Digitized by the Internet Archive in 2017 with funding from Getty Research Institute https://archive.org/details/elementarytreatiOOcodd «*• O P 1 I c s. AN ELEMENTARY TREATISE ON OPTICS. BY HENRY CODDINGTON, M. A. FELLOW OF TRINITY COLLEGE. SECOND EDITION. CAMBRIDGE: Printed by J. Smith, Printer to the University; FOR J. DEIGHTON & SONS; AND G. B. WHITTAKER, A VE-MARIA LANE, LONDON. PREFACE. It has been for some time a subject of complaint that there existed no easy Elementary work on Optics, suited to the present state of Mathematical knowledge. The works of Newton, of Harris, of Smith, contain, it is true, a vast deal of important information, but that information is conveyed in such a shape as hardly to be tangible to modern readers. Perhaps it may be permitted to say that objections of the same kind have been made to Dr. Wood’s elegant little Treatise, which being composed after the model of those mentioned, does not harmonize, if I may be allowed the expression, with the other mathematical works which are at present the object of study in our University. This consideration induced Mr. Whewell to draw up in the Spring of 1819 a Syllabus of those parts of the Science of Optics that are usually inquired into, and that Syllabus, with the instructions conveyed in his Lectures, which I had then the pleasure of attending, formed the basis of the little Treatise which, at the instigation of himself and others of my friends, I have ventured to offer to the public. PREFACE. «j iv Upon (lie whole, this Work is to be considered as a compilation rather than an original production, and all the merit 1 can lay claim to is that of having ar- ranged the materials which 1 found at hand, and endeavoured, with what success I leave it to the reader to judge, to present facts and reasonings in such language as should be at once concise and clear. I have studiously avoided long and diffuse investiga- tions and demonstrations, leaving many things to be supplied which an intelligent reader can readily sup- ply, in order that the several steps of a process might be seen as it were with one glance. For a similar reason I have often omitted to notice parts of Figures which did not seem to me to require being mentioned, as my object was not to give de- tailed descriptions from which Figures might be drawn, but merely to make intelligible those which accompany the Work. The Notation I have used, is in a great measure adopted from M. Biot, to whom I am indebted for an Appendix on the higher parts of the Science, which I have taken the liberty of translating from the e( Ad- ditions a l’Optique,” in his Edition of Fischer’s Physique Mecanique, p. 396. I have occasionally consulted Harris’s Optics ; of Smith’s I could make but little use ; Hayes’s Fluxions afforded me a good deal of information on the sub- PREFACE. V ject of Caustics. Dr. Wood’s Treatise I have naturally had almost constantly on my table ; but perhaps my greatest obligations are due to Dr. Young, whose Lectures do not need my praise or recommendation to those who wish to study this or almost any other branch of Mathematical Science. For one very elegant Article I am indebted to my excellent friend John C. S. Lefevre, Esq. M. Cuvier’s valuable Work on Comparative Anatomy, has furnished some details on the subject of the Eye, which I hope will not be thought misplaced. I am afraid it will be said that many of the Figures are too small, or not sufficiently clear : the student may in some measure correct this defect, by copying them for himself on a larger scale, which I hope he will find no difficulty in doing, with the assistance of the descriptions. In the present Edition, I have endeavoured to improve the Work in some degree, by amending some passages which I had not taken sufficient pains to make explicit, by supplying deficiencies and cor- recting inaccuracies in the plates, and by attending more carefully to errors of the press, of which I hope there will not be found many. Trinity College, Feb. 1 , 1825 . ' , ' CONTENTS Introductory Observations , p. 1. Page Eaws of Optics 1 Nature of Light 2 Gradual propagation of ditto ibid. Experiment illustrative of the Laws 3 Chap. I. Reflexion at Plane Surfaces , p. 4. Reflexion at one surface 4 two, successively ibid. Chap. II. Reflexion at Spherical Surfaces, p. 6 . Formula? ?' q^r q 1 J_ _ 2 _ 1 A + A ' ~r ~f 8 9 Fq : FE :: FE : FQ ibid. Corresponding values of A and A' ibid. Concave mirrors give convergence, convex ones divergence to parallel rays 13 Chap. III. Aberration in reflexion at Spherical Surfaces, p. 13. ?*/* qf q+f 1 (?+/)' * + (?+/)' . / (sec 6 — 1) — ( / mi 1 — 1 ,\ A = /«A ( 1 -| — tan 0* ) . \ 1 2 / Passage of a ray through a medium bounded by parallel plane surfaces 47 A"= A - m — 1 m T. Passage of a ray through a prism 4S h =

> Mr. Herschel’s definition of the power of a lens , ibid. Elegant enunciation following from that definition ibid. Successive refractions at several spherical surfaces 69 Chap. IX. Aberration in refraction at Spherical Surfaces , p. 70. a / / m « = ( A ( 3 m 1 A~ A 7 Case in which there is no aberration Aberration in a lens Lateral aberration Least circle of aberration Its diameter is equal to half the lateral aberration of the ex- treme ray 70 72 75 ibid. 76 Chap. X. Refraction at curved Surfaces not Spherical, p, 76. Accurate refraction in a spheroid and hyperbolic conoid .... 76 Newton’s Propositions 78 d u -j- m d v — 0 ibid. CONTENTS. IX Chap. XI. Caustics produced by Refraction , p. 80. Page Caustic to a plane refracting surface 80 Intersection of rays after oblique refraction .. 81 u r cos . tan

5 Crystals with two axes of double refraction 1 67 M. Biot’s machine for measuring extraordinary refraction. . . 169 Equations for double refraction .. 174, 175 Polarization of Light. 176 Original experiment exhibiting the phaenomenon ibid. M. Biot’s Apparatus ibid. Dr. Brewster’s observation, that the refracted ray when polarized is perpendicular to the reflected 180 Axes of polarization and translation 181 Moveable polarization 182 Table of densities and refractive and dispersive powers 188 Addenda. Ramsden’s Dynamometer 190 Method of determining the field of view of a Telescope... . 191 Miscellaneous Questions 192 \ \ INTRODUCTORY OBSERVATIONS 1. Concerning the nature of light, very little is known with any certainty ; fortunately it is not at all necessary in ma- thematical enquiries about it, to establish any thing about its constitution. The science of Optics reposes on three Laivs, as they are technically termed, which depend for their proof upon Observation and Induction. 2. In the first place, as it is observed that an object cannot be discerned if it be placed directly behind another not transparent, we conclude that the action of light takes place in straight lines. These straight lines are called raps, and are the sole object of dis- cussion in the following Treatise. 