November 29, 1901.] 



SCIENCE. 



855 



our national capital, might be a good place for 

 experimentation. 



Florian Cajori. 

 Colorado College, 

 Colorado Springs. 



astigmatic images of the bottom of a pool 



OF WATER. 



If light radiate from a point below the sur- 

 face of water, it can pass out through the surface 

 only within a circle forming the base of a right 

 cone whose semi-angle is the critical angle. 



Consider such rays lying in a vertical plane 



passing through the radiant point. The rays 



which have passed out into the air, if produced 



below the surface, are tangent to a virtual 



caustic. This caustic is a portion of the evo- 



lute of an ellipse, one cusp of which is in a 



vertical through the radiant point, and at a 



d 

 depth — , where d is the depth of the radiant 



point, and n is the index of refraction. The 

 branches of the caustic are tangent to the sur- 

 face in the circle determining the critical angle. 

 Successive sets of consecutive rays having an 

 increasing angle of incidence do not intersect 

 at a common point, but they intersect at con- 

 secutive points on the caustic. If the ver- 

 tical plane be rotated slightly in azimuth, the 

 rays from the same radiant point will inter- 

 sect in the caustic in its new position. This 

 caustic from the same radiant point will always 

 lie on a surface of revolution, formed by re- 

 volving the caustic in any vertical plane about 

 the vertical line through the radiant point. 



If the radiant point be viewed by an eye 

 placed at a fixed point, the pupil of the eye 

 may be conceived divided into vertical zonal 

 elements. Eays from the radiant point in these 

 various elements will intersect in a definite 

 area upon the surface of revolution. The point 

 would, therefore, appear as a hazy patch upon 

 the caustic surface. The text-books all repre- 

 sent the apparent position of a coin seen 

 through a water surface, as being lifted up and 

 towards the eye of the observer,, upon the 

 caustic surface. 



It is, however, evident that if the rays di- 

 verging from the radiant point in all azimuths, 

 and at a fixed angle of incidence, be produced 

 backwards after passing out into the air, they 



will all intersect in a common point upon the 

 vertical line through the radiant point. If, 

 therefore, the pupil of the eye be divided into 

 horizontal zonal elements, all the rays enter- 

 ing the eye will have a virtual intersection on 

 this vertical line. The focus of the upper zonal 

 elements of the eye will be slightly below those 

 of the lower. Nevertheless, the intersection 

 of all rays entering the eye from the radiant 

 point will be upon a line, instead of being 

 spread out over an area as in the other case. 

 The fact is that a plumb line deeply piercing 

 still water appears straight throughout. The 

 image upon the vertical line is much more dis- 

 tinct than that formed upon the caustic surface. 

 The latter image imparts a haziness to the ap- 

 pearance of the body viewed, but the apparent 

 position is determined by lines which intersect 

 in a common point, rather than by those which 

 do not. 



With this view of the matter the writer in 

 May, 1881, presented to the Academy of Science 

 of St. Louis a discussion of the apparent form 

 of the flat bottom of a pool as seen through 

 the surface.* The appearance was found to 

 be represented by a conchoid, which was related 

 in a simple way to the conchoid of Nicomedes. 

 The equations of both curves were deduced, 

 and several other cases were discussed. 



In a recent number of Annalen der Physik,-\- 

 Mattheissen has deduced the equations of these 

 two conchoids and has pointed out that the 

 surface produced an astigmatic effect. He 

 likewise deduces the equation for the nebulous 

 image due to intersection upon the caustic. 

 The minimum of this surface and that of the 

 conchoid are coincident and tangent to each 

 other, and they have the water surface as a 

 common asymptote. 



Francis E. Nipher. 



NOTES ON INORGANIC CEEMISTBY. 

 The earliest determinations of the density of 

 sulfur vapor were by Dumas and Mitscherlich, 

 and gave figures which pointed to the molecule 

 Sg, and this has passed current until quite 

 recent times. In 1860 Deville and Troost found 



* Trans. Acad, of Sc. of St. Louis, Vol. IV., No. 2, 

 p. 325. 



t No. 10, 1901, S. 347. 



