512 
Proceedings of the Royal Society of Edinburgh. [Sess. 
or not with the y set The illustrative example taken is the case where 
n = 2 , and the mode of stating it is that 
ftyy d// g 
1 \ dx, h dXj 
dXj dxj\dy a diyp 
dx i 0ajA [ 
typtyJ 1 
-1 
or 0 
according as a , f3 —p , q or not, it being understood that i , j is in succession 
1 
: 1,3; 
2,3; 
n 
n 
n - 1 , n. 
Donkin, W. F. (1854, February). 
[Demonstration of a theorem of Jacobi’s relative to functional 
determinants. Cambridge and Dub. Math. Journ., ix. pp. 
161-163.] 
The theorem or identity in question is that of the year 1844. The 
functions being u x , u 2 , ... , u n and the independent variables y 0 (jq j • • • } j 
Donkin says that the functional determinant may be represented by 
A. 
A 
A 
dx l 
0^o 
dx n 
0o 
A 
A 
cx v 
d x 2 
dx n 
0?t 
d n 
A 
dx l 
dx 2 
dx n 
it being understood that each symbol of differentiation is operative only 
upon that one of the functions which has the same suffix as the upper d of 
the symbol. As a consequence of this he considers that the non-zero 
member of the identity sought to be established would be 
0 
0 
0 
dx l 
dx 2 
dx n 
a. 
dx Y 
dx 2 
dx n 
d n 
0n 
dx x 
dx 2 
dx n 
(A) 
where the upper 0’s of the first row being now without a suffix are 
supposed to be no longer restricted in their effect. As, however, the 