3. When a small beam of light, admitted through a hole in the shutter into a dark room, falls upon a plane polished surface, such as that of a common mirrror, it is observed to be suddenly bent back, or reflected, according to the technical phrase, and as it has otherwise the same appearance as before, we conclude that each ray of light is bent at the point where it meets the surface, or that more properly for each ray that existed in the beam, we have now two, an incident and a reflected ray, meeting in the surface. Observation leads us to conclude that these rays are invariably in the same plane, and that they as invariably make equal angles with the reflecting surface, or with a line perpendicular to it at the point of reflexion : the angles which the incident and reflected rays respectively make with this perpendicular are called the angles of incidence and reflexion. 4. If again we present to the beam of light above-mentioned a very thick plate of glass or a vessel of water, or any other transparent substance, we shall find that part of the light is reflected on reach- A ing the surface, hut part enters the glass or water, not however without deviating from its former direction. It is in fact bent or refracted, so as to be more nearly perpendicular to the surface, so that the angle between an incident ray and a perpendicular to the surface, called as before, the angle of Incidence, is greater than that between the refracted ray and the perpendicular ; this latter angle is technically termed the angle of Refraction. Observations similar to those alluded to in the former case lead us to the conclusion that the angles of incidence and refraction are always in the same plane, and that though they do not bear an invariable ratio to one another, their sines do, provided the observations are confined to one medium , or transparent sub- stance. 5. We have then these three laws upon which to found our theory. 1. The rays of light are straight lines. 2. The angles of incidence and reflexion are in the same plane and equal. 3. The angles of incidence and refraction are in the same plane, and their sines bear an invariable ratio to one another for the same medium. Note. Sir I. Newton attempted to explain the Theory of Optics on the hypothesis that light is a material subtance emitted from luminous bodies, and that the minute particles of this matter are attracted by any substance on which they fall, so as to be di- verted from their natural straight course. He succeeded in demon- strating the laws above-mentioned upon that hypothesis, but not so as to set the question at rest. Other philosophers, probably with more truth, have supposed light to consist in undulations, or pulses propagated in a very rare and elastic medium which is supposed to pervade all space, and perhaps to have an intimate connexion with the electro-magnetic fluid. The action of light is by no means instantaneous. ’ It has been discovered by means of observations on eclipses of Jupiter’s sa- tellites, that light takes eight minutes, thirteen seconds of time, to come from the Sun to the Earth, 3 It will perhaps be as well to detail an experiment by which the Laws of Optics may be well illustrated. Let a square or rectangle A B (Fig. 1.) of wood, or any other convenient material, have its opposite sides bisected by lines CD, EF, and be correctly graduated along the top and bottom, so that the divisions, which must be equal on both lines, may be aliquot parts, tenths or hundredths, for instance, of GC or GD. Let this rectangle be immersed vertically in water up to the line EF in a dark room, so that a small beam of Sun-light admitted through the shutter may just shine along its surface in a line OG. There will then be observed a reflected beam along a line GP on the surface of the rectangle, and a refracted one GQ down through the water, also lying just along the surface of the rectangle. Now if the distances OC, CP, DQ be observed, it will be found that OC and CP are equal, and that OC and DQ, which are re- spectively the tangents of the angles OGC, DGQ to the radius g\ or GD, are so related, that if the sines of the same angles be tan calculated by the formula sin = — . ■ -- o ■ - = X rad. these sines /y/ rad * + tan 2 will be found to be in a certain ratio, which in the case of pure water is about that of 4 to 3 , or more correctly, 1,336 to 1. The experiment should be repeated several times when the Sun is at different heights, and the ratio of the sines of the angles OGC, DGQ will be found invariably the same. The fact of the incident, reflected and refracted rays, GO, GP and GQ, all lying precisely along the same plane surface, shows that those rays are all in the same plane, which is one circumstance mentioned in the Laws. It may be necessary to observe that it is indifferent as to the directions of connected rays, which way the light is proceeding, that is, whether forwards or backwards, as any causes that act to produce a deflection from the straight course in the one case, would produce corresponding effects in the other. 4 CHAP. T. REFLEXION AT PLANE SURFACES. 6. Prop. To find the direction which a ray of light , eman- ating from a given point, takes after refiexio)i at a plane mirror in a given position. Pet QR, (Fig. 2.) represent a ray of light, proceeding from the point Q; XY, the section of the reflecting surface by a plane perpendicular to it containing the line QR ; RS , the reflected ray making with XY an angle SRX equal to the angle QRY which QR makes with the same line: let QA be perpendicular to XY, and let SR meet it in q. Then since the angle QRA is equal to SRX, that is, to qRA, the right-angled triangles QAR, qAR, having the side AR in common, are equal in all respects. Therefore q A. is equal to AQ. Any other reflected ray R'S 1 will of course intersect QA in the same point q ; so that if several incident rays proceed from Q, the reflected rays will all appear to proceed from q, which as we have seen is at the same distance behind the mirror as Q is before it. 7- Suppose now that a ray QR (Fig. 3.) reflected into the direction RS by a plane mirror III, meet in S another mirror IK inclined to the former at an angle /. It will of course undergo a second reflexion, and returning to meet the first mirror be reflected again, and so on ; so that the course of the light will be the broken line QRSTUVX. Let perpendiculars be drawn to HR IK, at the points R, S, T, V, . . . meeting each other successively in L, M, N, O, . . . Each of the angles at these points will be equal to the angle at I, since for example RL 1= RIS — RIL — RIS — SI I — l IS- Let the angle of incidence at R ( QRm or mRS) be called (f, that at S. , (p > , and so on. Let also i represent the angle at I. Then rnRS — RSL = RLS, that is, — 0 2 = b similarly LST — STM = SMT, that is, 2 = l) 02 “ — l, 03 — 04 = t, 01 (pn ~ ^ 1 b or

— = — - , - ; sin ERq q sm < p sin (0 + = - , or — shows 2 A A r A! r A that — , and consequently A\ must be negative, that is, q goes to the other side of the reflector, and the reflected rays instead of converging, diverge. When <2 comes to A, q meets it there. As a converse to the last case but one, we may take that of rays converging to a point Q behind the reflector, and reflected to a focus q in front, (Fig. 10.) To accommodate the formula to this case, we must make A negative, and we have then 1 + A 1 A' 2 r a r 1 which shows A' to be essentially positive. The student will find no difficulty in examining particular cases; 11 the one most likely to occur, is that in which the radiant point is at the opposite point of the sphere from the centre of the mirror. Making A = 2r, we tint! here A / A ?■ _ 2 2A — r 3 It will occur to every one, that of the two foci Q, and q, that which lies between E and A moves much more slowly than the other, when their places are changed ; in fact, we have seen that by merely bringing up Q from E to F, q was sent from E to an infinite distance, and that when Q moved on from F towards A, q came back from an infinite distance on the reverse side of the reflector to meet Q at A. 13. We have hitherto considered only one species of spherical reflector, the concave; let us now take the convex, (l'ig. 1 1.) where as before, E is the centre, Q the radiant point, QR } RS an incident and a reflected ray, making equal angles with ER the radius or normal. Let SR cut AE in q. Then we have, keeping the same notation as before, ER sin EQR m = sin ERQ ; . r sin (7r — d>) — 9 sin ( 9 + (j ) ) that is, as before, - = : — : — ~ — - , q sin (p sin

