This paper considers the stability of two planar forms of the elastica: (1) A twisted circular hoop and (2) the classical noninflectional plane elastica. By means of the Kirchhoff thin-rod theory, the critical loads at which the planar forms become unstable and tend to “pop out” into spatial forms are determined. The results are related to the problems of coiling and kinking of submarine cable.

This article gives the results of an analytical and numerical study of a two‐gyro, gravity‐oriented communications satellite. The principal purpose of the study was to uncover and solve the analytical problems arising in the design of passive gravity‐gradient altitude control systems. Although the study was directed at satellite orientation, it is felt that many of the techniques developed have general use in the investigation of dynamical systems. We consider both small and large motions about the desired earth‐pointing orientation. In the small‐motion study, the goal is simultaneous optimization of the transient response and the forced response to perturbations caused by orbital eccentricity, magnetic torques, solar torques, thermal rod bending, and micrometeorite impact. In the large‐motion study, we enumerate all possible equilibrium positions of the satellite and then consider initial despin after injection into orbit, inversion of the satellite from one stable equilibrium position to another by switching of gyro bias torques, and the decay of transient motions resulting from large initial angular rates. As a specific numerical example, we have treated a 300‐lb satellite in a 6000‐nm orbit, stabilized by a 60‐ft extensible rod with a 20‐lb tip mass, and by two single‐degree‐of‐freedmn gyros, each with an angular momentum of 106 cgs units. Without a detailed discussion of hardware, it is concluded that such a system, having a total weight of 50 to 75 pounds including power supply, will provide a settling time for small disturbances of less than one orbit and will hold the antenna pointing error within a few degrees.

El£bieta Zaj^c CONSTRUCTIONS OF LEFT NORMAL BANDOIDS Introduct ionThe idea of investigation of left normal bandoids arises from the study of dissemilattices.

The last relation determines the manner in which T varies if static equilibrium exists. Thus the paradox appears to be resolved; the more difficult problem of determining the shape of the film remains.

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