Let us consider the problem of free vibration of an elastic circular plate under a given condition (such as the case of fixed edge-lines). We arrive at an infinite set of eigen-values (km) and corresponding set of eigen-functions Ym(r) (m=1, 2, 3,···). These functions are known to form complete set of orthogonal functions. Also, it is shown (by referring to theory of linear integral equation with symmetrical kernel, for example) that any given function F(r) can be expanded into form of an infinite series of Fourier-type, provided that F(r) satisfies certain required condition. In the present note, the author has given some considerations about the expansion of Fourier-type of this nature, when the given function F(r) does not satisfy the above mentioned requirement.