) + 9 sin ( 0 — d) ) that is, — = — — — = r— j ; q sm

, d r\ . 1 d ~q o 0 being made equal to 0 in every differential coefticient, or if we consider q' as a function of the versed sine of 9 , which we w ill call v, q = q' + + . . . . The brackets indicating that v is made = 0 in each coefficient. Now q ' - 7 / _ 7 f 7/ f-\-q cos 9 Z + yO — versing) q +f — qv d_l = ff_ dv (q +f- qv) 2 which when v — 0, becomes 1 fq 1 2yV 1.2 dv 14 1.2 (y +Z’— q v)' and so on, whence qf (y+ZT qf (q+ff q - sL + qf'- v a. qf- vl 4. ?V 1 /■ ,3 + + q+f (q+ff (y + /')’ (g+Z) This in geometrical terms amounts to + . _ QE- AN QE 3 TN- hq + QF ‘ ~ QF 3 ' 4 EF + The Aberration is represented by this series without its first term ; and when the angle 9, and a fortiori its versed sine, are but small, the second term of the series will give a near approximate value. Note. The above is perhaps the neatest way of obtaining the scries for the aberration : it is sometimes done by a method simpler in its principle 15 r ' _ 9/ 9/ 9 +/ 9 / A--22-V ? +/- 9 ^ 9 +/ 9 +/- i v 9 +/ v 9 - 9 / f. I 9^ , / V . - <7+? t + and tor u-j cos 0 dl p_ u p the proper function of u given by the equation to the original curve, and u be then eliminated, vve shall have an equation in u and p, which will be that of the caustic*. S < 25. In order to obtain an equation in rectangular co-ordinates, we may proceed as follows : Reasoning as before, since the caustic is formed* by the con- tinual intersections of the reflected rays, two of these are necessary to determine one point of the caustic, and the point where one of them meets the caustic, is that which it has in common with the next; so that if we refer the two reflected rays to the same abscissa, their ordinates, differing in general, coincide at this point, and as far as that point is concerned, a change in the point of the reflecting curve, or in its co-ordinates, takes place without any alteration in the co-ordinates of the reflected ray (Fig. 19-) We have therefore only to find the equation to the reflected ray belonging to an assumed point of the curve; to differentiate this, considering the co-ordinates of the curve as the only variables, and eliminate these co-ordinates between this equation, its primitive, and that of the curve. An Example will make this more intelligible. ~ To diminish the length of the process, we will confine ourselves to the simple case of parallel rays, and take one of them for our * For instance, if the reflecting curve be* a logarithmic spiral, and Q its pole, its equation is of the form p = mu , d u nd id . u v== 2 ~dp — Tu = u > v'—lmv *-/ 1 — m 2 ; ?<' 2 = 4?< 2 -4 — =4id(l-m 2 ), ~P~ « whence we find />' = »» «'; and the caustic is therefore another logarithmic spiral differing from the former only in position. 21 principal co-ordinate axis. Let AN (Fig. 20.) be this axis, AM, MP co-ordinates of the curve, PN a normal, QP, Pv an incident and a reflected ray. The question is first to determine the equation of the line Pv. Since this line passes through the point P whose co-ordinates are x, y , the equation must be Y — y = a ( X — x), a being the tangent of the angle Pv N. Now, tan PvN= — tan vPQ = — tan 2 NPQ = — tan 2 PNv. And since PN is a normal, _ dx 9 2 — T ,, T dx dii ' 2 dxdy ta " 1 NV = Ty ' •' ta " 2 FttV = —d? = ,/ v‘ ^ ./.* ' dy 1 The equation is therefore Y-y + 2 and we have to put for dxdy dy 3 — dx dxdy dy 1 — dx 2 (X-x) = 0 (1); r its value in terms of the co- ordinates given by the equation of the curve, and eliminate x and y between this, the equation (l), and its derivative. 26 . The process is sometimes facilitated by taking for the variable a function of the angle PNM, as its tangent which is d x equal to — . The quantity we have called a is the tangent of twice this angle, and if we put 6 for this angle, the equation to the reflected ray is , , tan 6 Y-y + 2. — (A — x) =0. 17 1— tanfF Exa.wple. Suppose the curve to be a common parabola , its equation is y Q = 4 ax, tan 9 = dx 1/ d y 2 ad 22 y y — 2a tan 9 ; x = — = a tan 0 . 4 a Then if we put t for tan 9, 2t Y- c 2at+~ -fX-af) = 0; Y (1 - * 2 ) + 2(X — a)i = 0. Then differentiating with respect to X— a .( 1 ). 1 t 1 -f A — a — 0 ; . " . t — ( 2 ); and substituting this value of t in (l), we have (X-a)\ , f(X-a) 0 -^) + 2 , or Y~ + (X— af = 0 ; the equation to a point, namely, the focus, where X = a. 2/. The method pursued in the following example is per- haps less elegant than that just given, but it has often the advantage of being simpler and less prolix. Required the form of the caustic, when the reflecting curve is a common parabola, and the incident rays are perpendicular to the axis. Let P (Fig. 21.) be a point of the curve MP, Pq an incident and a reflected ray. Then, taking for granted that when q is a point of the caustic, Pq is one-fourth of the chord of the circle of curvature at P, we have the following easy method of determining the co-ordinates of q. Let sin AM = x, MP = y, A n — X, nq = Y, Pq = v, MPG = (j>; J 1 +y ' / _ (l y y = y~ = dx d~ y ff cos \/ 1 +/" Sill 2iy 1 — y ' (p = r ; cos 2 (b — — —7?, ; * 1 +y 1 +/: ?/ A" = a + a sin 2 d> = x — — , V V = y + 1 -J/ 2y" So far all is general ; in the particular example proposed, i/ = 4ax] y = 2a^x^, v/ = a z x _z , y'— — \a i x~ A=ar + 2,r = 3,r, -1 T , 1 — a x *■ , , a 1 F = 2/ + x — zrr = 2 — — a a" — « 2 a a 2 3 a a 2 — a 2 « 2 9 a — A 1 = 7 r A 2 . 3 V 3« 2 From this it appears that F = 0, or the curve crosses the axis, where X = 9 a, which answers to the point in the parabola for which a = 3 a, dy _ 1 (9 a — X dx 3/3 « 2 l 2 A* 1 9a -3X 2 ‘} 6 VSaS ’ A 2 When a = 0 this is infinite, so that the caustic like the reflecting curve is perpendicular to the axis at its origin : when d Y 1 —18 a — 1 a = 9«> T= 0; d X 6 V oci 2 3 a 2 Vo The angle at which the caustic afterwards cuts the axis is -I therefore that having for its natural tangent — 7— , which shows it 0 . n Vs to be one of 30 °. The curve extends without limit in the same directions with its generating parabola. 