176 Proceedings of Royal Society of Edinburgh. [sess. 
Still further importance is given to the second proposition by 
assigning the next section (§ 12, pp. 341-343) to the consideration 
of certain deductions therefrom. First there is taken the special 
case where 
4 > ~ **' J 4*1 = **1 > * * * 5 4 > )n — % m 5 
and where therefore 
34>l _ 'ST' + d4>m+ 2 
“ dx dx ± dx n ^ ~ dx m+l dx m+2 dx n * 
the result clearly being that If f, f 15 . . . , f n be functions of 
x , Xj , ... , x m , 4 > m+1 n <£ m+1 , . . . , 4> a be functions 
of x , Xj , . . . , x n , then the functional determinant of f , f t , ... , f n 
with respect to x , x x , ... , x n 
V + ¥ t ¥l 
^ ~dx dx Y 
tym ' 4fm+l <rfn # ^ + ^m+1 
* 8</>m+l d4> n ' dx m+l 
Here the first factor would reduce to 
0^ n ‘ 
Y + ^ . ¥1 ¥3 
J ~ 0£ 0aq 0a? m 
if it were possible to put 
4 > m + 1 fm+l ? 4*m+2 ~Jm+2 1 ’ ‘ > 4*n ~fn } 
hence there follows the further proposition, which Jacobi speaks 
of as “ prae ceteris memorabilis,” that the functional determinant 
of f , f 1 , ... , f n with respect to x , x x , ... , x n is equal to 
m. 
, (Vi) . . . 
( of m \ V S/m+1 
^/m+2 
W 
\dx J 
^m+2 
if the brackets in the first factor be taken to mean that the functions 
therein occurring , viz. , f , f T , . . . , f m are considered to be expressed 
in terms of x 1 , x 1 , ... , x m , f m _i , f m+2 , . . . s f n . An extreme 
case of this, viz., where the first determinant factor is of the order 
1, has already been given. 
In order that we may be able to substitute 
1 
2± 
0r 
vu 'm + 1 
iwi 
0X m + 9 
^4 > m+2 
Y + ^4*171+2 
7) r U sLmi ~ 7) r ‘ 0r 
( ^n {Jd "m+2 
$4>n 
Hn 
fan’ 
in the former of these two propositions it is necessary that from 
the equations which give </> m+1 , <£ m+2 , . . . , 4>n in terms of x, 
