  We derive formulae for some ratios of the Macdonald functions, which are
simpler and easier to treat than known formulae. The result gives two
applications in probability theory. One is the formula for the L{\'e}vy measure
of the distribution of the first hitting time of a Bessel process and the other
is an explicit form for the expected volume of the Wiener sausage for an even
dimensional Brownian motion. Moreover, the result enables us to write down the
algebraic equations whose roots are the zeros of Macdonald functions.