94 28. Required the form of the caustic when the refecting curve is an ellipse , and the radiating point its centre. Fig. 22, &c. The polar equation to an ellipse about its centre being o a 2 b" P = ~ we have dV and du = - ~ + v = a 2 + b 2 - u l> a°b 2 u ( a'-j-b 2 -uj 2 u 3 u 2 -b*-n‘ ~ u ( a a~ + if - 2 - U Hence, when u = a, v = and when u — b, v = 3 u — ( a 2 + b 2 ) b 2 ii. 2 | / 9 2o 3 + /> o Cl 2b n --a' b. The former of these values is always essentially positive, since a is supposed to represent the semi-axis major , and therefore 2 a" must be greater than b~ ; but 2 b~ may be equal to, or greater than a 1 , so that when u = 6, r may be infinite or negative. When 2 b 2 > a 2 , the form of the caustic is such as that shewn in Fig. 22. \/3 When b = a, u = b gives v = 2 b, and the curve is that of 2 s / Fig. 23. When 2 b~ = a 2 , we have infinite branches asymptotic to the axis minor, as in Fig. 24. When 2 b 2 > a~, there are asymptotes inclined to that line (Fig. 25.) 29 . There are some simple cases in which it is easy to deter- mine the nature of the caustic bv geometrical investigation. Prob. To find the form of the caustic, when parallel rays are refected by a spherical mirror. (Fig. 26.) Taking as usual a section of the mirror, let Pp be one of the reflected rays, touching the caustic in p. Then, since we know that in this case Pp is one quarter of the chord, if EP be bisected in O, and Op be joined, OpP will be a right angle, and if a circle be described through the points Pp 0, OP will be the diameter of it. Let R be the centre, join Rp. Then since OPp = EPQ — PEA, if a circle be described with centre E and radius EO, cutting the axis EA in F, the principal focus, the arc OF which measures the angle PEA to the radius EF, must be equal to the arc Op, which measures twice the angle OPp to the radius OR, which is half of the other radius EO. It is plain therefore that the curve Cp F must be an epicycloid described by the revolution of a circle equal to Pp O, on that of which FO is a part : and that there must be a similar epicycloid on the other side of the axis ; moreover that if the other part of the circle, CBc, represent a convex mirror, there will be a similar pair of epicycloids formed by the intersection of the reflected rays, considering them to extend behind the mirror without limit as all straight lines are supposed to do in the higher analysis. Prob. To describe the caustic given by a spherical reflector , when the radiating point is at the extremity of the diameter. We shall see that the section of the caustic consists again of two epicycloids. Let Q, (Fig. 28.) be the radiating point. Pp a reflected ray touching the caustic in p. 11 2 It appears from the equation - T - = , that Pp is m u v r cos cp this case one-third of the chord, (since u = 2 r cos tt, 7 r — 6 or 2 n -f 1 > . If Q 12 "’ | je the last image, we must have HO U ) 7r, or 2u > 6 the same expression as before, 2 n+ 1 being the number of images in the one case, and 2n in the other. In like manner we should find, that the number of images in the other series is the least whole number greater than — — — . Tf , , .6 6 ' It t be a measure of 7 r, since ~ and — are proper tractions, L L the number of images in each series must be — , and therefore the 1 2 TV whole number of images — . In this case, however, two images £ 34 of the different series will coincide ; for if - be an even number, i some one of the distances 00" or 00„(2i). OO iv or OO iv (4t)j&c. will become equal to ir, so that 0 12 " 1 and O l2n) meet at the opposite . . . 7 r point of the circle from O ; and if - be an odd number, we must i have (2 n + ]) t = 7r, that is, 4 in + 2t = 2?r, that is, 2«i +20 + 2»t + 2 0' = 27 t, that is, 00 (2 " + 1> + 00 + =27 r. It appears then upon the whole, 'that taking in the object 0, the 271 - whole number of points visible will be — . i 42. Let us now r consider the images produced by spherical reflectors. We must of course find the focus of reflected rays corresponding to each point in the object, and consider the figure which all such foci compose by their aggregation. As a first instance, suppose there be presented to a spherical mirror, a portion of a sphere concentric with itself, (Fig. 38.) All the points of this are equidistant from the centre of the sphere, and if they be considered as foci of incident rays, the foci of reflected rays will all be at some other distance from the centre, on the same radii with them, so that the image will be a portion of a sphere like the object. 43. Prop. Let a plane bbject be placed in front of a sphe- rical mirror , so as to be perpendicular to its axis ; required the form of the image. Fig. 39 represents a section of the object and mirror, (a concave one) through the axis. PQ is a part of the section of the object having for its image pq, q being the focus of reflected rays answer- ing to Q, and consequently lying on the same diameter with it. In order to determine the form of the image p q, we will con- sider it as a spiral curve referred to the centre E as a pole, and EA as an axis. 35 Then supposing the object PQ to be beyond E the centre of the mirror, Let represent the radius EA, 0 angle PEQ, c line EP, q ( = -^) eq, V cos 0/ Eq. Then, referring to page 8, \Ve find 1 11 i 1 _ — — ~ + - cos 0. <1 j <1 J c Now the polar equation to an ellipse referred to the near focus is 1 1 , * - = —r. 57 H T. 37 COS 0, p a (l — e ) a (1 — e ) which coincides with the above, provided a (1 — e") =f, and 'iSLzil =Ci that is , c= / e c and ‘ = (T^) = rz = ^ c l It is also necessary, that e be less than unity, that is, c>f. 44. There are some things to be remarked in this elliptic image. We found the quantity a (1 — e~) to be constantly equal to the half radius. Now a (l — e l ) is half the latus rectum of the ellipse, which is also the radius of curvature at the vertex. It appears then, that although the place of the vertex p, and the magnitude of the ellipse, depend on the place of the point P, yet the curvature of the image at the vertex is invariable, as is the 36 latiis rectum of the ellipse, which passes through E, and is always equal to the radius. In order to account for this geometrically, we must observe, that rays proceeding from a point at an infinite distance from a mirror, are reflected to the principal focus which is at the middle of the radius containing that point in its prolongation. Now PQ being perpendicular to TP, a line drawn from £ to a point in PQ infinitely distant from P or E must be perpendicular to AE, and the focus for rays proceeding from such a point will necessarily b c j ', the middle point of the radius E V which is perpendicular to AE. It appears, that supposing the line PQ to be infinitely ex- tended both ways from P, and to be placed in AE produced, at a distance from E greater than half the radius, the image is a portion of an ellipse*, extending from the extremity of the axis major to those of the latus rectum ; we shall see hereafter how the ellipse may be supposed to be completed. It is, however, necessary that we examine what change takes place in the image when P is brought within the limit assigned above, namely, when EP is not less than half the radius, or further when P is placed on the other side of E. In the first place, when EP is half the radius, we have 6 ' =./; • • e a = = 00 • In this case then the ellipse changes to a parabola, (Fig. 40.) Suppose now EP be less than half the radius, c< f; a= — c e — I Here we have then a portion of an hyperbola. When P is at the centre, c = 0, e= co . comes a straight line coincident with PQ. - _ c ~f f-c~' (Fig. 41.) The hyperbola bc- * When P is infinitely distant, the image is a semi-circle far e — 0. 