1900-1.] Dr Muir on a Proposition given by Jacobi. 
425 
r)/ 7)f r)/ 
Using these equivalents for • • • > we transform the 
da; dx'j 0a? w 
Jacobian 
into 
^ + Jr 
^r~ + zrx 
■U 1 + 
2, + - 
^ dx 
3/i . 
dx 1 
‘0/» 
dx n 
, A, 
d A, 
bfn 
dx 
dx 
dx 
dx 
A + . . . 
, i, 
3/2 , . . 
Vni 
dx x 
dXj 
dx 1 
dx 1 
dx n 
, A, 
A, . , 
. A 
(IX„ 
dx n 
dx n 
and this on having its first column diminished by multiples of the 
other columns becomes 
30 t 3/i 
dx dx 1 
tyn 
dX„ 
as was to be proved. 
(6) Jacobi then proceeds with the case where two of the 
functions are altered, his exact words being — 
“ Si per aequationes 
fit 
0 — °) f 2 — a 2’ J 3 ~ a 8’ 
fi = 0n 
fn — ^r 
eodem modo probas fieri 
yp + ¥* 3/i 
^ ~ dx dx , 
0/n = y + 5 0 00! _ ?/ 2 §/» 
0X„, ^ ~ 0$ 0fl? 1 0^ 2 0^ 
unde etiam 
V + ^ . % . . . ^ = V + . Ml . % . . . 0/n 
~ dx dx T dx n dx dx l dx 2 dx n 
This practically concludes his reasoning, for he merely adds “ Sic 
pergendo sequitur generaliter and gives the second of the 
two enunciations above quoted. 
(7) Now what he here really proves is — If f, f l5 . . . , f n be 
functions of x, x 1? . . . , x n and by legitimate operations the 
functions f 15 . . . , f n be introduced into the expression for f which 
