424 Proceedings of Royal Society of Edinburgh. [sess. 
viewed in one set of differentiations as being dependent on 
x, x 1} ... , and in another set as being constant with respect to 
x, x 1} . . . This amounts to saying that it might he preferable 
to write /= a instead of f=a. 
(3) In the next place, by “ the change of f into by means 
of the equations /= a, f 1 = a 1 , . . . , /S ='o i _ 1 , f i+1 - a m , . . 
fn = a n ” we are to understand the performance of substitutions 
whereby a, a 1} ... , a^, a i+1 , . . . , a n make their appearance in 
the expression off, this new expression, called being therefore 
such that by resubstituting in it for a, a u . . . we are led back to 
f. After the second enunciation above quoted Jacobi himself 
says “Nimirum restituendo in omnibus — 1 pro Constantibus 
dx K 
a, oq, a 2 , . . . , a n functiones f f lf f 2 , . . . , f n Determinans 
functionale alterum in alterum identice redit.” 
(4) The proposition may therefore be conveniently enunciated 
as follows without making use of the a’s : — If f, f 1 . . . , f n be 
functions of x, x l5 . . . , x n , and by legitimate operations the 
functions f, f 1} . . . , f i _ 1 , f i+1 , . . . , f n be introduced into the 
expression for / which thereby takes the form of <£ i} then 
"V + ^ ^ + .... 5 
^ ~ dx dx 1 dx n ^ ~ dx dx x dx n 
it being understood that in the differentiations of </>, ef> v • • • > <£ n we 
are to view f, f 15 . . . , f n as constants. 
(5) Jacobi’s proof for the case where only one of the functions 
is changed, viz. / into ef>, is irrefragable. He says, in effect, that 
as ef> is a function of x, x v ... , x n , f v . . . , f n , and / is what </> 
becomes when for/ 15 . . . , f n we substitute their expressions in 
terms of x, x lt . . . , x n , it follows that 
II 
roMco 
dej> 
dx 
+ 
d A. d A 
df dx 
+ 
def> ^ df 
df 2 dx 
+ • 
deb 
' ’ + . Wn 
df i 
dx 
df_ 
bx 1 - 
def> 
dx 1 
+ 
deb . d A 
0/i H 
+ 
d<b df 2 
df 2 ' Saij 
+ * 
deb 
' ’ + ¥n 
A, 
dx 1 
II 
1 s 
S-J 5 * 
coMoo 
def> 
dx n 
+ 
df ) df 
df dx n 
+ 
■H. % 
0/ 2 ° X n 
+ • 
def> 
’ * + ^7" 
df n 
ffn. 
dx n 