37 When P is between A anti E, EP or c is negative; our formula then becomes 1 _ 1 7 ~ f which answers to - cos Q , c 1 1 or - P a (l — e ') a (1 — e~) 1 e w COS 6, + a (e“ — 1 ) a (e 2 — 1 ) cos 0, the former of which is the equation to an ellipse, when the angle 6 is measured from the farther vertex, and is consequently the sup- plement of that used in the former cases ; the latter is the equation to a pair of hyperbolas. This latter case we will now' examine, as it comes first in order. Let then P, (Fig. 42.) be between E and F, (the principal focus). In the first place, we know that if k be that point in PQ for which Ek — EF, the image of K must-be infinitely distant, so that the line Ek must be parallel to one of the asymptotes. Again, if g be the point where PQ cuts the circle, its image coincides with it. Every point between k and g has its image without the circle, the distances of these images diminishing gradually from infinity to the radius. It appears then that the image in this case consists of an hyperbola, and its conjugate, wanting the part betw'een the vertex and the extremities of the latus rectum. When P coincides with F, (Fig. 43.) the image is a parabola, wanting the part about the vertex extending to the extremities of the latus rectum. The equation in this case takes the form ~ — -cos 0 = -'(l — cos 0), 9 f / - j / _ / which we know to be that of a parabola, in which the angle 0 is measured from the axis, not beginning at the vertex. When P is between F and A, (Fig. 44.) c > f ; e < 1, and the image is part of an ellipse, namely, all but that part which we found in the first case. If we suppose P to go outside of the circle beyond A, (Fig. 46.) we shall be led to the case of a convex mirror. Our equation will still be and the image will in all cases be a part of an ellipse, turning its convexity towards P. When P is on the circle at A, (Fig. 45.) the image extends from that point both ways to f, the bisection of the radii EN, EN', which are parallel to PQ. When P is at an infinite distance from A, the image is a semi- circle with centre E, and radius EF. 45 We may now show how the curve of the image, which we have in different cases found to be a part of a conic section, may be supposed to be completed. Supposing in all cases the line to be infinite in extent each way. In the first place, when P is at an infinite distance, (Fig. 47.) the semi-circle NAN' representing a concave mirror gives a semi- circular imag efFf ' ; and the convex mirror represented by NA' N' gives the image fF’f', which completes the circle. When P is at a finite distance outside the circle, the concave and convex parts give together a complete ellipse pfp'f', (Fig. 48.) When P is on the circle at A', the ellipse is such as represented in Fig. 49, where Ep is two-thirds of EF. When P is between A! and F', the ellipse cuts the circle, (Fig. 50.) When P is at F' the middle point of EA', (Fig. 51.) the two parts of the circle divided by the line, unite to produce a complete parabola. 39 When P is between E and F', the reflexion of the whole circle gives two complete hyperbolas as in Fig. 52. In the first place, the semi-circle NAN ' gives the portion of hyperbola fpf ■ The part g A' g gives the infinite branches gh, g'h', and the- conjugate hyperbola mp m\ and the former hyperbola is completed by the reflexions at Ng, N' g , considered as convex mirrors. The part kJc of the object has for its images the hyper- bola mp'm and part of fpf, namely, upf; the infinite parts of the line outside the circle are represented by fg, f'g, and by fy, f ' y \ the remaining parts kg, k' g, have for images only yu, and / / y k . 46 . In all that has preceded, we have confined our attention to sections of the mirror, object, and image ; but of course the reader will not find the smallest difficulty in inferring that the image of a plane object, made by a spherical mirror, is, according to cir- cumstances, a portion of a sphere, a spheroid, a parabolic or hy- perbolic conoid, or a plane. 47 . By referring to the figures, it will readily be seen that when the mirror is concave, the image is, in most cases*, inverted with respect to the object: a convex mirror always gives an erect image. 48. It will also be seen, that when the image is inverted, it is what is called a real image : when erect, it is imaginary . 4 9 . Let the object presented to a concave mirror be a portion of its own sphere, (Fig. 53.) Since rays proceeding from P, the extremity of the diameter, are reflected top, making Ep two-thirds of EF, and that all points of the object are equally distant from the centre, it will readily be seen that the image of the portion of sphere represented by PQ, is a corresponding portion of sphere, pq, having its radius one-third of that of the mirror. 50. Suppose now the object be a portion of any other sphere. * When the object is between the mirror and the principal focus, the object is erect, otherwise not. 40 Let AEOPj (Fig. 54J be the line joining the centres, which we .will consider as the axis of the mirror. PQ a, part of the object; p q its image. Let OP = 6; EO = c;EQ = q ; Eq — q ; QEP, or pEq=9, EF=f, Ep — p. Then we know that q = c . cos 9 + s/ b~ —c~ sin 9 2 , and if to simplify the problem, we suppose PQ and 9 to be very small, we may put for this q = c (' + b — — - 9~ = (6 + c) . (\ ‘2 b V 1 1 , 1 1 Then - =- +- =“ + <1 S ( 1 J b-j-c \ 2 b J 1 _ i 1 P ~ 7 ^ b + c ‘ It is not easy to determine any thing about the form of the curve from this, but we may deduce one rather remarkable conclusion. pq' . The diameter of curvature of pq , is equal to if p r be a qr tangent at p . Now this is equal to p sec 9 — q Hence, calling the radius of curvature p, we have 1 p sec 0 — q <2p p- P ~ ( ! , P-\ A* p- 9- ‘ / 9 l p~ r i pq 6 ~ + — - , nearly <2p J (1 \q' p' 9~ 2 p c _1_ I 2 b (b + c) + 2/ + 2 (A +c) 1 1 <2b + 2 /” 41 Now b is the radius of the object, andy' is that of the image of a straight line at the vertex; moreover the curvature of a line is aptly measured by the reciprocal of its radius of curvature. It appears then that the curvature of the image of a small portion of a sphere, is equal to that of the object, together with that of the colloidal image of a small plane object, at the same place*. 51. With regard to the magnitude of the image produced by a spherical mirror, it is easy to see that as it subtends the same angle at the centre of the mirror, that the object does, if we suppose them to be plane, an hypothesis which agrees very well with ordi- nary cases of experiment, the linear magnitudes, that is, the lengths or breadths, of the object and image, will be in direct proportion to their distances from the centre, so that, if we put L, l, for the lengths of object and image, q 0 , qj for the distances EP, Ep ; 7 o _ , 1 2 , 1 , c r /=C.-^-=L- 0 + -). 