In the insertion-loss design method for filters, compensating networks and other frequency-selective 4-terminal devices, the final step is the synthesis of a network of coils and capacitors which possesses a set of characteristic functions of frequency, in the form of one of several alternative kinds of matrix, previously deduced from the desired overall frequency behaviour. The present paper is chiefly concerned with this step.Practical experience has shown that the most satisfactory way of synthesizing a complicated network is to use a chain of sub-networks connected in cascade. For such a synthesis the most convenient matrix is the so-called “chain matrix,” since the chain matrix of a chain of networks is simply the product of the individual chain matrices. Conversely, a specified chain matrix can be synthesized by factorizing it into sub-matrices, each of sufficiently small degree to allow of direct synthesis, and then connecting all the corresponding networks in cascade.In previous methods of factorizing a given chain matrix it has always been necessary to split off small network sections one at a time. In the present method sections of arbitrary size can be split off in each stage. The importance of this is that the larger the size of section to be directly synthesized, the greater is the likelihood of being able to avoid coupled coils in the design.The method depends on a new theorem of a novel and far-reaching character concerning reactance and impedance functions. In proving and discussing this, reference has to be made to certain properties of such functions which are of great importance in themselves but have not hitherto been presented in a connected manner. Accordingly, a systematic treatment of these is first given, including a number of results believed to be new.A possible extension of the method, potentially of considerable value, to general dissipative networks is indicated.