OPTICS. 



L U then the index of refraction, and we ihould have A'- .A, in 



_ P% 



which cue A' U leu than A. Thii explain* why the bottom of a clear 



i nearer to the surface than it really is by about one-fourth 

 of its true depth. 



The image of a straight line in vacuo seen from such a medium will 

 be another straight line ; for let A A' be such a line, produce it to p, and 

 join D a, then since a' B' : A' B' : a B : A B : : : 1 ; therefore a' is the 

 focus conjugate to A', and consequently a a' is the image of A A'. . 

 must however be observed that A A' must be of small dimensions, in 

 order that the rays reaching the observer's eye may be considered as 

 nearly perpendicular to DE, otherwise the above proportion would 

 require to be modified, and the image would be curved. In the above 

 case the image a a' is more remote from the surface D E than the 

 object ; but the contrary happens when the object is in the medium, 

 when the image will be nearer the surface than the object is. Hence 



Thus, when a 

 immersed 



..___. will cause 



thYittidTto appear bent or broken, as well as shorter than it really is ; 

 but when immersed perpendicularly to the surface, the stick appears to 

 be only contracted about one-fourth of the part immersed, for the imago 

 and object are then in one straight line. 



As refracting media bounded by a single curved surface rarely if ever 

 can occur in practice, we shall proceed to consider lenses, particularly 

 the double convex lens, as known most generally. For their various 

 species and for further details see LENS. 



Let D B E tf represent a plane section of a double convex lens, that 

 plane including the centres c c" of the bounding surfaces DUE and D B'E ; 

 let A (in the axit c d) be the focus of incident, and a of the emergent 



rays. Let m be the index of refraction for incident, and therefore 

 * m 



tor emergent rays ; and let A o K a represent the track of a ray near the 

 axis; letcB = r; dtf=r > , AB = A, an' = A,and the thickness BB'=<, 

 we have to determine the relation existing between these quantities. 

 First we have sin AOC=W sin K.OC, let QAO=, OCA = O, and the 

 inclination of K o to the axis be <f> ; while 8 is the distance of the point 

 at which o K cuts the axis from B ; then the above equation U the same 



1) 



, from which by trigonometry we deduce 

 sin_mcos-ooi NQW nJ = oO ( 

 rin a coca sina AO 



mate value, when a U at B, U -? = , for the same reason the ultimate 



AB A 



value of "" ^ is -, and the ultimate values of cos t, cos $ and cos a, 

 sin a S 



an each unity ; therefore we get - -^=m-l, or -I -^.^ni ; 



A o A o r 



hence J, which determines the focus of the first set of refracted rays, is 

 known ; and therefore also 4 + f , which is its distance from the second 

 surface. Now, since the ray would traverse the same course if we sup- 

 posed it to commence at a, and proceed through a K o to A, it follows 



in the same way that _+-!!L 

 A o -f t 



-, from whence A' in known. 



OPTICS. 4 



If we neglect ( as being small, we may eliminate ^, and thence obtain 

 1_ + J_=(m 1)( - 1 , + LV- the spherical aberrations may be found 



by a similar prooew to that we have employed for reflection, and the 

 inverse or erect positions of the images ascertained by the like method. 

 When we have one side plane, we have only to suppose r infinite, 

 and when concave to suppose r negative, and thus one general formula, 

 by proper attention to the signs, may be made to apply to all forms 

 of lenses. 



There is a cause of aberrration for refracted light, which does not 

 exist for reflected rays, and it is of more consequence in deforming and 

 colouring images than all the effects of spherical aberration. The chro- 

 matic dispersion [DISPERSION] arises from the fact that all the coloured 

 rays which compose solar or other light have different refractive indices 

 for one and the same refracting medium ; hence the prismatic spectrum, 

 which only consists of successive circular images of the sun, of the 

 different colours of the rays, overlapping each other. This aberration 

 has, by the successive labours of Dollond, Fraunhofer, and others, been 

 successfully combated. [OPTICS, PRACTICAL.] 