2c-j-r V r ' or if f = ~ , ■ 2 being the principal focal distance, 52. It might be expected that we should treat of images pro- duced by reflexion at surfaces not spherical, but the subject is in * We here suppose that the curvatures of the object and mirror are opposed to each other, that is, are of the same kind ; if the radii lie in the same direction, so that the one be convex, and the other concave, the curva- ture of the image will be less than in the case of a right line, the expression beinff then - — - — 7 . p f h F 42 general too difficult for an elementary Treatise like the present* As far as common practical results are concerned, we shall find it sufficient to substitute for surfaces of revolution, portions of spheres having the same curvature, and as to others, plane sections will generally give all the information desired. Suppose, for instance, the mirror were cylindrical, and convex, and the object a circle placed directly in front of it. It will easily be seen that that diameter of the circle, which is parallel to the axis of the cylinder, will not be altered in the reflexion, but that the diameter perpendicular to that axis with all chords parallel to it, will have for their images portions of conic sections of less breadth than themselves, so that the image will appear diminished in breadth, and distorted into a form like that of the bowl of a spoon. CHAP. VII OF REFRACTION AT PLANE SURFACES. N 53. We must here recall to the Student’s recollection, the details in the introduction about the manner or law of refraction, which is, that when light enters a transparent medium, its course is bent or broken in such a manner, that the sine of the angle of inci- dence, bears to the sine of the angle of refraction, a certain ratio which is the same however the angle of incidence be varied. The full and correct statement of the case is, that when a ray of light passes from one medium into another, as from air into water or glass, the refraction above described takes place at their common surface. 54. In either of the instances just mentioned, the angle of re- fraction is less than that of incidence. If the passage of the light were from water or glass into air, the contrary would be the case, and in general it is observed that when light passes from a rarer into a denser medium, the ray is brought nearer the perpendicular to the surface bounding them : when from a denser into a rarer, the converse. 43 55. There is a remarkable exception or modification to this rule in the case of combustible substances, (in which the diamond is included) which always refract much more than other substances of like densities, that is, in cases when the inclination of a ray to the perpendicular is diminished by the refraction, it becomes less for a combustible, than for an incombustible substance, but when the angle of refraction is greater than the angle of incidence, it is less increased in passing into a combustible, than into another substance of equal density. 56. Between media of equal refracting powers there is uo refraction, and in general the superiority of action of one medium over another is well expressed by the invariable ratio of the sines of incidence and refraction, when a ray passes from one into the other. 5/» We may here take occasion to observe that since when a ray of light passes out of a denser into a rarer medium, that is, out of one of a stronger into ojie of a weaker refracting power, the angle of refraction is greater than that of incidence, there is some angle of incidence for which the angle of refraction is a right angle (v. Fig. 55 .) Past that point there can be no refraction, for though we might fancy an angle of incidence greater than a right angle, there is no angle whose sine is greater than the radius. For instance, when a ray of light passes out of glass into air, the ratio of the sines of the angle of incidence and refraction is about that of 2 to 3. Here then we have sin (p = ^ sin (p r , or sin (p ' = | sin (p, and since sin (p ' cannot exceed 1, sin (p cannot be greater than |, or

// + t = 7r ; cp' + \js' = 1 , From these equations, knowing 1 and in \p = mi — (p, o = m i — t = (m — 1 ) / . 69 . From this equation we may find the value of m, known, and S observed, for if t be m — 1 = - ; 1 8 5+i rn — 1 i — — 1 1 70 . The reader may observe, that in the figure we have been using, the ray is, by the refraction, bent away from the angle I of the prism ; this is universally the case, as we may easily show. Let us take the three different cases. (1) When the angle IRQ is an obtuse angle, (Fig. Gl.) (2) When it is a right angle, (Fig. 62.) (3) When it is an acute angle, (Fig. 63.) In the first case, IRS and ISR aye both acute angles, and it is clear, that the bending of R ST is from the perpendicular Sy, that is, from the angle I. In the second there is no inflexion at R 3 but ISK is an acute angle, and therefore the emergent ray is on the far side of the perpendicular from /, and of course declining from it. \ In the third case, the deviation is at first towards the angle, afterwards from it, and we must show that the second deviation exceeds the first. G 50 Supposing the ray to proceed botli ways out of the prism at R and S, the angle of incidence at R, SER, is less than that at S, namely, pSR, the latter being the exterior angle of the triangle SER, the former an interior and opposite angle. Now the greater the angle of incidence, the greater is that of deviation, for if cp, cp' be the angles of incidence and refraction, -<£' is the deviation. Then since sin cp = m sin COS fi') and d$ = d(b h “ 7 . Vf • I cos cos yj Now this will clearly be equal to nothing, when (^) and v// are equal, as their cosines and those of — m sin 1 ; m — — — — . sin ( 53 with the incident ray, and that inside the liquid prism, although the relative positions of the eye and object will not be the same, as if there were no refraction but that of this latter prism. We may sometimes save the trouble of grinding a solid sub- stance into a prismatic form, by placing it in a fluid of the same refracting power as itself, which, in fact, amounts to using instead of the substance, another of the same refracting power, and appears to involve a petition of the point in question, namely, the refracting power of the substance ; but it is not so in practice, because by placing a bit of any transparent substance in a dense fluid nearly of the same colour, and diluting this with a rarer fluid, we shall soon see when the fluid is reduced to the same density as the solid, by there being no irregular refraction caused in light passing through the liquid and the solid, which, in fact, will become in many cases quite invisible in the liquid. It is evident, that it is quite indifferent in making the optical experiment afterwards, whether the light pass through the bit of the solid substance or not. Canada balsam, diluted with spirit, is a convenient liquid to use for solid substances of small densities. For further particulars, I beg leave to refer the reader to Biot’s Physique, vol. III. Dioptrique, Chap. I, or to Dr. Brew- ster’s Treatise on new 7 Philosophical Instruments. CHAP. VIII. REFRACTION AT SPHERICAL SURFACES. 75. Prop. A RAY of light is refracted at a spherical sur- face, bounding two different media; given the point where it meets the axis; required the point where the refracted ray meets the axis. Figs. 66, 67, 68, 69 , represent four different cases. (1) A denser refracting medium with a concave surface. Fig. 66. (2) A denser medium with a convex -surface. Fig. 67. 54 (3) A rarer medium with a concave surface, Fig. 6S. (4) A rarer medium with a convex surface, Fig. 69. In all these figures QR is the incident ray meeting the axis AE in Q ; RS the refracted ray meeting AE in q. E is the centre of the surface. Let r = EA or ER, 9 = z REA , (p = l ERQ or GRQ, the angle of incidence,

