For strong homogeneous fields the ideal loop and solenoid elements of a prototype system may be expanded into thick cylindrical coils whose current density is (a) uniform or (b) inversely proportional to the cylindrical radius. Error coefficients of the zonal harmonic series for field or gradient that are cancelled in any given prototype continue to vanish as the elements are expanded, even to fusion, if at each stage the coil boundaries are adjusted by an iterative method. This involves the use of (a) the harmonic source functionsUn* or (b) the Legendre functionsPn; either set is best computed by a recursion formula. An alternative method based on Lyle's principle requires no iterations and only trivial calculations. Though it also is valid for systems of any order, it can only expand a set of loops into coils with square cross sections of limited area. Also, the true null coefficients are replaced by small residues proportional to 4th and higher powers of the section dimensions when the second method is used.The number of tabulated prototype systems of the 6th and 8th orders is increased to more than 200. Since multiply infinite sets of prototypes can be computed for every order higher than the 8th, univariate tables would be inadequate and only a few illustrative examples of such systems are given. Some 150 thick‐walled systems of the 6th or 8th order are listed in several tables, but such a selection can only suggest the wide range of possibilities. These systems have more degrees of freedom than do the corresponding prototypes, and even the 6th order is multivariant. At the 8th order and beyond, the growing redundancy of variables permits the design of systems with a common current density in all coils and with all section dimensions proportional to a set of small integers. Systems of order 4nhave paired coils with gaps for lateral access, an advantage that is lacking in the 6th‐order designs.Efficient computer programs have been written to design systems of all the types so far enumerated, including multiple sets of both prototypes and fully adjusted systems of thick coils for orders from 8 to 20 in steps of 4. Since adequate tabulation of results is ruled out by their diversity and sheer bulk, in view of the wide range of potential applications the programs themselves must be considered the basic research tool. They will be described in detail elsewhere.As a measure of field inhomogeneity, including contributions from all orders and from both axial and radial field components, the concept of total vector error is developed. The tables include error limits for all systems, and some 40 systems or their error contours are shown in the figures.