplies 
every row by a n , and performs the operations which we may 
indicate by 
a r 
cob + 
— ^col, 
a n 
cob 
dcol 
2 + ?2=5col 1; 
a„ 
cob 
- x coL + 
jlJco 1 2 + — — -col^ , 
thus obtaining 
a n 
®n- 1 
&n- 2 
0 
a n 
M'n—l 
0 
0 
a n 
0 
a n* 0 
a n s i + fl„_i s 0 
a n S 0 
a n s x + a n _ x s Q 
a n s 2 + a n _ 1 s 1 + a n _ 2 s 0 
a n s i 
a n S2 + a n _ x s x 
ci n s 3 + a n _ x s 2 + a n _ 2 s x 
PROC. ROY. SOC. EDIN. — YOL. XXYI. 25 
