1901—2.] 
Dr Muir on the Theory of Jacobians. 
167 
¥_ 
dx 
¥1 
dx 
Vn 
dx 
(bf\ 
•lr + 
(bf_\ 
,A 
+ * ■ 
, • J- 
(¥\ 
A 
Wi> 
W 2 y 
' dx 
• -q- 
wJ 
" dx 
(bf ^ 
,¥ . 
(df_\ 
t 
. 4- 
(¥\ 
M r 
Wi ) 
} dx x 
\^/ 2 y 
'3a, 
T 
• T 
WJ 
1 dx 1 
M . 
/df ^ 
4- . . 
. . 4- 
(¥) 
, d A. 
hx n 
W 2 y 
’dx n 
i 
. -f- 
WJ 
dx n 
From these, on multiplying both sides of the first by A, both sides 
of the second by A 1? and so on, and then adding in columns, there 
is obtained, with the help of the n + 1 equations immediately 
preceding, 
Y+ d l. d A 
^ dx dx 1 dx n \dx ) 
which is what was to be proved.* 
A + 0 + 0 + • • 
• + 0 ,. 
Now suppose that in any way it has been established that if 
’ ' * * =6, the functions involved are not mutually 
independent, and that the next higher case is to be investigated, viz. y 
where "V ±~ • • • • • ^ = 0 . Of the n + 1 functions in- 
^ dx dx 1 ox n 
volved in the latter determinant the last n of them must either be 
independent or not. If they are not independent, there is nothing 
more to be proved. And if they be mutually independent, then 
the lemma gives 
!)-»• 
* Using this theorem upon itself we have 
§£. . . . . bfn _ 
^ dx dx x dxn \dx J\dx 1 ) ^ dx 2 
dfn 
dx n 
provided that on the right / is expressed as a function of x , , / 2 , . . 
and /i as a function of x, ,/ 2 , ... , fn ; and ultimately 
. jh .... 
^ dx dxj dx r , 
(A 
m\.. 
. . (A\ 
\dx) 
wj 
\dXn) 
provided that in every instance on the right /& is expressed as a function of 
x , x 1 } x 2 , ... t x k , f k+1 , ... ,f n . 
A theorem like this ultimate case Jacobi enunciates and proves quite 
independently at the end of his memoir (v. § 18). The one, however, is seen 
to include the other if we note the simple fact that 
Y , 3 /; 3 /l dfn _ y ■ dfn dfn-x ... . df 
dxdx-L dxn ^ dxn dxn-i dx 
