Solutions of the relativistic equations of motion of a charged particle in a constant electromagnetic field are obtained in terms of a Lorentz transformation determined by the tensor describing the field. Some results of O. Veblen, J. W. Givens, and the author on Lorentz transformations are summarized in Appendiz A and are used to obtain explicit expressions for the orbits. The methods used may be applied to the orbit of a charged particle in the field of a plane wave. The latter orbits are discussed in detail in Appendix B.