1901-2.] Dr Muir on the Theory of Jacobians. 193 
reference being to the 5th theorem of his paper of the year 1833, 
already dealt with. 
The next section, the 18th, and penultimate, shows how by 
modifications made upon the functions, the functional determinant 
may reduce to one term. The first function / being expressed in 
terms of x , x 1 , % 2 , . . . . , x n , it follows that theoretically x is 
expressible in terms of /, x } , -x 2 , . . . , x n , and that therefore by 
substitution so also are f, f , . . . , f n . Similarly , being now a 
function of /, x 1 , x 2 ,...', x n , we conclude that theoretically x 1 is 
expressible in terms of /, , f 2 , . . . , x n , and that therefore by 
substitution so also are / 2 , / 3 , . . . , f n . Supposing this process to 
be completed, Jacobi denotes the new forms of / x , / 2 , • • • , f n hy 
f i (y? ) *^2 ’ • • • ’ ^ ) j 
A (/j f\') X 2 ) • * • 5 "O 5 
A (/, A* ■ • • A-l> ^n), 
and the difference between any one of the old forms and the 
corresponding new by F with the appropriate suffix. He thus has 
n + 1 equations 
0 
= F = f - 
(•£ 3 A J 
% . 
. . , aj„) , 
0 
= = A - 
a (A ^i, 
^2, . 
• • j x n) 3 
0 
= f 2 = / 2 - 
A (A A, 
aJ 2 » • 
. . , ® n ) , 
0 
= F n = f n ~ 
/„(/,/!, 
• ' ‘ J 
A-i > A, 
connecting 2n + 2 variables 
X j X j . . . 
» x n , j , y ] 
? • • 
• , A 
now viewed as independent. By a previous theorem there is thus 
obtained 
y + d J . d h ... d l* 
y d A ( pn+i dx ' * * ' dx n 
Z-i — ' 3F 9Fj 3Fj * 
± Tf ' W ^ Vn 
PROC. ROY. SOC. EDIN. — VOL. XXIV. 
13 
