668 PHILOSOPHICAL TRANSACTIONS. [ANNO 1800. 



ture, some straight line of union, as ao, must be supposed. But, from whatever 

 cause the first line ao is inferred, by the same cause other intermediate lines mp, 

 tq, &c. will be produced, and curves defm, mtsr, will be ultimately formed, having 

 a point of contrary flexure at m. The form of the curve does not appear to admit 

 of accurate investigation, nor is it of importance to the subsequent reasoning if 

 wholly unknown. We may however form some judgment of its nature; for whe- 

 ther the densities depend on different specific gravities of different fluids, or on un- 

 equal temperatures of different portions of the same fluid, the curves will be nearly 

 alike. In each of these cases, to whatever small distance pc, fig. 3, the mutual 

 attraction is sufficient to occasion intimate union of the fluids, the density mn of 

 the mixture will be an arithmetic mean; and, for the same reason, at any inter- 

 mediate smaller distances, there will be a series of arithmetic means ef, gh, &c. 

 interposed, and the line ao, uniting the ordinates, will be straight. By progressive 

 effect of this attraction, and successive interpolations, in fig. 2, curves defm, rstm, 

 will be formed; of which the straight lines mp, tq, &c. are tangents. The at- 

 tracting distances np, oq, &c. are subtangents; and if it be admitted that these 

 are every where equal, the curves so produced are logarithmic, and the increment 

 of the ordinate greatest at m, where they meet. 



Prop. 3. If parallel rays pass through a medium varying according to the pre- 

 ceding proposition, those above the point of contrary flexure will be made to diverge, 

 and those below the same point will converge, after their passage through it. For, 

 since the deviation of each ray depends on the increment of density where it passes, 

 and since the increment of density is greatest at the point of contrary flexure, any 

 rays, as ab, fig. 4, passing near to that point, will be refracted more towards the 

 denser medium than those at cd, which move in a higher stratum, and will diverge 

 from them, but will be refracted towards and meet those at ef, which pass nearer 

 to the denser medium, where the increments of density are also less. 



Cor. Hence, adjacent portions of the converging rays will form a focus, beyond which 

 they will diverge again ; and the varied medium will produce effects similar to those 

 caused by a medium of uniform density*, having a surface similar to the curve of 

 densities, since convergence or divergence will be produced, according as the curve 

 of densities is convex or concave; consequently, by tracing backwards, to the 

 extremities of an object, the progress of the visual rays, or axes of the pencils re- 

 ceived by the eye, it will be manifest that, any object seen through the inclined 

 concave part rm, fig. 6, would appear elevated, erect, and somewhat diminished. 

 An object seen through md, where it is convex and inclined, would be elevated; 

 and if situated beyond the focus of visual rays from the eye, it would appear in- 



* In the varied medium, be and Dm, fig. 5, the corresponding increments of the abscissa and or- 

 dinate, are to each other as radius to the tangent of the angle c. Therefore the tangent of deviation, 

 which is as the increment of the ordinate, varies as the tangent of the angle c. So also, in the uniform 

 medium, since the sines of refraction and incidence are in a given ratio, their differences will bear a 

 given ratio to either of them ; and when the angles are small, the tangent of deviation will vary as the 

 tangent of incidence, or as the tangent of the angle c, which is equal to it. — Orig, 



