Proceedings of Royal Society of Edinburgh. 
156 
du 
du 
du 
du 
dx ’ 
dxf 
dx 2 ’ 
’ ' ’ dx ’ 
v ^n- 1 
du x 
du x 
du x 
du x 
dx ’ 
dxf 
dxf 
’ ' ’ a^_i’ 
du 2 
du 2 
du 2 
du 2 
dx ’ 
dx 1 ’ 
dxf 
’ dx n _ x ’ 
dUn- 1 
du n _ i 
du n _ i 
d“„-i 
dx 5 
dx x ’ 
dx 2 ’ 
dx n _ x ' 
Erit e.g. pro tribus functionibus u , u 1 , u 2 , tribusque variabil- 
ibus x , y , z : 
du 
du x 
du 2 
du 
du x 
du 2 
du x 
. 0W 2 
du 
dx 
dig 
dz 
dx 
dz 
dy 
dy 
dx 
dz 
du 2 
du 
du x 
du 
+ — ■ 
du x 
du 2 
du 
i - 
du x 
du 2 
dz ~ 
dy 
dx 
dy 
dz 
dx 
dx 
' 32/ 
quarn patet expressionem casu, quo u , , u 2 , sunt expres- 
siones lineares, in expressionem ipsius A supra exhibitam 
redire. Quibus positis dico, siquidem a? = £> , x 2 = p 2 , 
...., Wn-i — Pn-i satisfaciant aequationibus u = t, u 1 = t', 
u 2 = t" , , u n _ x — t (n ~ x) , producti 
A^ 
(u - t) {u x - t') (u 2 - t") . . . (u n _ x - ’ 
dictum in modum evoluti, partem earn, quae omnium simul 
elementorum x , x x , ... dignitates negativas neque ullius 
positivas continet, ut supra in casu multo simpliciore, fieri 
(x -p) (x i -p 1 ) (x 2 - p 2 ) (x n _ x Pn—i) ' 
It will be observed that Jacobi looks temporarily upon the 
ordinary determinant {aVc . . .) as the particular case of the 
Jacobian in which the involved functions are linear in all the 
Variables concerned. 
Jacobi (1830). 
i [De resolutione aequationum per series infinitas. Crelle’s 
Journ ., vi. pp. 257-286.] 
