Conjugate points in stadium and circle billiards

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In the semiclassical approximation of the Green's function, the Maslov index is obtained by counting the number of conjugate points along classical orbits. We prove that if an orbit starts from the boundary of a stadium or a circle billiard, a conjugate point can never land on the boundary and that there is a conjugate point after a bounce off the boundary if and only if the bounce occurs on a curved side. We demonstrate exceptions to this simple rule when the orbit starts from the interior of the billiard domain. These results are useful for semiclassical calculations involving stadium and circle billiards.