427 
1900-1.] Dr Muir on a Proposition given by Jacobi. 
Altering u Y by introducing into it u 2 and we have 
u 1 = u 2 + u 3 — 2yz, u 2 = y{z + x), u z — z(x + y) 
the Jacobian of which is 
-2 z -2 y 
y z + £ y 
z z x + y . 
That this is the same as the previous Jacobian is readily seen by 
increasing its first row by the sum of the second and third rows. 
If now, however, we alter u 2 and u 8 in the same way as u v we 
have 
u 1 =-u 2 + u z - 2yz, u 2 = u z -\-u x ~ 2 zx, u z = u x + u 2 - 2 xy, 
and the Jacobian becomes 
-2z -2 y 
-2 z . -2x 
-2 y -2x 
which is not ixyz but - 1 6xyz. In the sense here given to it, 
therefore, Jacobi’s proposition does not hold when more than one 
of the functions is changed. 
