1 



OPTICS. 



OPTICS. 



63 



be the object, A o will be the image, and their proportion maybe easily 

 calculated, for A G : a g : : c A : c a, that is, as p : p', or as A : A , which 

 we have seen is the same ratio. 



Example 4. In the concave speculum of two-foot radius, an object 

 is placed within 6 inches of its interior surface : how far will the 

 image appear at the back of the speculum, and how much will it seem 

 enlarged .' 



1 I ' X 



Here r = 2, in feet, A'= , and since , - = - we have 



2 = 1, therefore A = I, or the image will appear a foot behind the 



convex side and will be enlarged in linear dimensions as A to A', that 

 is, as 2 to 1 ; in surface 4 : 1 ; in volume 8 to 1. 



Example 5. An object is placed 10 feet distance from a convex 

 speculum of 3 feet radius ; find the position and magnitude of its 



112 1 23 t 



Here r=3, A=10, therefore / TQ = 31 whence^ = g^, therefore 



7 

 A' = 1 55 feet, or 1 foot 3 inches, 8 parts nearly, at which distance in 



|M 



the concavity of the speculum the image will seem to be, and (in linear 



30 

 dimensions) Object : Image : : A : A' : : 10 : ^jj, that ia as 23 : 3 ; the 



surfaces as 529 : 9, Ac. Thus the reader with only a moderate know- 

 ledge of simple equations will be able to solve all questions relative to 

 the images of objects formed by spherical specula, concave or convex. 

 The images in the last two examples are erect. Generally the image 

 will be erect or inverted according as one of the conjugate foci is 

 between the principal focus and surface, or between that point and the 

 centre ; and this will include all cases, for it is easily seen that in 

 every case one of the foci is in some part of the radius between the 

 centre and surface. 



In the preceding calculations, we have confined ourselves to such 

 rays as fall nearly perpendicularly on the reflecting surfaces. The rays 

 which are at a considerable distance from the axis of a spherical 

 speculum are not reflected accurately to the same point as those 

 incident near the axis ; hence arises a diffusion of the reflected rays 

 arising from the sphericity of the speculum and denominated the 

 spherical aberration ; and when measured along the axis, it is called the 

 longitudinal aberration ; but when perpendicular to it, through the 

 focus, the lateral aberration. It will be sufficient in this article to 

 calculate the amount of these aberrations in the most usual case when 

 the incident rays are parallel, as those which proceed from the heavenly 

 bodies. 



Let 8 D represent a ray falling parallel to the axis c B ; B D being the 



Flir.S 



intermediate arc of the section of the speculum, D a the reflected ray; 

 if this figure revolve round c B, it is evident that all rays incident on 

 the annulus through which o moves will likewise be reflected to a, 

 which is therefore strictly the focus of that annulus. Now r, the 

 middle point of CB, is the point to which rays falling near the 

 axis are reflected; hence or is the longitudinal and tb the lateral 

 aberration corresponding to the above annulus. To calculate 

 the amount of these we may observe that the angle SDC (of inci 

 dence) is equal to CD a (of reflection), and also to DC a (by the 

 theory of parallels) ; and since the angles a O C, a c D, are thus 

 equal, therefore c a = a D. Let D T be a tangent at D, thcu a o T anc 

 o T D, being respectively the complements of a D c and a c D, are also 

 equal, whence aT = ar>, but also co = ou, therefore a is the middle 

 point of c T ; and since r is the middle of c B, it follows that a P is the 

 half of B T ; thus the longitudinal aberration is known ; and since the 

 angle r a >> is the double of D c B, the lateral aberration is from thence 

 known. Let the angle D c B = 9, and radius c B = r, then c T = r sec. 9 

 and BT = r (sec. 8 1), hence we obtain the exact values of the two 



aberrations, namely, the longitudinal = j (sec. 61), and the latera 



rtan. 2 9 

 16= o (sec. 91). Hence in order that the aberrations may 



DO inconsiderable, we ought to have the extreme magnitude of 6, 

 namely, the angle B o d (in fiy. 3), also small. On this supposition 

 ormultc sufficiently approximate may be deduced from the above and 



t 

 >etter adapted for practice. For sec. 9 put 1 + 5 , and for tan. 2 9 



put 29, which are respectively the approximate values; thence 



r.e- B D* 

 ve get, longitudinal aberration =7- = i^7> and lateral aberration 



r.9 3 B D 3 



= n J3 1 both of which are evidently very small, particularly 



,he latter. The least circle of aberration is the smallest that would be 

 iormed on a card placed perpendicular to the axis near the focus F to 

 receive the reflected rays ; now if the intersection g of a reflected ray 

 D rj with the final one d ij be taken the most remote possible from the 

 axis c B, it is evident that all the other reflected rays will pass between 

 g and the axis, and hence the perpendicular distance from g to the 

 axis is the radius of the circle of least aberration or diffusion. The 

 question is thus reduced to one of maxima and minima, and may 



easily solved in the usual manner by means of the Differential 

 Calculus. 



