A minimum‐ordered set of elliptic integral equations is given for magnetic vector potential, axial and radial fields, the mixed gradient ∂*B*_{ρ}/∂*z* for axially symmetric iron‐free current systems, and for mutual inductance and force between coaxial units. The units may be circular loops, cylindrical or plane annular current sheets, or coils of thick section. With full use of correlations, six basic equations suffice for all properties of loops, solenoids, or combinations. The axial solenoid field is calculated by a superior new method. Thick coils are analyzed with the solenoid rather than the loop as element. Magnetic properties are then integrated in radial depth by Gaussian numerical quadrature of variable order. This method can deal with coils whose sections are not rectangular, and whose current densities are a function of the cylindrical radius.

No tables are used, and the procedure is simpler and faster than previous methods, for computer use or for solving smaller problems by hand. It has been coded as a unified set of programs for the IBM 7090. These will be described elsewhere; they can compute fields, and trace lines of force even within the windings, or force and mutual inductance between coils in contact, also self‐inductance of ideal solenoids or of thick cylindrical coils. Errors are normally less than one part per million, in extreme cases less than one per thousand.

## REFERENCES

*On the Direct Numerical Calculation of Elliptic Functions and Integrals*(Cambridge University Press, Cambridge, England, 1924).

*Tables of the Internal Magnetic Source Functions $Un$ for Thick Solenoids and Disk Coils*(Edwards Brothers, Inc., Ann Arbor, Michigan, 1953). These tables are available upon request from the Thermonuclear Division of Oak Ridge National Laboratory.

*Electricity and Magnetism*(Clarendon Press, Oxford, England, 1873). Third edition (1891) reprinted by Dover Publications, Inc., New York, 1954. Sec. 694.

*Approximations for Digital Computers*(Princeton University Press, Princeton, New Jersey, 1955), Sheets 48 and 51. As

*k*approaches zero,

*K*and

*E*approach the common limit $\pi /2$, and significant figures are lost from $K\u2212E$. Since Hastings’ error curves for

*K*and

*E*are nearly identical, while both sets of coefficients are given to 11 decimals, it was decided to compute

*K‐E*directly, in addition to

*K*, using the coefficients obtained by subtraction.

^{4}and to Bartky,

^{5}which in turn are based on Landens’ transformation.