We have hitherto considered only such :rays as fall nearly perpen dicularly on the reflecting surface ; but since rays fall at all incidences from 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 forma 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 caticau.sties 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 theu produce the equation to the caustic. In the general case of a system of rays no longer symmetrical about au 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 • eine of incidence to that of refraction to be A11:7/4 : I, 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 1 will be as 1 : in, and will be the refractive index; at is evidently greater than unity, since the ray after entering the medium is turned 1 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 an be the index of refraction from vacuum into one medium which we may call a, and ve that from vacuum into a different 7s, medium it, then — is the index when the light passes from the medium A to B. [Limn.] 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, n BE the surface of the medium, n a perpendicular on DE, A a an incident ray near this perpendicular, • (A sa produced) the course of the ray if unretracted, c c its actual course nearer to the perpendicular than a c, then co 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 A B= A, an= Ade, and M. be the index of refraction ; then n A 6, the complement of A a D, is equal to the angle of incidence, and B a o to that of refraction. Let these angles be respectively denoted by I and • and s a = k ; then k=a tan. 1, and also to A' tan. te, therefore a' tan. 1=8" cos.— but when C is very near n, the A tan. it sin. It cos. I cos. I angles i and it are exceedingly small, and their cosines may bo taken as units, in which case = m, therefore A'=a1.1; and since nt is greater than unity, A' is greater than A in the ratio of nt :1. Conversely, if a ray from a medium bounded by a plane surface pass into vacuum, then the index of refraction, and wo should have Le= . A, in es ra which case es' is less than A. This explains why the bottom of a clear river seems 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 Is, and join 13 a, then since a' n' B' :an:an:: : 1 ; therefore a' is the focus conjugate to A', and consequently a a' is the imago of A A'. It 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 D e, 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 many familiar optical phenomena may be understood. Thus, when a straight stick is partly immersed in water, the image of the immersed part being raised nearer to the surface than the true object, will cause the stick to 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 image 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 represent a plane section of a double convex lens, that plane including the centres c c' of the bounding surfaces D n e and D n'E; let A (in the axis c o') be the focus of incident, and a of the emergent rays. Let vs be the index of refraction for incident, and therefore 1 en for emergent rays ; and let aotta represent the track of a ray near the axis; let CD=r; de=r',AB=A,aII'=4, and the thickness is IV= t, we have to determine the relation existing between these quantities. First we have sin a o c=m sin xoc, let GA C=0, c a= a, and the inclination of K a to the axis be se ; while 8 is the distance of the point at which o it cuts the axis from a; then the above equation is the same u sin (a + ci)= nt air (ep + a), from which by trigonometry we deduce sin 0—st sin cos sa—cos 0 „, sin 0 c o , ll esow of which the .