1901—2.] Dr Muir on the Theory of Jacobians. 
171 
Of course, having obtained a solution by using the peculiar 
eliminating multipliers 
dx k 
w 
0/1’ 
¥n 
and being aware of the existence of a general set of such multi- 
pliers, Jacobi had ready to hand the means, which he did not 
fail to use, of finding an important identity. For, if the functional 
determinant he denoted by R, it had long been known that 
i 
J3R 
df 
-s + 
0R 
^ $1 
dx k %x k 
s, + 
, 0R 
Jfn" 
l 
and a comparison of this with the value of r h above obtained gives 
at once 
1 0R 
a particular case being 
i-Z 
a M 
% 
¥1 
dx-i 
dXj c 
¥* 
Vn 
dx„ 
dx 
¥ 
This result may be viewed as giving any one of the differential- 
dx, 
quotients ^ in terms of the differential-quotients ; or, if we 
vfi vXj. 
multiply both sides by R, it may be viewed as giving an expression 
for the cofactor of any element of the functional determinant. 
Before leaving this part of the subject it may be noted that 
Jacobi discusses with equal fulness a set of linear equations in 
which the determinant of the coefficients is the conjugate (as it 
afterwards came to be called) of the functional determinant, with 
the result that the practical ‘ rule ’ above given is shown toehold 
here also. 
The expression obtained for the cofactor of any element of the 
functional determinant is utilised (§ 9) to find an equally interesting 
expression for the differential-quotient of the functional deter- 
minant with respect to a quantity a which may he x , x 1 , . . . or 
any other involved in the functions. As R can be considered 
'df 0R 
a function of its (n + 1) 2 elements — , it is clear that — is 
dxy. da 
