1900-1.] Dr Muir on a Proposition given by Jacobi. 
423 
Note on a Proposition given by Jacobi in his “ De deter- 
minantibus functionalibus.” By Thomas Mnir, LL.D. 
(Read July 1, 1901.) 
(1) The proposition in question is stated as follows*: — 
“ Pondmus ( enim ) inter quantitates , x, x 15 . . . , x n datas esse 
totidem aequationes 
f — a j f\~ a n * • • j f n = 
in quibus a, cq, . . . sint Gonstantes : dico Determinans 
y ±¥. d A . . . . d A 
dx dx l dx n 
non mutare valorem si functiones f, f 15 . . . , f n varias subeant 
mutationes quotes per aequationes propositus subire possunt, ita 
tamen ut functioni alicui f, transmutandae non ipsa adhibeatur 
aequatio = cq.” 
If we were dependent on this alone for Jacobi’s meaning there 
might be some difficulty in regard to the interpretation. Fortu- 
nately, however, at the conclusion of his demonstration he restates 
the proposition in another form, viz. “ Si per aequationes 
f = a 5 f\~ a l5 fi-l = a i- 15 fi+l ~ a H 1J • • • • j fn — a n 
fiat 
per aequationes 
f=d 
fore 
y + d l. d A . 
^ dx dx l 
fi 
f 1 = a l> 
• • • 5 fn ~ a n 
Vn 
= V 9^1 
dx n 
~dx dx 1 
dx n 
(2) The expression “ Ponamus inter quantitates x, x v . . , 
x n datas esse totidem aequationes f=a,f 1 = a 1 , . . . , f n = a n in 
quibus a, a v ... . sint Constantes ” is particularly unfortunate, 
for it is certainly not intended that n + 1 equations are given, by 
the solution of which the independent variables x, x v ... , x n 
may be shown to be constants ! In fact, a, a l5 . . . are simply 
alternative symbols for two symbols being deemed 
desirable for each function because the function requires to be 
* Crelle’s Journ., xxii. p. 345. 
2 E 
VOL. XXIII. 
