1901-2.] Dr Muir on the Theory of Jacobians. 
191 
by reason of the deduction from f— 0. 
us the ‘ formula memorabilis ’ 
Division by ^ thus gives 
m+ 1 
m + 1 
. dx . dx 
Air A, — 
l dx 1 ox C/ 
• - A 
dx 
Vo °m+l 
where A = ^ 
dx , 
dj\ 
dx o 
'm+l 
and A* is derivable from A by putting differentiation with respect 
to x in place of differentiation with respect to x kf the order x , 
x 1} x 2 , ... , x 1i _ li x n , ... x m _ 1 being observed. 
Jacobi adds “ Formula inter egregia inventa illustrissimi 
Lagrange censetur,” and asserts that for the purposes of proof it is 
not necessary to put /= 0.* 
This is immediately followed by the theorem — If u , u x , u 2 , . . . ? 
u n be functions of x 1? x 2 , . . . , x n then 
Y du^u~ l dugi -1 du n u~ l 1 ^ + du 2 du n 
" ~ dXj dx 2 dx n u n+l ^ ~ U dx 1 dx 2 dx u 1 
the connection being somewhat distant, and probably lying in the 
fact that in both the 6’s are of the 2nd order. The mode of proof 
is curious. In the first place, by putting 
= M 
ae t ‘ ‘ 
* No indication is given of where Lagrange published the theorem 
attributed to him. As for the particular way in which x is given as a 
function of x x , x 2 , ... , x m +i , whether by the equation f=0 or not, it is 
clear that this cannot affect the truth of the result, because the latter con- 
tains no /at all, the requisites for validity being (1) that ,/ 2 , ... , / m+1 
are functions of x , x x , x 2 , ... , a? m +i ; (2) that x is a function of x 1 , x 2 , 
. . . , x m+1 ; (3) that on the left the/’s have by substitution been freed of x 
before differentiation ; (4) that on the right this has not been done, but they 
are there differentiated as if x were a constant. The subsidiary result 
d fdfj+i _ 3 / / df+x \ 
^ dx 'dx k+1 dx\dx k+1 J 
is certainly true whether f=Q or not, all that is requisite (see p. 176 above) 
being that fi+± on the right shall be what f i+i becomes by substituting for 
x its value in terms of /, x x , x 2 , ... , x m + 1 , and that in the differentiation 
of it / shall be viewed as constant. 
