Starting with the energy functions of a general, linear, bilateral, resistanceless, lumped network the canonical equations of the circuit are obtained. These equations are solved in a general form taking into account applied electromotive forces and initial charges and currents by a matrix multiplication process. It is shown how the matrix multiplication process replaces the usual ``frequency equation'' which determines the normal modes of the system. Initial charges and currents in the various component parts of the network are considered for generality. The method given appears to have some computational advantages in certain classes of problems over the method of the ``Heaviside calculus.'' The discussion is concluded with a discussion of normal coordinates and an illustrative numerical example is given. Since oscillatory networks are used extensively in communication engineering it is hoped that the present discussion will be of value. Since, if the circuit has a small amount of resistance present it may be shown that periods of oscillation are affected only by the squares of small quantities, it is seen that the method given is applicable to many technical problems. A method is given by which the attenuation due to the presence of small resistances in the system may be readily calculated.