128 Proceedings of Royal Society of Edinburgh. [sess. 
1 2 
a a 
1 2 ' 
n - m - 1 y 
.a a' 
n-m-l n-m n-m + 1 
.Vi 
^ ( c h-a^){a 2 -a 0 ) . . . (a n -a n - 1 ) 
— 2 db Cy-f TO _7 l Cy 1 4- m _ 1 J,_ 1 . . . Cy -n, 
where the % in the first case refers to permutation of 
a 0 , a v . . ., a n , and in the second case to permutation of 
y, y p . . y m . In a couple of lines it is next pointed out that 
the putting of m = 0, m=l, . . . in this suggested identity gives 
1 2 
a i tt 2 * 
n - 1 y 
• * a n-i a n n 
1 
1 2 
P “ U 
n-2 y yi 
a,a n . 
. . a 
1 2 
n-2 n - 1 n 
i 
P 
y-ni 
= c 
'y+\-v\>yi 
Cv, - 
'yi+l-nQy-nj 
then, rather unexpectedly, there is given a mere restatement of 
the identity itself, viz. : — 
“ Generaliter cequatur quotiens propositus 
n-m- L v Vi 
a ,a a 
n-m - 1 n — m n-m + 1 
P 
a ym 
n 
determinanti quod pertinet ad sy sterna quantitatum 
Cy-fm -n Cyj-j -m-n 
Gy+m-n - 1 Qyi+m-n-l 
• • Cy TO + m -% 
Cy_ TO Cy x -n .... Cy m -n- 
This is the last result of the memoir, the few additional lines 
used being merely for the purpose of showing how the deter- 
minant just mentioned may be simplified. The simplification 
consists in leaving out the element a n in forming the C’s of the 
second row from the end, the elements a n , a n _\ in forming the 
C’s of the third row from the end, and so on. The reason in the 
first case is that this will have the same effect as subtracting from 
each element of the row a n times the corresponding element of 
the last row, and the reason in other cases is similar. If C' be 
