This paper gives the outline of a new method for the calculation of electromagnetic transients on transmission lines above homogeneous ground. The basic idea is that line transfer functions which normally are calculated in terms of hyperbolic functions of \omega are now expressed in terms of s = \sigma + j \omega . This is possible due to the existence of analytical expressions of line impedances and admittances in terms of j \omega , or s in the new approach. It is then possible to find the poles of the transfer function by the secant method and then the corresponding residues. The partial fraction approximation of the transfer functions is incomplete and a correction may be used to account for the missing poles. The calculations are performed numerically only for samples of s needed for the calculation of poles and residues. The end result is a set of state equations with \upsilon and i as inputs/outputs at the switch terminals. Inputs and outputs are interchanged as the switch poles open or close. The integration can be numerical or closed form for sinusoidal inputs (generator voltages) and the state variables carry the past information from one stage of switching to the next. Essential parts of the procedures are illustrated by numerical results.