MR imaging and spectroscopy requires RF fields of high homogeneity. Quadrature volume coils meeting this requirement include the birdcage coil and the TEM resonator. We describe here a quadrature volume coil designed around a modified coaxial transmission line in which multiple inner conductors are arrayed on a circular perimeter and surrounded by a common shield. Current antinodes are established at appropriate points on the inner conductors by series transmission line stubs, either open circuit or short circuit, which terminate the line. Quadrature phasing is generated by a novel circuit constructed from a pair of high-performance current baluns and a commercial quadrature hybrid. The coil is a "pure" TEM coil as it is simply a resonant transmission line. There is no mode structure to consider. The construction of a prototype quadrature four-element coil is described and preliminary test results from this coil in a 4.7T horizontal bore magnet are reported. A related double nuclear coil design for (3)He and (1)H in which two linear transmission line coils are arrayed on a circular perimeter and simultaneously tuned and matched to their respective frequencies is also described. Preliminary tests from this coil in a 1.5T clinical scanner are reported.