The purpose of this paper is to determine eigensolutions of a rotationally periodic structure (P) of period (N) in terms of those of the (i) th substructure (S^{(i)}) of (P). Let (\mathbf{y}^{(i)}) be the generalized displacement vector and (\mathbf{f}^{(i)}) be the generalized force vector at interface (I^{(i)}), where substructures (S^{(i)}) and (S^{(i-1)}) are connected. Transfer function formulation of (S^{(i)}) implies that there exists a linear operator (\mathbf{G}) mapping (\mathbf{y}^{(i)}) and (\mathbf{f}^{(i)}) at interface (I^{(i)}) to (\mathbf{y}^{(i+1)}) and (\mathbf{f}^{(i+1)}) at interface (I^{(i+1)}). In addition, the operator (\mathbf{G}) depends on transfer functions (and therefore eigensolutions) of (S^{(i)}). The periodicity of (P) then requires that (\mathbf{G}^{N}) be an identity map. This results in (N) self-adjoint Fredholm integral equations, the non-trivial solutions of which predict eigensolutions of (P). As a consequence, a periodic structure (P) with period (N) will have exactly (N) eigenvalues lying between two consecutive eigenvalues of the substructure (S^{(i)}), if the interface (I^{(i)}) contains only one degree of freedom. Finally, three examples are illustrated. The first example considers anti-plane strain vibration of a linear elastic solid containing four slots of infinitesimal width. The second example derives exact eigensolutions of a string-mass periodic structure. The third example illustrates how eigensolutions of an axisymmetric structure can be recovered under the formulation of rotationally periodic structures.