VOL. LIU.] I'HILOSOPHICAL TRANSACTIONS. TTJ 



the extreme rays. And the angle of refraction of the mean ray is 77° l6' ig^ 

 By mean ray is understood the ray whose sine of refraction is a geometrical 

 mean between the sines of refraction of the extreme rays, the common radius 

 being unity. 



Let now the same rays be refracted the contrary way by a surface of water 

 WT ; then, to make the mean emergent ray parallel to the incident pencil, its 

 angle of incidence must be 86° 37'-!^ : and the extreme rays will now converge at 

 an angle of 10\ minutes, nearly. Through the point of convergence o, draw 

 (by the Lemma) a plane wt, to terminate the water, and unite all the rays into a 

 colourless pencil os : and this emergent pencil will be found to make, with a 

 perpendicular to the terminating surface, an angle of 49° Q\, and will be in- 

 clined to the first incident pencil in an angle of 14° 28' 10". Nor is there any 

 other plane besides this which will thus unite the rays. If planes parallel to it 

 cut the rays any where but in their point of convergence, they will be parallel to 

 each other, but exhibiting their several colours. And planes not parallel to it, 

 will every where give a coloured image, excepting only when they pass through 

 the point of convergence ; but then the rays having crossed at that point, will 

 thenceforth diverge from one another, and give a coloured spectrum. 



From all which it appears, that light refracted through different media may 

 emerge colourless, though its first direction be considerably altered. And that 

 its mean direction may remain the same, though its extremities be sensibly 

 tinged with colours. Positions which, I know not by what mishap, have been 

 deemed paradoxes in Sir Isaac Newton's theory of light, but which are really the 

 necessary consequences of it. 

 Of Telescopical Object-Glasses giving an Image free from Colours. Fig. 8 andQ. 



If the extreme rays, the red and violet, after one or more refractions, diverge 

 from points d and d, the distance of the point of divergence of the least re- 

 frangible from the lens, being greater than that of the most refrangible, such a 

 semidiameter of the last spherical surface, from which they are to pass into the 

 air, may be assigned, as shall unite the extreme, and all the intermediate rays, 

 in the same focus f ; neglecting the aberration from the figure. The Rule is 

 this : For the distances of the points of divergence from the lens, write d the 

 greater, and d the least ; the semidiameter of any of the given surfaces being as- 

 sumed for unity: and -, - expressing the ratios of the sines of incidence and 

 refraction, of the violet and red rays, out of air into the last medium whose sur- 

 face is required; the semidiameter of that surface will be ~~d~ ' ^® "^^^ 



be easily demonstrated from a theorem of Dr. Smith, in the remarks subjoined 

 to his Optics. Thus, if the last medium be glass, the semidiameter of the sur- 



