188 Proceedings of Royal Society of Edinburgh. [sess. 
the right-hand side 
the cofactor of — — is ( - l) ,K 'S ) B / a.. . 
bx n+k 
The connection between the A’s and the fs is thus 
k = 
so that 
A. b + + . 
+ h m b, m = ( — ])?».(»+i)B m 1 (/a& + /* 1 & 1 + . . . + f^ m b m ). 
The left-hand member here, however, being equal to the left-hand 
member of the identity with which we started, it follows that the 
two right-hand members must also be equal, and therefore that 
fib + + . . . + p m b m = B 
v ± d f 
^ ill 
d A 
dx m 
bfn 
dx„ 
Of course this shows, exactly as the original identity did, that if 
b = b l = . . . . =b n = 0 and B ^ 0 
then 
'V + ^ ?fi .... _ q 
^ ~ cx m dx m+1 bx m+n 
— that is to say, the functional determinant of /, f 1 , ... , f n with 
respect to another set of n+ 1 variables vanishes also. Jacobi, 
however, does not at once say this, but drawing his reader’s 
attention to the fact that the new set of variables contains n-m 
taken from x , aq , ... , x n _ x and m + 1 others, viz., x n , x n+1 , , 
x n +m ) he affirms that the identity reached shows how the functional 
determinant of /, f , . . . , f n with respect to any set of n + 1 
variables is expressible in terms of the m + 1 functional deter- 
minants whose variables are 
**T ? • 
. . . , x n _ 
-1 : 
i '^n 5 
3 , 
x l , . 
. . . , x n _ 
-1 ’ 
X , 
£^1 , . 
. . . , x n . 
-1 » 
'A • 
His words are — 
“Unde formula docet quomodo e functionum /, f , . . . , f n 
Determinantibus b k per idoneos factores multiplicatis et 
additis proveniat earundem functionum Determinans quarum- 
cunque variabilium respectu formatum atque per ipsum B 
multi plicatum. Hinc bene patet, quod § pr. demonstravi, 
