Networks in this paper consist of non-commensurate transmission lines with branches and branching resistors at junctions. When signals on a transmission line are divided multiple ways at the junctions of branched lines, multiple reflection waves occur by the impedance mismatching. For the analysis of multiple reflections and network design, lattice diagrams have been used so far. However, the expansions of network transfer functions provide an easier way for the same purpose as in the case of lattice diagram. The output transient responses can be directly calculated from the expansions of network transfer functions or can be numerically calculated by software such as the fast Laplace transform. Therefore, once the network transfer functions are given, calculation of transient responses can be carried out quite easily. In this paper, the expansions of network transfer functions have been derived with respect to delay elements ξi=exp(-sτi) by formularizing the propagation of multiple reflection waves, and then the multi-variable rational network transfer functions have been obtained from the expansions. As an example, a 3-port transmission line network with normalized characteristic impedances 1, 1, 6 and normalized branching resistors 1/23, 1/23, 126/23 has been taken up. As the terminal resistances at output ports can be determined from the relation of the first arriving wave to the steady state, the design of 3-port transmission line networks which will furnish output waveforms similar to the waveform of the input within given tolerances has been considered. The output waveforms have been calculated for pure terminal resistances and for the pure terminal resistances plus parasitic parallel capacitances.