Astronomical and Nautical Collections, 211 



surface. In the same manner, if we consider two elementary 

 refracted rays, proceeding from the points G and D, and tend- 

 ing to a point infinitely distant, in the directions GK and DL, 

 and suppose DM to be perpendicular to them, GM will be 

 the portion of the path that the ray GK must describe, be- 

 yond its companion, reckoning from the surface, in order to 

 arrive at the point of meeting. These rays will therefore 

 arrive at this point in equal times, if the light describes the 

 distance GM in the same time as ID : now it is clear that 

 this can only happen if the two spaces are proportional to the 

 lengths of the undulations of light in the respective me- 

 diums ; or, representing the lengths of the undulations by d 

 and d, if we have GM. -.Til^d' : d. But, taking GD for 

 the radius, GM will be the sine of the angle GDM, and ID 

 the sine of the angle IGD : but IGD is equal to the angle of 

 incidence IDP, and GM to the angle of refraction QDL; 

 consequently the sine of the angle of refraction must be to 

 that of the angle of incidence as d is to d, in order that the 

 two elementary refracted rays, which we are considering, 

 may perfectly agree at their point of meeting : and this con- 

 dition being equally fulfilled by all other elementary rays 

 proceeding from the different points of the surface AB, which 

 are reunited at the same point, all their undulations will per- 

 fectly coincide in this point, and will co-operate in their 

 effect. It would be otherwise with the elementary rays G^ 

 and D/, tending also to a remote point, but in a different 

 direction ; for then Gm, being greater or smaller than GM, 

 is not described in the same interval of time as ID, and one 

 of these rays must necessarily be in advance of the other ; 

 now G may always be taken at such a distance from D, that 

 this difference in their paths may be precisely equal to half 

 the length of an undulation : so that for every elementary 

 ray D/, which departs from the direction DL, there is always 

 another ray G^ directed towards the same point, which dif- 

 fers from it in the length of its route, by half an undulation;, 

 and whatever may be the law of variation of the intensities of 

 the elementary rays which would, originate in the agitations 

 at G and at D, as proceeding in difierent directions, when 

 separately considered, it is clear that, the circumstances being 



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