OF DIOPTRICS AXD CATaPTRICS. 



73 



faces, that — r:-T+-,or — zz— ■ — - (421} ; and e— 

 c J a e J a 



f±d 



423. Definition. The centre of a lens 

 is a point, between which and the centres of 

 the surfaces, segments of the axis are inter- 

 cepted, proportional to the respective radii, 

 and lying on the concave or convex sides of 

 bolh surfaces. 



424. Theorem. All rays, which in their 

 passage through the lens, tend to the cen- 

 tre, are transmitted in a direction parallel to 

 their original direction. 



LetABpass *. A \ // 



through the g ANy(^^ jj cVi-LJL 



centre C, and 

 join AD and 

 BE;thensince 



CD : E : ; AD : BE, AD is psrallel to BE ; and the sur- 

 faces at A and B being also parallel, the ray is equally re- 

 fracted in contrary directions at A and B. 



Scholium. In some cases, the optical centre may be 

 without the lens, but no practical inconvenience results 

 from considering it as always within the lens, especially 

 when the thickness is evanescent ; and then the two pa- 

 rallel directions of the rays passing through it must coincide 

 in the same line. Now when the focus of incident rays is 

 removed a httle from the axis, the inclination of each ray to 

 the surface being increased or diminished nearly alike, their 

 mutual inclination after refraction or reflection remains but 

 little changed, and the conjugate focus is nearly at the same 

 distance as before. Hence we may find the place of the 

 conjugate focus of a point without the axis ; for since the 

 ray, which passes through the centre of the surface or lens, 

 preserves its rectifmear direction, the focus must necessarily 

 be in this line, and at the distance already determined for 

 rays in the direction of the axis : and thus we have the 

 magnitude, as well as the place, of the image of any object, 

 sufficiently near the truth for common purposes. 



425. Theorem. When a ray of light is 

 refracted at the surface of a sphere, the inter- 

 sections of the incident ray with a concen- 

 tric sphere of which the diameter is greater in 

 the ratio of the index of refraction to unity, 



VOL. II. 



and of the refracted ray with another con- 

 centric sphere which is smaller in the same 

 proportion, are in the same radius. 



LetAB:AC::AC:AD:: 1 :r; 

 then the triangles ABC, ACD are 

 equiangular, and ^ACD~ABC. 

 But sin. ABC : sin. .\CB :: AC : 

 AB :: T : 1, and ACB is the angle of refraction correspond- 

 ing to the angle of incidence ACD. This theorem affords 

 an easy method of constructing problems relative to sphe- 

 rical refraction. 



Scholium. It may easily be shown, that if the ray CD 

 were reflected at D, it would meet the ray CE at E ; and 

 supposing the velocity greater in the rarer medium, in the 

 ratio of the densities, it would arrive there in the same time ; 

 and if DE were again reflected at E, it would coincide 

 with CE again refracted. 



426. Definition. When a pencil of 

 rays falls obliquely on the surface of a 

 sphere, the point towards which those rays, 

 which are situated in any plane passing 

 through the axis, are made to converge, may 

 be called the peripheric focus. 



Scholium. These points form a line of concourse, 

 which is a part of the circumference of a circle; and this 

 is the focus at which the image of a circular circumference 

 becomes most distinct. It has hitherto been in general ex- 

 clusively considered, under the name of the geometrical 

 focus of oblique rays. 



427. Definition. The focus of colla- 

 teral rays, situated in a conical surface hav- 

 ing the same axis with the sphere, may be 

 called the radial focus. 



Scholium. It is obvious that the rays of the collateral 

 planes, which are always perpendicular to the surfoce of the 

 sphere, can only meet in the axis: therefore the points in 

 which the collateral rays of a pencil meet, constitute a 

 portion of the axis. The image of any radiating lines, cross- 

 ing the axis, must evidently be most distinct at the radial 

 focus. 



428. Theorem. When rays fall ob- 

 liquely on a spherical surface, the index of 

 refraction being r, the actual cosine of in- 

 cidence t, the cosine of refraction u, and the 



