The exact solution for the impedance of a manifold of differing but uniform transmission lines is derived. The number of uniform structures contained in the manifold is arbitrary and can be increased without bound. The exact solution to the manifold is used to obtain the impedance of a continuously tapered structure having a characteristic impedance and a velocity of propagation that varies along the length of the structure. This solution to the continuously tapered structure can be made as accurate as desired. An approximation to the continuous solution is derived that is valid provided the maximum value of the product of the internal reflection coefficient and the wavelength is much less than unity. The application at this approximate solution to the synthesis of tapered structures is discussed.