1901-2.] Dr Muir on the Theory of Jacobians. 179 
and also, as usual, the occasion is utilised to draw attention to the 
analogy between the differential-quotient and the functional- 
determinant. “ Quam formulam si cum generali comparas, et 
hie vides perfectam locum habere analogiam inter differentiale 
primum functionis unius variabilis atque Determinans system atis 
functionum plurium variabilium. 5, 
The theorem next brought forward (§ 14) is said to be useful 
in connection with the preceding general theorem for finding the 
functional determinant when the functions ‘ quocumque modo 
implicito dantur,’ and is also spoken of as being in itself ‘prae 
ceteris memorabilis.’ It is — If f , fj , . . . , f n be functions of 
x , x l , ... , x n and there be given the equations 
f= a -,fl = a i , • • • , fn = a n 
in which a, cq , < . . , a n are constants , then the functional determinant 
will not be altered by any tranformation made upon the f’s by the 
use of the given equations , it being understood, of course, that the 
equation f, = a, is not used in the transformation of f. 
Taking first the case where only one of the functions, say f is 
transformed, this becoming <f> by the use of the equations 
f\ — a l 5 fl = a 2 5 • • ■ fn — a n » 
we see that <j> in addition to x , aq , ... , x n may involve a T , 
'2 > * * 
• 5 J 
and that therefore we 
have 
V 
deft 
rf 
dcf> 
. 3 A 
I 
dcf> , 
% 
dx 
• • + 
0C f> 
bfn 
dx 
dx 
da 1 
dx 
T 
da 2 
da n 
* 0a; 5 
df 
def> 
+ 
d<f> 
. d A 
I 
def) 
% + . 
dx 1 
. . _L_ 
def> 
a/» 
dx 1 
dx l 
da 
dx 1 
i • 
0a 2 
da n 
‘ 0aq ’ 
df_ 
def) 
1 
0c p 
, d A 
0c f) 
d A 
• • + 
def) 
. d A 
dx n 
dX n 
T 
0a 1 
dx„ 
+ 
da 2 
dx n 
da n 
' dx n ' 
By substituting in the functional determinant 
V + jf . A .... A 
~ dx dx 1 dx n ’ 
