507 
1908-9.] Dr Muir on the Theory of Jacobian s. 
sums of multiples of the increments of f \ , f 2 , . . . , f p will also be zero ; and 
thus the whole of the first row 
A l/l A l / 2 \fn 
will be composed of zeros, and the determinant to which it belongs will 
vanish. On the other hand the determinant of the increments of the x’s 
can at the same time be made different from zero, the increments in the 
first row being not necessarily all zero and those in the other rows being 
what we please. The ratio of the two determinants therefore vanishes. 
Following this are given the theorem regarding the relation of 
y/ + |i ¥2 ¥n\ to y/ + ^i^ >ii . 
^ V “ dx 1 dx 2 dxj \ ~ 0/j 0/ 2 ' ' ' c/J ' 
the theorem for finding the functional determinant when the f s are 
given mediately as functions of the x’s, that is to say, as functions of 
</>lD 1 , £»2 > • • • » X n)> <p 2 ( x i > X 2> ■ ■ ’ X n) ; • • • • , 4>p( X l > X 2’ • ■ • > X n) ' ai1 d the 
corresponding theorem when the functions are only implicitly given, 
that is to say, by means of connecting equations 
FiA'i , 
x, 
2 5 
X, 
, fi , /a , ■ • • > fn) = 9 , 
Xc 
2 5 
X n j ,t \ 5 f 2 5 • ■ • j fn) — 9 • 
The mode of treatment will be readily guessed from what has gone before. 
The same cannot, however, be confidently affirmed in connection with 
the theorem which expresses the functional determinant as a single 
product. This is found grouped under the heading “ Diverses formes que 
l’on peut donner a un determinant ” (fonctionnel), the said forms being 
obtainable on varying the systems of increments assigned to the variables. 
In the first example, the array of increments of the x’s is taken to be 
a A 
0 
0 
0 
A. 2 x 2 . . . , 
0 
0 
0 . . . . 
A n x n 
; other array to be 
dx 1 1 1 
dx L 
d .fn A 
' ’ dx l r 1 
1 4 x 
Sx , 22 
* 
3x„ 2 2 ' ' 
x 
■ ■ dx 2 2 2 
x 
ox n 
|- 2A „x„ . . 
fan 
■ • |^A„x„ 
fan 
