166 
Proceedings of Royal Society of Edinburgh. [sess. 
The proof of the converse proposition, Jacobi owns, is ‘paullo 
prolixior.’ It is of the kind improperly known as £ inductive,’ and 
the first part (§ 7) of it goes to show that if the proposition holds 
in the case of ^ ^ ~ it will also hold in the case 
' dx 1 ox 2 dx n 
o tZ±l-°£ 
. d A 
dx dx 1 
the lemma that- 
As a preliminary, there is established 
If i t fi , f 2 , . . . , fn are mutually independent, then 
V +<Ok . .[.Ml ' = (AW + d A .Mi ... A 
^ dx dx^ dx n \dx/^ dx l dx 2 dx n 1 
where the brackets enclosing a differential-quotient are meant to 
indicate that f there is to be taken as a function of x , f x , f 2 , ... , 
f n - 
Denoting the cofactors of 
df_ ^ 
, 0 /- 
dx 
dx 1 
’ ton 
the determinant ^ 
+ d J . A . . . 
dx dx Y 
Vn 
dx n 
by 
A, 
An 
> A n 
i have 
of 
course 
df A 
dx A 
+ 
d £p + ■ 
0/ A - 
+ - 
.Ml. 
dx 1 
. . d A ' 
dx n 
A A 
dx 
+ 
+ ■ 
+ Mia = 
dx n An 
0 
>• 
dx A 
+ 
I; a . + - 
+ = 
dx n 
0 
But since f Y , f 2 , ... , /„ are mutually independent it is possible 
by solution to obtain x 1 , x 2 , ... , x n each in terms of the 
remaining variable x and f 1 , f 2 , . . . , /„ ; and as a consequence 
it is possible by substituting for x 1 , x 2 , . . . , x n to obtain / in 
terms also of x, f , f 2 , . . . , f n . Differentiating / with respect to 
x , x 1 , . . . , x n , and using brackets to indicate that the / within 
them is to be viewed as a function not of x, x 19 , but of 
»,/n • • • ,/», we have 
