Transverse gradient coils with co-axial return paths offer reduced acoustic noise compared with standard cylindrical gradient coils, due to local force balancing, and can also easily be made to have a length to diameter ratio that is less than one. Analytic expressions for the magnetic field and vector potential generated by this type of coil are described here, along with a formula for calculating the coil inductance. It is shown that these expressions allow the implementation of powerful analytic methods of coil design, as well as the incorporation of active magnetic screening. It is also demonstrated how the mathematics specifies the best parameters to use when designing coils with small numbers of elements. A head gradient coil for use at 3.0 T has been designed using the analytic approach described here. The process of coil design and construction is outlined and the performance of the coil in comparison with a similar standard cylindrical coil is described.