1901-2.] Dr Muir on the Theory of Jacobians. 
181 
functions involve is m greater than the number of functions. 
First it is noted that if the functions be not mutually independent, 
they are not independent with respect to any n + 1 of the variables, 
and therefore each functional determinant formed with respect to 
n + 1 of the n + m + 1 variables must vanish. Then the converse 
proposition is taken up, viz., that if all these determinants vanish, 
the functions are not independent. The method of proof is that 
known as mathematical induction, that is to say, the assumption 
being made that the proposition holds for n functions f, f lt ... 
f n _ x it is shown to hold for n-+ 1 . Clearly we may start by 
viewing f,f lt ... , f n -i as being independent, for if they be not, 
there is nothing to prove; and this being the case, the various 
determinants of these functions with respect to n of the variables 
cannot vanish. Denoting the first of the said determinants, viz., 
'V + xL . A 
J — dx dx 1 
Wn - 1 
dx n _ l 
by B, 
and choosing from the given vanishing determinants of the (n + l) th 
order those having n of their variables the same as those of B, viz., 
the m + 1 determinants, 
'V 4 - ^ 
. d A . . 
¥n 
2 — dx 
dx ± 
dx n _ i 
* 0 V 
y J +%- 
.A . . 
dx 1 
bjn- 1 
dx n __ 1 
. 
dx n+ 1 
y j ± i f ■ 
dx 
. A . . 
dx 1 
'b.tn — l 
dx n _ i 
. 8 A 
^*®n+n i 
we see that from a previous proposition these are respectively 
equal to 
where the operation indicated within brackets is meant to be 
performed on f n as expressed in terms of 
/>/ i 
, fn- 
\ j •X'n ) ' Ay n + 1 5 
• ) • /L n+rn • 
As B does not vanish, it follows from this that 
= 0 , 
¥n.\_ 
ox. 
= 0 , • • 
