24 2 Dr. Wollaston on double Images 



The attracting distances ?ip, 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. in. If parallel rays pass through a medium varying 

 according to the preceding 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. 3, passing near to that point, will be refracted more to- 

 wards 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 pro- 

 duced, according as the curve of densities is convex or concave; 

 consequently, by tracing backwards, to the extremities of an 



* In the varied medium, be and bm, Fig. 4, the corresponding increments of the 

 abscissa and ordinate, are to each other as radius to the tangent of the angle c. There- 

 fore, the tangent of deviation, which is as the increment of the ordinate, varies as the 

 tangent of the angle c. 



So also, in the ui iform 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. 





