  We demonstrate that the formalism for the calculation of the Jacobi last
multiplier for a one-degree-of-freedom system extends naturally to systems of
more than one degree of freedom thereby extending results of Whittaker dating
from more than a century ago and Rao ({\it Proc Ben Math Soc} {\bf 2} (1940)
53-59) dating from almost seventy years ago. We illustrate the theory with an
application taken from the theory of coupled oscillators. We indicate how many
Lagrangians can be obtained for such a system.