We have hitherto considered only such [rays as fall nearly perpen- 

 dicularly on the reflecting surface ; but since raya fall at all incidences 

 Erom a luminous point, each pencil of rays, at whatever incidence, 

 when in a plane passing through the centre of the spherical reflecting 

 surface, after reflection converges to or diverges from some point in 

 that plane (unless reflected parallel) ; a line of light containing all such 

 points in that plane forms a caustic line, and in all possible planes con- 

 stitutes a caustic surface. Caustics formed by reflection have been 

 distinguished from those produced by refraction by giving the name of 

 diacaustics to the latter and catacaustics to the former, in the same 

 manner in which that part of optics connected with refraction has 

 been denominated dioptrics, and that with reflection catoptrics. In 

 both cases the caustic line is the curve which each reflected or refracted 

 ray touches : hence the equation to the caustic curve, whether produced 

 by parallel, diverging, or converging rays, is easily obtained by taking 

 the equation to one of the reflected or refracted rays, and then apply- 

 ing the differential calculus to find the curve touched by all such 

 straight lines ; namely, differentiate the equation to the reflected or 

 refracted ray relatively to the constant in its equation, and then 

 eliminate the constant between the two equations, which will then 

 produce the equation to the caustic. In the general case of a system 

 of rays no longer symmetrical about an axis we still have caustic sur- 

 faces, but the problem is then one of greater complexity. If the sun 

 or a candle shine on a vessel containing liquid, and polished in the 

 interior (such as a china cup of tea), a caustic line of light will be 

 observable on the surface of the liquid, which line is a horizontal 

 section of the caustic surface by that of the liquid. 



We now come to the consideration of refracting transparent media ; 

 and if we suppose the constant ratio of the sine of incidence to that 

 of refraction to be as^m : 1, then m is called the index of refraction for 

 the particular medium employed, the incident light being supposed to 

 pass from vacuum. But if the light pass from the medium into 

 vacuum, then the ratio of the sine of incidence to that of refraction 



will be as 1 : m, and will be the refractive index ; m is evidently 

 greater than unity, since the ray after entering the medium is turned 

 towards the perpendicular, and is less than unity, because after 



the ray emerges from the medium into vacuum, it is turned from the 

 perpendicular. If m be the index of refraction from vacuum into one 

 medium which we may call A, and m' that from vacuum into a different 



medium B, then ia the index when the light passes from the 



wi 



medium A to B. [LioiiT.] A table of refractive indices is given in 

 OPTICS, PRACTICAL. 



Diverging rays fall from vacuum on the plane surface of a uniform 

 and transparent medium : it is required to find the relation between 

 the conjugate foci. 



Let A be the focus of incident rays, D B E the surface of the medium, 

 AB a perpendicular on DE, A o an incident ray near this perpendicular, 

 a c (A o produced) the course of the ray if unrefracted, a c its actual 

 course nearer to the perpendicular than G c, then c a to an eye placed 

 in the medium will appear to proceed from the point a, the con- 

 jugate focus ; the question is to determine the relative situations of 

 a and A. 



Let AB = A, OB=A', and m be the index of refraction ; then BAG, 

 the complement of A o B, is equal to the angle of incidence, and B a a to 

 that of refraction. Let these angles be respectively denoted by I and 

 B, and Ba = t; then i=A tan. I, and also to A' tan. n, therefore 

 Atansini cosB cosR . 



A tan. R sin. R cos. I cos. I 



angles I and R are exceedingly small, and their cosmos may be taken as 



unite, in which case ^ =m, therefore A'=mA; and since m is greater 



A 



than unity, A' is greater than A in the ratio of m : 1. Conversely, 

 if a ray from a medium bounded by a plane surface pass into vacuum, 



