1901 - 2 .] Dr Muir on the Theory of Jacobians. 
169 
The general theorem being thus established, Jacobi refers in a 
line or two to the corresponding contrapositive proposition, viz., that 
if the functional determinant do not vanish , the functions are mutu- 
ally independent , and to the contrapositive of the converse, viz., that 
if the functions be mutually independent , the functional determinant 
cannot vanish. Finally, in recalling the ultimate case where the 
determinant is of order 1, he notes that, as in that case, so generally 
the four propositions may be combined in a single enunciation, viz., 
According as the functions f , f x , . . . , f n of x, x 15 . . . , x n are not 
or are mutually independent, the functional determinant does or does 
not vanish. 
The next subject taken up (§ 8) is the solution of a set of linear 
•equations 
df r 
+ 
y. 
+ • • 
f 
d f r 
dx 
dx 1 1 
* T 
dx/ n 
fr 
+ 
¥> r 
dx x 1 
+ • • 
• + 
dfi 
~ r n 
dx n 
dx 
+ 
dx 1 1 
+ • • 
• + 
dfn 
T n 
dX n 
in which the determinant of the coefficients of the unknowns is 
the functional determinant 
^ 4. 
" ~ dx dx l dx n 
Of course a condition of solution is that this determinant does not 
vanish, and therefore that the functions/, j\ , . . . , f n are mutually 
independent. But this being the case it follows that x, . . . , 
x n are expressible in terms of f,f 19 ... , f n , and, as a consequence 
of substitution, that any other function of x, x Y , . . . , x n is ex- 
pressible in terms of the same, thus giving us 
dcj> dcf> dx d(f> dx-^ d<h dx n . 
dfk ~ dx ' df k + dx 1 ’ df k + + dx n ' df k ’ ^ * 
Now if we turn to our set of equations, and multiply by 
dx k dx k dx k 
W ’ ¥ 1 ’ ■ ■ ' ’ % 
