A minimum‐ordered set of elliptic integral equations is given for magnetic vector potential, axial and radial fields, the mixed gradient ∂Bρ/∂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.

1.
The geometrical problem of calculating the efficiency with which a round window collects particles or quanta from a point source is the same as that of calculating the axial field of an ideal solenoid. An extensive literature has grown up around each application independently of the other. (See Ref. 2.)
2.
M. W.
Garrett
,
Rev. Sci. Instr.
25
,
1208
(
1954
).
3.
J. V.
Jones
,
Proc. Roy. Soc. (London)
63
,
192
(
1898
).
4.
L. V. King, On the Direct Numerical Calculation of Elliptic Functions and Integrals (Cambridge University Press, Cambridge, England, 1924).
5.
Walter
Bartky
,
Rev. Mod. Phys.
10
,
264
(
1938
).
6.
M. W.
Garrett
,
J. Appl. Phys.
22
,
1091
(
2951
).
7.
M. W. Garrett, 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.
8.
M. W. Garrett, “Computer Programs using Zonal Harmonics for Magnetic Properties of Current Systems,” Oak Ridge National Laboratory Report, ORNL‐3318 (1962).
9.
The history of these equations by no means parallels the idealized sequence of the table. Here the complete set is formally regarded as derivable from Chester Snow’s formula [Eq. (8)], when the expression $M(S,L)$ has been replaced by the right side of Eq. (9). Snow’s paper, is so far as it touches the present discussion, is concerned only with Eqs. (8) and (9). (See Ref. 10.) Snow credits both equations to Viriamu Jones, but the writer has not found Eq. (8) in Jones’s work.
10.
Chester
Snow
,
J. Res. Natl. Bur. Std.
22
,
239
(
1939
), RP 1178.
11.
J. C. Maxwell, Electricity and Magnetism (Clarendon Press, Oxford, England, 1873). Third edition (1891) reprinted by Dover Publications, Inc., New York, 1954. Sec. 694.
12.
A. V. H. Masket R. L. Macklin, and H. W. Schmitt, (a) “Tables of Solid Angles and Activatations,” Oak Ridge National Laboratory Report, ORNL‐2170 (1956), TID‐14,975 (1962);
(b)
Rev. Sci. Instr.
28
,
189
(
1957
).
13.
N. B. Alexander and A. C. Downing, “Tables for a Semi‐Infinite Current Sheet,” Oak Ridge National Laboratory Report, ORNL‐2828 (1959). The tables include $Bz,$$Bρ,$ and the vector potential.
14.
The relation between $Bz(S)$ and the solid angle, including the discontinuity, is most simply visualized in terms of the following very general theorem. Consider a homogeneous cylindrical sheet whose currents circulate in planes normal to the generators (the section may be any arbitrary closed curve). Then $Bz$ at any point in space is proportional to the total solid angle Ω subtended by the cylindrical surface, evaluated precisely as if the flux through the surface were to be computed from a positive charge at the point. That is, for a positive sheet the element $dΩ$ is counted plus when the line of sight passes outward through the surface, minus if inward, and zero if it passes through an open end or cuts the surface twice.
15.
C. Hastings, 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 $π/2$, and significant figures are lost from $K−E$. 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.
16.
J. L.
Thomas
,
C.
Peterson
,
I. L.
Cooter
, and
F. R.
Cotter
,
J. Res. Natl. Bur. Std.
43
,
311
(
1949
), Rp 2029. The method is that of the third example (p. 314). It combines procedures due to King4 and to Bartky,5 which in turn are based on Landens’ transformation.
17.
For the Gaussian abscissas and weight factors, see
A. N.
Lowan
,
N.
Davids
, and
A.
Levenson
,
Natl. Bur. Std. Appl. Math.
Ser.
37
,
185
(
1954
).
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