«+ 1) AE or EQ=m . EA. (Fig. 66.) ■ n • r» ri QE QE Rq t hen since ER = EA, — — =m= ; ER QR Eq QR_Rq RE qE " RE EA 1 The triangles QRE, RqE are therefore similar, and Eq — ER . - , EQ EQ and is consequently the same for all points R. 55 mA (A' — ;■) = A' ( A — r ) ; A' — m A r m — 1 A + r (Case 1 , 4.) m — 1 or — = m ■ ] A' A + r m r A' -h m A A ’ A' + r ’ m A (A' + ? ’) = A' (A + r ) ; A' = m A r m ■ — m— 1 A -j- r or — ; = A' mr + rn A 77 • There is another expression sometimes used, in which the distances are measured from the centre, (Fig. 70 .) Let EQ = q, Eq — (/, QE A q _ q r — q' AQ Eq r — q ’ q q (r ~ q) — mq (r — q); qr 9 = mr — m — 1 5 1 Wi — q r 1 in - + - 78. It will be observed^ that we have taken m to represent the ratio of the sines of incidence and refraction in all cases, whether the passage of the light be into a denser or a rarer medium ; if we chuse that m should always represent the ratio of the sines of incidence and refraction out of the rarer into the denser, we must, in Cases 3 and 4, put — for m, < r rn A + r * A' 1 1 _ m — 1 m 1 lien — - — . = — , and 7 = d . A A +r rn A r A 56 - We may now tabulate our results as follows : Case. Refracting Medium. Surface. - Equation. 1. Denser, Concave, 1 A'“ m — 1 1 + . • mr m A O Denser, Convex, 1 A ,_ m — 1 j 1 mr m A 3. Rarer, Concave, 1 A'“ m — 1 m r + A - 4. Rarer, Convex, 1 A'“ m — i m r + A' 79. The distance Aq being independent of the angle RQA^ provided that angle be extremely small, we may consider q as the focus in which the refracted rays meet when several incident rays proceed from Q in an extremely small pencil nearly coincident with the axis. 80. In order to find the principal focal distance , which we call f, as in Chap. II, we have of course only to make A infinite in the equations just given; we have then in Case 1, o “9 1 7 1 7 1 1 4, - t m — 1 mr m — 1 mr or/ = m r. in — 1 or/ = m. r. m — 1 m — 1 5 r m — 1 r or f = or/ = 1 r. m — 1 1 r. m — 1 We might of course easily have found this directly; thus, let QR, (Figs. 71 — 74.) be an incident ray parallel to the axis slE, RS the refracted ray cutting the axis in F the principal focus. 57 Then RF EF sin REF sin ERF sin ERQ 1 sin ERF = M ’ ° r m and putting AF for RF as, before, A F = m . EF, or — . EF, m whence AF= + AE, or + AE, as above. “ m - 1 m - 1 It is important to observe, that in all cases, the distance (AF) of the principal focus from the surface is to its distance (EF) from the centre as the sine of incidence to the sine of refraction. 81. If we introduce the distance^ - into the formulae, we shall have in 1 1 1 Cases 1 and 2, — - = - + , A J m A , 1 1 rn 3 and 4, , = A / A 82. A spherical refracting surface may, in fact, be said to have two principal foci, one for rays proceeding, parallel to the axis, from the rarer into the denser medium, the other for parallel rays proceeding in the contrary direction. They are on opposite sides of the surface, and at different distances from it,- as may easily be seen from the formulae, for in Cases 1 and 4, f is positive, that is, F lies on the side whence the light proceeds; in Cases 2 and 3, f is negative. In Figs. 1b and 7G, F and f are the two principal foci above described, F for parallel rays entering the denser medium, f for those proceeding out of it into the rarer one. 83. We will now proceed to examine the varieties of position that Q and q, the conjugate foci, are capable of. Case 1 . In the first place, when Q is at an infinite distance, the place of q is F, (Fig. 71.) When Q is at E, q is likewise at E. In all intermediate cases, that is, when Q is beyond E, q lies between E and F, -(Fig. 63.) 11 58 When Q is between A and E, q is between Q and E. This may easily be seen from the geometrical construction, (Fig. 77 •) or it may be shown from the formula : for 1 ! _ / | 1 \ J to — 1 to — 1 / \ 1 \ A A' V m/ A mr m V A r/ , which shows that A is greater or less than A / according as it is greater or less than r. When Q comes to A, q coincides with it. By differentiating the equation - — - = + , we find A m r m A cl A' Ja mr 1 A_ m & (to — 1 A + ?T which shows that the distance Qq is at a maximum in the space between A and E when mr z = (m— 1 A + r) 2 , or Vm — 1 m — 1 d A' for when A — A / is at a maximum dA — d A =0: and = 1. d A If we place Q on the other side of A, (Fig. 78.) or make A negative, we shall have 1 m - ! 1 A" to r m A’ whence we collect that as long as A < — , A' is negative and to - 1 ° r increasing : that when A = , or Q is at f , A' is infinite, and ° to - 1 J that afterwards it becomes positive, or that q goes to the other side of A. Obs. It will probably have occurred to the reader, that by placing Q within the denser medium, we have virtually passed from the first case to the fourth, with the only difference that the places of Q and q are inverted. I have, however, purposely placed Q in all possible positions, in order to illustrate the connexion between the 59 cases , and to show that the conjugate foci are convertible, as in reflexion, and that what are incident rays in one point of view, may be considered in another as refracted, and vice versa. 84. It will be observed that in this, and in all other cases of refraction, the conjugate foci move in the same direction, whereas in reflexion they always come towards, or recede from each other. The following are corresponding values of A and A A = cc, /•, 0, r m — 1 oo . m r A = , r, 0, m — 1 mr m — 1 Case 2. Here we have m — t 1 + ; whence it m r in A appears that as long as A > , or Q beyond J, (Fig. 76.) A is negative, or q on the contrary side of A from Q. r When A = , or Q is at A is infinite. ni — 1 When A < , or Q is between A and f, A ' is positive; so m — 1 that Q and q are on the same side of A : q is at first infinitely distant, and its change of place must be very much quicker than that of Q, for while this moves from / to A, q comes from an infinite distance to the same point. When A is negative, or Q within the denser medium, Fig. 79- 1 _ m — J 1 A ' mr in A ’ A' is then necessarily negative, as we might expect, the two foci moving together from A in the same direction. Aq is at first greater than AQ, but the two points coincide in E, and afterwards Q gets beyond q , and, in fact, it moves from E to an infinite distance while q goes from E to F. \ 60 The following therefore are corresponding values, , ?' A = A'= — m — 1 in r rn — 1 5 0 , 0, — r. m— 1 Cases 3 and 4 have, in fact, been discussed in the two others, we will therefore only exhibit the principal corresponding values of A and A'. Case 3. A = CTj , mr m — 1 v, 0, oc A' = r r, 0, — r m — 1 ’ ^ ' m — i Case 4. A = CO , 0, - r, mr m — 1 ’ ® * A' — V 0, - r. CO, r LI — , y m— 1 m — i Upon the whole we may collect the following results. In Case 1, divergency is given to incident rays, except when they proceed from a point between the centre and the surface. In Case 2, convergency is given. In Case 3, convergency, except when the focus of incident rays is between the centre and surface. In Case 4, divergency in all cases. Of course we except the case of rays proceeding from, or to the centre of a surface, which are not refracted at all. 85. We now pass on to a more useful part of this subject, which treats of Lenses, that is, of refracting media terminated by two spherical surfaces, or by a plane and spherical surface. There are several kinds of these: 1. The double convex , of which Fig. 80. represents a section through the axis. 61 2. The plano-convex, Fig. 81, which may be considered as a variety of this, the radius of one of the spheres becoming infinite. 