Using continued fractions we study the ground state properties of the spin-½ Ising chain in a transverse field with periodically varying interaction strengths and external fields. We consider in detail the chain having a period-2 modulation of the interaction strengths and compare the results obtained with those corresponding to the spin-½ isotropic XY chain in a transverse field. In contrast to the behaviour of the transverse XY chain, the transverse Ising chain does not exhibit a step-like magnetization versus field dependence caused by the alternation of bonds. Its susceptibility exhibits a logarithmic singularity at a field determined by interaction strengths, and it is stable with respect to spin-Peierls dimerization.