§ 1. The vibrations that may be set up and maintained in the central filament of a single vortex ring of small but finite section have been investi gated by Thomson* and others, but no corresponding investigation appears to have been undertaken for a system of parallel rings, although the matter is of some importance in connection with the state of motion behind a moving body. In a previous paperf the authors have examined the stability of an infinite system of equal vortex rings situated in parallel planes with their centres evenly spaced along an infinite line and with their planes at right angles to that line. Instability was there found to occur for disturbances confined to displacements of the centre of each ring along the central axis, the filament of each ring still remaining circular. In the present paper the investigation is extended to deformation of the vortex filaments, and some interesting conclusions are drawn regarding natural modes of vibration of the infinite system of vortex rings, such as may occur without the longitudinal instability referred to in the previous paper becoming apparent. It is found, for example, that for any given ratio of radius of ring section to radius of ring there exists a critical ratio of ring spacing to radius, separating the region of stable oscillation from that of instability, a result in some respects closely analogous to that found by Karman for the stability of two infinite parallel rows of rectilinear vortices. J § 2. The deformations of the filaments will be taken to be of the first order of small quantities compared to the radius of the rings, and the latter will be assumed to perform such oscillations as will give rise to no change in the position of the centre of mass, to the same order. The analysis then resolves itself into an investigation of the conditions in which the presumed oscillations will persist unchanged.
A conventional method to measure the position of a buried line has been used. This method is based on the measurements of magnetic field around a line which conducts a current. However, the pitfall is that the magnetic field generated by an induced current in a secondary line allows considerable measurement errors.
This paper describes a novel method to exclude such errors caused by the induced current. While the magnetic field generated by the current in the primary line is in phase with the current, that generated by the induced current delays more than 90° compared to that of the current in the primary line because of a reluctance component of the secondary line. Hence, the 90° component of the magnetic field results from the induced current. This component is measured by a solenoid coil and a lock‐in amplifier; the 0° component is measured also and corrected by using the 90° component. Thus, the net magnetic field generated by the current in the primary line is calculated. It is used to determine the correct position of the primary line. The new method is tested by using two buriéd lines and is proved to be useful.
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