Non-rotational Symmetry Research Articles

Complex potentials in shallow shells of revolution

AbstractBy introducing complex variable techniques in the theory of shallow shells, the problem is reduced to the solution of one fourth-order differential equation, and the displacement components and stress resultants and couples are expressed in terms of four complex potentials. Two of these complex potentials are those of thin plate theory, thus admitting of deducing the solution of a shallow shell problem from the limiting case of the circular flat plate. These results are applied to a case of non-rotational symmetry, a special case of which has practical significance in the design of tanks and boilers.

Correction of Electrostatic Lenses by Departure from Rotational Symmetry

It is known that departures from rotational symmetry may offer correction possibilities for certain aberrations which limit the resolving power of present-day electron microscopes. The geometrical optics of such departures in electrostatic systems and the general correction conditions are investigated. The minimum number of design parameters is taken, in order to ensure a simple structure, and a theory is developed for the correction of third-order spherical aberration in the particular type of objective lens evolved. This theory is simple enough for computation, yet applies to lenses with four electrodes capable of a high magnification. An example shows a tolerance of 1/5 micron for the correcting electrode if the resolution limit is to be reduced to 10% of its rotationally symmetric value. A better example could probably bring this within engineering practice, but first-order astigmatic electrodes are also needed, and the tolerances of these are one or two factors of ten more exacting. It is therefore here that the difficulties will arise in the application of non-rotational symmetry.

The Effect of a Bend and Other Discontinuities on a Two-Wire Transmission Line

The theoretical problem of a bend on a two-wire transmission line is analyzed by the vector-potential method. The equivalent circuit elements are obtained by comparing the variableline parameters near the bend with the conventional line parameters found on an infinite line. The effect of the nonrotational symmetry of the axial vector potential on the conductor surfaces is treated by an approximate method to obtain one of the equivalent series elements. Experimental values of the circuit elements for the bend were found to be in good agreement with the theoretical. Good agreement was also found between experimental and theoretical values for open-ended and bridged-end lines.