12 



OPTICAL INSTRUMENTS. 



I C3 \: this circle is called the aberra- 



\bd 2 ) 



tion of latitude. 



(20.) The aberration produced by a 

 lens with a spherical surface is shown 



in fig. 18, where aB c is a section of a 

 plano-convex lens. Let the plane side 

 be exposed to parallel rays, and let a A 

 be an extreme pencil of rays ; D the 

 centre of curvature ; D / the axis of the 



Fig 18. 



lens ; and F the focus of a slender pen- 

 cil of incident rays, at an infinitely 

 smaller distance from the centre. Now, 

 as the extreme ray a A is perpen- 

 dicular to the plane surface, it will 

 pass directly through to the convex 

 side, where it will be refracted to /, 

 crossing the axis in that point, for D A 

 is perpendicular to the curve at a, and 

 a D the sine of incidence, n D the sine 

 of refraction ; hence, an image of the 

 object will be formed at F by the cen- 

 tral rays, and another image of the same 

 object will be formed at / by the ex- 

 treme rays ; while, from F toy, images 

 of the same object will be formed by the 

 intermediate portion of the lens. The 

 longitudinal aberration F/ bears a cer- 

 tain ratio to the thickness or versed sine 

 B P ; and when the lens is placed in 

 the position shown in the figure, it is 

 equal to | or 4& times B P ; this quan- 

 tity will be decreased, when the curved 

 surface of the lens is exposed to paral- 

 lel rays, that is, when the refraction of 

 the first surface is made nearer the per- 

 pendicular, or when the ray is bent in 

 passing from a rare into a dense me- 

 dium, and this difference out of air into 

 glass, will be in the proportion of 27 to 

 7 ; so that when the convex side is 

 placed next the radiant, the longitudinal 

 aberration will be only I of the thick- 

 ness BP, or 1.166. 



When a crossed convex lens is used, 

 the proportions of the radii of whose 

 surfaces are as 1 to 6, and the most 

 convex side is exposed to the distant 

 radiant, the longitudinal aberration will 

 be the least possible quantity ; viz. if. 

 or 1.0714 of the thickness of the lens, 

 When the radii of a double-convex 



lens are equal, the aberration is | of its 

 thickness ; therefore, this lens is not so 

 good as a plano-convex of the same 

 thickness, in its best position. The 

 longitudinal aberration F / increases as 

 the square of the aperture, when the 

 curvature of the lens is not altered; 

 and is inversely as the focal distance, 

 when the aperture is constant. 



The lateral aberration, which is the 

 actual confusion of the image at the 

 focus of central rays, is equal to the 

 longitudinal aberration F/, multiplied 



by~ , or the aperture of the lens di- 

 13 r 



vided by the focal distance, which is 

 equal to K H. Now, if rays are drawn 

 from the different parts of the lens, it 

 will be found that they will be refracted 

 through a small circular space I R, 

 whose diameter will be \ of K H ; hence, 

 this point must be considered as the 

 focus of the lens. The lateral aberra- 

 tion of lenses increases as the cube of 

 the aperture, if the radius remain the 

 same, or inversely as the square of the 

 radius when the apertures are the same. 

 These laws may be considered as de- 

 termining the relative aberration of all 

 lenses ; yet it is found that if we employ 

 media of different refractive powers, 

 and form each into lenses of like curva- 

 ture, the separation or spreading out of 

 the rays at their focal point will be dif- 

 ferent ; that possessing the highest 

 refractive power producing the least 

 aberration, though its amplifying power 

 will be greatest : thus,' if three lenses 

 were ground in the same tool, one of 

 plate glass, and the others of sapphire 

 and diamond, they would possess very 



