1901—2.] Dr Muir on the Theory of Jacobians. 
177 
x 1 , ... , we obtain x m+1 , x m _ 2 , ... , x n in terms of the other 
sds and , <£ m+2 , . . . , </> n . Consequently, we have the 
proposition, If f , f x , ... , f n x m+1 , x m+2 , . . . , x n be ex- 
pressed in terms of x, x 19 ... , x m , <£ m+1 , <£ m+2 <f> n the 
functional determinant of f , f x , ... , f n with respect to x , x x , 
... , x n is equal to 
Y + d l. d A 
^ dx dx } 
3 /m ^ ¥m+l 
dx m 8c p m+l 
bx m+ -y bx m+2 
" “ 3^m+l 3</) m+2 
3/m+2 
b < hm + 2 
% 
b(f) n 
Similarly, in order to be able to substitute 
1 ~ ^ 3/m. 
3^m+l 3^ m+2 
cte B 
3 f n 
for 
3/™ 
3^m+l bx m+2 
Vn 
dx n 
3/m +1 3/ +m2 
in the other proposition it is necessary that from the equations 
which give f m+1 , / m+2 , . . . , f n in terms of x, x 1 , . . . , a?„ we 
obtain x m+1 , 2 , ... , x n in terms of the other x’s and 
fm + 1 j / m +2 } 
f n . We therefore have the result — If 
/j /]. »'•••> fm 5 1 J 
•^m+2 > • 
• • j «^n 
be expressed in terms of 
x , x^ , . . . , , f m +^ , 
J m+2 > • * 
• 5 /n > 
the functional determinant off,f , . . . 
, f n with respect x, 
is equal to 
+1 
• ffm 
dx m 
2 
dx^ 
3 / 
m+i 
0JC, 
3/ w 
m+2 
dx n 
Wn 
Leaving this, Jacobi harks back (§ 13) to an earlier proposition 
with a view to a generalisation now possible, viz., the proposition 
where the functions are given implicitly in terms of the independent 
variables by means of n + 1 equations 
F = 0 , F 1 = 0 , . . . , F n = 0 . 
the extension arises from the number of equations now given 
being n + m, viz., 
Y = 0 1 , F x = 0 , . . . , F n+m = 0 , 
PROC. ROY. SOC. EDIN. — YOL. XXIV. 
12 
