506 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
then the limiting value of the ratio of the determinant of the second array 
to the determinant of the first array as the elements of the latter array are 
indefinitely diminished is called the determinant of the n functions. Since 
in the circumstances mentioned 
dx-x 
A kfi — Z~^k x i + ~ + 
d A 
BXr 
dfi . 
+ ~ — &k X n 
OX„ 
for all values of h and i not greater than n, it follows from the multiplica- 
tion-theorem that the aforesaid limiting value is equal to 
d fl 
S/i 
dx x 
dx 2 
dx n 
Vs 
V, 
d/ 2 
dx 1 
dx 2 
dx n 
¥n 
3/» 
Vn 
dx x 
cx 2 
dx n 
and, this determinant being independent of the increments given to the 
independent variables, it is held that the definition is legitimised. It 
might have been added that the name assigned to the limiting value is 
also thereby justified. 
The more important of Jacobi s results, eight or nine in number, are 
then re-established, precedence being given to those regarding the 
vanishing of the determinant. Supposing, first, that the functions are 
independent of one another, he asserts that x 1 , x 2 , . . . , x n may be conceived 
as expressed in terms of f x , f 2 , ... , f n ; and, the latter being viewed as 
independent variables, the determinant of their increments can be con- 
sidered as completely arbitrary and can thus have a value different from 
zero. Further, in respect to this determinant the determinant of the 
increments of y } * * * ? IX cannot be infinitely great, because the terms 
of both determinants have the same number of infinitesimal factors of the 
first order. It thus follows that their quotient — that is, the functional 
determinant — is not zero. Next, supposing that the functions are not all 
independent of one another, but that f p+1 , f p+2 are functions of 
/i , / 2 , . . . , f p , and that the latter alone are mutually independent, Bertrand 
asserts that we may suppose 
\fi = 6 , Aj/g = 0 , . . . . , A J p = 0 , 
this in fact being possible in an infinite number of ways, because only p 
relations are thereby established between the increments Aptq , , 
A x x n . It will then result that the increments of f p+ 1 ,f p+ 2 , . . . , f n being 