The possibility of correcting the aberration arising from the unequal 

 refrangibility of rays of light in the refracting telescope, by means of a 

 couiiKnmd object-glass, permits the employment of object-glasses of a 

 much larger diameter, in comparison with then- focal length, than nmld 

 otherwise have been tolerated. Now for a lens of given focal length, 

 the spherical aberration increases rapidly with the aperture. It 

 becomes therefore of importance to correct as far as possible this 

 aberration by giving appropriate forms to the surfaces, or by combining 

 together two or more lenses. 



Writers on optics show that a curve-line by whose revolution about 

 an axis there shall be described a surface which, bong tliat of 

 a refracting medium, will cause all rays incident uiwn it, when they 

 diverge from or converge to one point, to be refracted so as to converge 

 to or diverge from one point is, in its most general form, of the fourth 

 order : but when the radiant point is at an infinite diet-iuce from the 

 refracting surface, as when it is at a celestial body, the form of the 

 surface, supposing the density of the refracting medium to be greater 

 than that of the medium which surrounds it and in which are the 

 incident rays, is proved to be that of a spheroid : and rays falling on 

 its convexity, parallel to the major axis of the spheroid, would, within 

 the medium, converge accurately to the focus most remote from the 

 place of incidence. If the refracting medium were less dense than that 

 in which are the incident rays, the surface would be that of an hyper- 

 boloid. The semi-transverse axis both of the spheroid and hyper- 

 boloid must be, to the excentricity, as the sign of the angle of incidence 

 is to the sign of the angle of refraction : in the former case the 

 refraction is from the surrounding medium into the spheroid ; and in 

 the latter, from the concave surface of the hyperboloid into the 

 surrounding medium. It follows therefore that if a meniscus lens 

 denser than the surrounding medium have it* anterior surface sphe- 

 roidal, and its posterior surface that of a sphere whose centre is at the 

 further focus of the spheroid; since the rays will then suffer no 

 refraction in pausing through the posterior surface, it will be aplanatic. 

 Also if the anterior surface of a medium be plane, so that the parallel 

 rays incident perpendicularly on it may suffer no refraction in entering, 

 and the other surface be part of a hyperboloid, the medium between 

 the surface being denser than that which surrounds it, the pkne lens 

 thus formed will be aplanatic ; the refracted rays converging to the 

 opposite focus of the hyperbola. 



The form of the expression for aberration, when parallel rays are 

 incident on a lens of moderate aperture having spherical surfaces, is 

 such that the aberration cannot be mode to vanish with any real values 

 of the radii of those surfaces unless the index of refraction in tho 

 medium be equal to or loss than 0-25. But there is in nature no 

 medium which has such a refractive index , therefore, tailing in 

 making spherical lenses strictly aplanatic, and it having been found 

 impossible, hitherto, to form them with surfaces produced by the 

 revolutions of conic sections, mathematicians have investigated expres- 

 sions for the form under which, with a given refractive index, the 

 aberration of the focus shall be a minimum ; see the article " Light " 

 in the ' Encyclopaxlia Mutropolitana ' (art. 305), where the 

 between the radii of the surfaces, in this statu, is given. From that 

 ratio it is shown that, when the index of refraction is 1'5 the lens 

 should be of the double convex form, having the radius of tho posterior 

 surface six times as long as that of the anterior surface, or that which 

 is nearest to the radiant point. 



But although tin- sphericd aberration for a pencil of paralli 

 cannot be made to vanish in a single lens bounded by spherical 

 surfaces, the compound lens employed in an achromatic object glass, 

 designed in the first instance to correct the chromatic aberration, h.-n 

 thefurther advantage of enabling us to render insensible the spherical 

 aberration. Sm-h a lens present* us with four spherical surfaces, the 

 three independent ratios between the radii of which enable us to 

 three conditions, while the scale of the system is determined l.y tho 

 focal length of tho combination. Tho most important condition to 

 satisfy is, that the chromatic aberration shall be corrected ; and it we, 

 further assume that the spherical aberration (or rather its loading 



