If the metric of an n-dimensional space is taken in the form ds2 = u2dτ2+dσ2, where dτ2 and dσ2 are cartesian metrics of r and (n−r) dimensions, respectively, the various forms of u for flat space are quite simple. The study of accelerated motions in special relativity by various authors has led to four dimensional metrics of this form. Those in which u = ±1 at the space origin for all values of time are of particular interest. They are locally cartesian at the accelerated observer, and so the coordinates in the neighbourhood of the observer correspond directly to physical measurements. Hence, such metrics provide convenient means of describing physical conditions experienced by accelerated observers. If the τ-space contains the time direction and is of one or two dimensions, arbitrary rectilinear motions are allowed.