1888.] Prof. Anglin on Theorems onainly Alternants. 395 
1 a ... a 
{bed . . 
■ iV 
TT {c d e . . 
■ l)r 
= ^-(a he. . . 1) 
The second identity may be readily deduced from the first by 
the relation 
{ahe . . . = a{h cd . . . l\ + {h ed . . . , 
which for convenience may be written 
We have 
1 a a? . . . , 
(bed . . 
lyl 
\+x 
7t(c d e . 
■ . l)r 
1 a a? . . . (P,.+i - ap,Y he . . . h\ 
~^{ahc . . . 
1 ap^{aprf . . . (aprY^ % - ap,.f-^ 
7r{ahc . . . h) 
by(X) 
1 aiJr . • • {apYf^'^ 
0-0 + 0- . . . +(-l)”-^ . . 
= l^-{ahe . . . 1). 
■^Tr{ahc . . . h)^ 
It may also be established independently in a similar manner to the 
first. Por 
(^h c d . . . {a e d . . . hjy j ] 
= h{ed . . .l\ + {cd . . . l)r+i -a{cd ... l%- {cd .. . Z)^+i 
= (h-a){cd.. .l\ (/5). 
Now consider the alternant 
i J i^r+l? • • • Pr-\-l ’ 
