This note is devoted to the plane motion of a multiple pendulum consisting of a massless, inextensible string with a number of concentrated masses attached to it. It is shown that, for a pendulum of given total length, the sum of the squared reciprocal frequencies of small oscillations does not depend on the distribution of the masses along the string. In the special case of a pendulum with equal and uniformly spaced masses, this statement is related to a property of the zeros of the Laguerre polynomials. Passing to the continuum limit, one obtains a corresponding property of the zeros of the Bessel function J0. For a multi-body pendulum, the theorem remains valid, provided each member of the pendulum is suspended at a certain point of its predecessor.