  In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the
solution of a Painlev\'e equation in terms of Fredholm determinants of integral
operators. This equation is $\psi''(t)+t^{-1}\psi'(t)=(1/2) \sinh 2\psi+2\alpha
t^{-1} \sinh\psi$, a special case of the Painlev\'e III equation. The proof in
the cited paper is complicated, and the purpose of this note is to give a more
straightforward one. First we give an equivalent formulation of the solution in
terms of the kernel ${e^{-t (x+x^{-1})/2}\over x+y}\Big|{x-1\over
x+1}\Big|^{2\alpha}$. There are already in the literature relatively simple
proofs of the fact that when $\alpha=0$ Fredholm determinants of this kernel
give solutions to the equation. We extend this result here to general $\alpha$.
