Dr Muir on the Theory of Jacobians. 
509 
1908-9 
and 
\A 
^ 1/3 
• • • ^\fm 
0 
9 
9 
Ao/l 
Ao/ 3 
(M 
<] 
9 
9 
9 
A 3/ 2 
A 3/3 * ' 
■ • • A 3 / m 
0 
0 
9 
±mfl 
A w ,/ 3 
• • • A m / m 
9 
9 
9 
A m+l.f 1 
A m+l.f 2 
• • * m 
A 77l-|-l./" ; 777-}- 1 
Ar/7-fl./7n-|-2 • • 
■ Attj-I-I f n 
A 771 + 2 ./ 1 
^m+2 J 2 
A 771 + 2/3 
• • • A m+’Z.fm 
A777-)-2ym-}-l 
A 777 - 1-2 J 777+2 • • 
• • ^m+2.f n 
&nfl 
A n.t 2 
A n / 3 
• • • A n y m 
A njm+l 
A 71./ 777 -f-2 • • 
■ ■ A „/„ 
from which the functional determinant is obtained in the form 
+ 
¥m + 1 Of m .+ 2 
0-^m+l ^'^m+2 
¥n) 
dx n J 
where, from looking as before at corresponding rows of the two arrays, we 
see that in the first determinant f x , / 2 , . . . , f m are to be viewed as functions 
of x x , x 2 , . . . , x m , f m+1 , / m+2 and in the second determinant / m+1 , 
/m+ 2 , • • • , f m are t° be viewed as functions of 00 - ^ ? * * * ? * 
The next section is still more interesting, as it concerns the proposition 
which Jacobi stated incorrectly in his original memoir of 1841 and returned 
to in 1844. The data according to Bertrand are the usual n functions 
5 f n each dependent on 00 ? t ^2 y • • • y ? with the addition that the 
said functions when expressed in terms of x ± , x 2 , ... , x n , f 1 , f 2 , . . . ,f n 
become 0 X , 0 2 . . . . , <f> n : and the problem he sets himself is to find the 
relation between 
V (±¥l ... S 4\ and y(±^ ^ 
“ < -* J \ dx x dx 2 dx n ) ^ \ dXj dx 2 dx n J 
the differentiations in the latter determinant being performed on the 
understanding that the /’ s there occurring in the c p’s are to be viewed as 
constants. The equations 
<^>1 f y 9 , 0 2 ./ 2 9 , . . . , <fi n J n 9 , 
may be held to give implicitly the f s as functions of the x’s , and therefore 
by a previous result 
y + /0</>l 0</>7 
2 ± 
d A ¥? 
dx Y dx 0 
Vr 
dx n J 
) « (-i) 
dx x dx 2 
dx rj 
0^1 _ 1 00 L 
001 
a/i a/ 2 
¥n 
00 2 00 2 _ ^ 
002 
9 /i 3/2 
0/4 
' 'l J u ‘ n 
S/j 3/2 
d( t>n , 
‘ ' 0A ; 
