Critical Mean Curvature of Liquid Surfaces of Revolution. 35 



Peak, Colorado, 14,147 feet above the sea, and nearly double 

 the height of Adam's Peak. There, towards sunset, the 

 shadow of the mountain creeps along the level prairie to 

 the horizon, and there begins to rise up in the sky till the sun 

 has just gone down, and the anticrepuscular shadow rises too 

 high to catch the outline of the Peak. The author only 

 witnessed a portion of this sequence, for just about the time 

 that the shadow stretched to the horizon, clouds obscured the 

 sun, and the rise of the shadow could not be observed ; but 

 from all the descriptions he heard, there can be no doubt 

 that the character of the shadow is identical with that of 

 Adam's Peak, only that, as the order of sequence is reversed, 

 it is more easy to follow the origin of the shadows. 



Since the above was written, the author's attention has 

 been called to the sketch of the shadow exhibited by the well- 

 known traveller Miss C. F. Gordon Cumming, in the Colonial 

 Exhibition. This picture represents the shadow lying down, 

 but not raised, on an irregular surface of white mist and 

 mountain tops. The most interesting thing is a prismatic 

 fringe of colour along the straight outside edges of the shadow ; 

 but there is no trace of a bow round its point. 



When we consider how much the appearance of the shadow 

 depends on the height, size, and aggregation of the mist, we 

 need not be surprised at the numerous phases of reflection and 

 refraction that have been described by travellers ; but the 

 general principles which have been laid down in this paper 

 appear to govern all. 



IV. On the Critical Mean Curvature of Liquid Surfaces of 

 Revolution. By A. W. Rocker, M.A., F.R.S* 



LET a weightless mass of liquid, or a liquid film, be attached 

 to two equal circular rings, the planes of which are per- 

 pendicular to the line joining their centres. It will form a 

 surface of revolution ; and if it is in stable equilibrium, the 

 longest or the shortest diameter will be half way between the 

 rings. It is convenient to call this the principal diameter. 

 At all points on the surface the sum of the reciprocals of the 

 two principal radii of curvature is constant. Half this quan- 

 tity may be called the mean curvature. Maxwell has, in his 

 article on Capillary Action (Enc. Brit., 9th edition), given a 

 simple proof of the fact that if the film is a cylinder, a slight 

 bulge will cause an increase or decrease in the mean curvature 

 according as the distance between the rings is less or greater 



* Communicated by the Physical Society : read November 27, 1886. 



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