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722 I'HILOSOPHICAL TRANSACTIONS. [aNNO 1763, 



thence, the distance vr being given, with the angles of refraction at the 2d sur- 

 face, the points p or/, to which the rays converge, or from which they diverge, 

 will be given. And their locus, or the curve in which all these points are found, 

 may be assigned; whether the angle of the prism be constant, and the angle of 

 incidence be variable, or the contrary; and whether the rays are refracted, or at 

 a certain obliquity come to be reflected by the 2d plane. 



If it be further required, that the extreme, and all the intermediate rays, which 

 meet at p, in fig. 3, should thenceforth remain united in a colourless pencil : 

 through the point of convergence p draw (by the lemma) the line zx, making 

 the angles zpr, zfv, such that their difference rfv being the given angle of con- 

 vergence, their sines may be as the sines of refraction of the red and violet rays, 

 when they pass from a given denser medium gkh into the air, at a common 

 angle of incidence: and hfg, perpendicular to zx, will be the line in which the 

 surface of that medium must cut the plane of refraction, when the rays rf, vp, 

 are refracted into the same line pn. And if the medium be terminated on the 

 other side by any plane kn, to which fn is perpendicular, the pencil ny, conti- 

 nued in the air, will remain colourless. For instance, if the medium gk be 

 glass, and the angle rpv be 1 4' 26", zfr the angle of incidence of the red rays 

 will be found of 17° 54' 14"; and the angle of refraction xfn, common to all the 

 rays, will be 12° 6' 34"4-. But if the plane hg, to which zx is perpendicular, 

 pass not through the point of reunion f, but on this or the other side of it ; the 

 rays in their passage through the medium, though parallel to each other, will be 

 laterally separated. 



Let a ray sol, fig. 5, of a mean degree of refrangibility be refracted by ab, the 

 side of a glass prism arc, so that the refracted ray om may be perpendicular to 

 the side of the prism ac ; it is required to apply to this, another prism of a dif- 

 ferently refracting substance, as of water, so that the ray mo being refracted at o, 

 by the side dc, the refracted ray so may be parallel to os. The angle of inci- 

 dence SOP, and the refractive power of the glass, being given, the angle som, 

 and its supplement lom, be given, produce mo to w ; and because os is to be pa- 

 rallel to LO, take for the difference of the angles in the lemma, the given angle nos 

 (= lom), and through the point o draw rop, so that the sine of pon may be to 

 the sine of pos, as the sine of incidence to that of refraction, when a meanly- 

 refrangible ray passes from water into air ; and doc, perpendicular to r/>, will be 

 the position of the side required. 



We have here supposed the ray so to be homogeneous, of a mean refrangi- 

 bility; but if it be a ray from the sun, the image at s will be very much tinged. 

 The colours will have been separated at o ; a small matter more at m, but they 

 will diverge very considerably at o; for setting aside the refractions at o and m; 

 that is, supposing a pencil mo to pass uTirefracted in water, till it fall on a sur- 



