Stability of oscillations in systems obeying Mathieu's equation

Numerous relatively simple physical systems give rise under appropriate circumstances to oscillations which obey the equation y″ + ?(1 + k cos t)y = 0 (Mathieu's equation). These oscillations may be either stable, periodic, or unstable, depending upon parameters of the physical system as expressed by the parameters ? and k in the basic equation. It has been customary to distinguish between the stable and unstable states by diagrams of the type of Fig. 1, from which it is possible to tell whether a given set of values of the parameters ?, k will yield a stable or unstable solution. In this paper are given curves which not only present this information, but in addition give for an important part of the stable state the values of the characteristic exponent μ. The solution of the equation y″ + ?(1 + k cos t)y = 0 depends to a large extent on this exponent, and the availability of values of μ should greatly facilitate the practical application of the equation.     

The elimination of hypo from photographic images

It is very difficult, if not impossible, to remove the last traces of hypo from photographic papers by any known procedure of washing. The sulfur in the residual hypo ultimately, and especially under abnormal conditions of temperature and humidity, combines with the silver image to form yellowish brown silver sulfide. This phenomenon is known as sulfiding or “fading” of the image. The various factors which affect the rate of fading of images and the washing out of hypo from films and papers are outlined.Chemical methods of hypo elimination have been proposed from time to time but the majority of these have not been satisfactory because they tend to leave substances such as thionates in the photographic material, which are equally as difficult to wash out as hypo and which also tend to sulfide or fade the silver image. A new hypo eliminator is recommended consisting of two volatile chemicals, hydrogen peroxide and ammonia. This eliminator oxidizes the hypo to sodium sulfate, which is inert and soluble in water, while any excess eliminator evaporates on drying.Two formulas and treatments are proposed: (1) Complete elimination of hypo for use by the professional, advanced amateur, and photofinisher who demand the highest standard of photographic quality in their prints.(2) Almost complete elimination of hypo (less than o.oi milligram per square inch).Since the conditions to which prints will be subjected are rarely known in advance, use of the “complete elimination treatment” is advised in all cases.     

Some effects of variable excitation on synchronous motor oscillation

In this paper are presented the results of a study of synchronous motor oscillations resulting from sudden application of mechanical load. By means of the Differential Analyzer at the Moore School of Electrical Engineering of the University of Pennsylvania, theoretical synchronous machine equations were solved for conditions of suddenly applied load, and the effect on the resulting oscillations of variations in field excitation was determined for the most practical cases. The analyzer results were checked by a number of experimental observations, and it was found that the magnitude and duration of the oscillations could be greatly decreased by using the proper type of excitation variation.