3. The double-concave , Fig. 82. 4. The plano-concave , Fig. 83. 5. The meniscus, Fig. 84, bounded by a concave and a convex surface which meet. 6. The concavo-convex, Fig. 85, in which the surfaces do not meet. 86*. Prop. To Jind the direction of a ray after refraction through a lens. The method we shall follow here is to consider a ray refracted at the first surface, as incident on the secosid, and there again refracted ; we shall have occasion to add to the letters hiherto- used A" for the distance of the focus after the second refraction, t ... . the thickness of the lens; r ... . the radius of the second surface. Then taking, for instance, the concavo-convex lens in which both the centres are on the same side, (Fig. 86.) 1 1 . m- i . , _ . — = j — tor the first retraction, A m A mr ] A " + t tn A' + f m — 1 / 3 r for the second, t being added to A / and A" as the distances are now to be mea- sured from the second surface. However, in order to simplify the expressions, it is usual to suppose the thickness of the lens incon- siderable in comparison of A and A", in which case we may write m m — 1 A' 7 ~ 1 m— 1 m — 1 1 A * % 1 A 62 87- Now in the first place it will be immediately seen that this expression gives the principal focal distance, which we will call F, by leaving out the last term, which is equivalent to making — = 0, or A infinite : we have thus A 1 , - ( 1 — = (m - 1 ) {- r r i and then, 1 A" ~ F + A " It appears from the former of these that F is positive or negative 11 . ' . . according as , is so : let us examine what sign this is affected with in different cases. In the concavo-convex lens placed as in Fig. 86, r < r and F is positive. When this lense is turned the contrary way, r > r , but they are both negative, we have then j, = (m— 1) H _ n Ir r) ’ and F is positive as before. In the meniscus, either r > r, both being positive, and then 1= ~ ( ” _1) i; or r r , and both are negative: so that - (’"-‘'I; - ;-}• (1 _ X? r / ’ * It is often found convenient to put some symbol such as - for - — , which gives j, = , or F — y . When the radh are equal in a V double concave or convex lens p = - . 6:i In the double-concave lens f is negative, 1 fill p = !){,: + /} • In the double-convex r is negative. 1 = - (m— 1 ) l 1 + In the plano-concave either r is infinite, or r is infinite, and r negative ; therefore putting r for the single radius 1 in — 1 f = r In the plane-convex, I in — 1 „ r F = m — l F = F r 5 in — 1 When in the clouble-concave , or double-convex lens the radii are equal, L = + ( m - 1 ). r or F = ± i r 2 (m — 1; 88. It appears from all this, that the place of the principal focus is the same, whichever side of a lens is turned towards the incident light, and that The concavo-convex f the double-concave and the plano-concave The meniscus the double-convex and the plano-convex make parallel rays diverge. make parallel rays converge. * If m = - which is nearly the case in glass, F = + r, or the principal focal length is equal to the radius of sphericity. t See Fig. 87, for the relation between these different kinds of lens. Those placed together are equivalent. 64 l i = - + -,°,A A F A + F’ 89- The equation ] A 7 when put into geometrical language, gives rise to the following proportion, (Fig. 88.) Aq : AQ :: AF : AQ + AF, or if Af = AF, that is, if f be the principal focus of rays incident on the contrary side of the lens to Q, Aq : AQ :: Af : fQ, which it is more convenient to state thus Qf : /A :: QA : Aq. From this we derive another useful proportion, Qf : QA :: QA : Qq. S From either the equations or the proportions it will be easy to prove that when the distance of Q from the lens is varied, that is, when the place of Q is changed, the lens remaining fixed, the two foci move in the same direction. The following are corresponding values of A and A ', for a concave lens : oo ..QF..F.J ...0 - - . . — F. . -2F. . -SF. oo 2 2 F F 0 2 3 -F .... co ... .2 F. . . . . F. The following are for a convex one F 3 , — - , — F,-F. 3 2 4 ’ 90 . The distance Qq between the foci is represented by A — A", or A + &"■> according as the lens is concave or convex, 65 but as the equation gives A” negative in the latter case,, we may take A — A!' as its general value. Now A- A" = A- A F A + F A 5 A + T’ / . ^ QA\ ^that is, Qcj = Q-p J This quantity evidently admits of a minimum value. To find this, we will equate to 0 the differential of its logarithm, which gives 2 A 1 A +F \ = 0 ; A = -2 F. The negative sign shows that the incident rays are convergent towards a point beyond the lens. 91. To return to the original question: if it be not thought proper k to neglect t, the thickness of the lens, we may make the calculation rather simpler by measuring A" from the second sur- face. Then, in — 1 + m r A' + 1 m — 1 ( m A r 7 h rn | r I m r + (m — 1 ) A + t\ m — 1 m r + (?«-- 1 ) A f in r 4- ( m — 1 ) A 1_ — — < 1 -j- . — v Art m A r The binomial in the second term may be expanded, and as many terms taken as thought proper. If we consider only parallel incident rays, the equation becomes of course much simpler; I 6t> In any particular case it is easy to put the proper values of Ml, t, r, and r in the equations, and determine accurately the value of A ' or F, but no simple general expression can be obtained for them. 92 . The sphere may be considered as a sort of lens. In fact, it is a particular species of double convex, in which the thickness is twice the radius. In investigating its focal length, it will be most convenient to refer the distances to the centre, as in Art. 77- In Fig. 89, if EQ = q, Eq—q , ET=q", ER = r, we have I m— 1 m EA and EQ being in the same direction. ] 1 1 m 1 . . — — -f" ( Here r is negative.) q r m q _ in — 1 m— 11 m r m r q mi — 1 1 = — 2 . h - . m r q The principal focal length is of course — m r , the nega- 2 (mi — 1 ) tive sign meaning that the focus is on the opposite side from that whence the light proceeds. 3 3 If the sphere be of glass, and placed in air, = - , and F — - r, if of water, = - , an d F—2 r. 93. There is one case in which a ray will pass through a lens without deviation, that is, the emergent ray will be parallel to the incident : it is when the surfaces at which it enters and emerges , are parallel. 6? This is shown in Figs. 90 , 9L 92, 93, where QRST is the course of the light, and it will easily be seen that the spherical surfaces at R, S, can be parallel only when the radii ER, E'S, are so. The point O, where the refracted ray RS cuts the axis, is called by some writers, the centre of the lens; it is within the lens in the cases of double-concave and double-convex lenses, but without, in the meniscus and concavo-convex. The point O is invariably the same at whatever angle the parallel radii be drawn, for Fli r EO = EE' . -~ ~jV = (r + / + t) . -=-7 . ER + E S r + r The point m where the incident ray cuts the axis is easily found : we have only to put the value of TO for A 'in the equation, for the single refracting surface, and find A. AO = EO — A E — (r + r'±t) - jl. - — r= , . r + r r + r 1 - m — \ 1 rn — 1 r + r Then — = ± 1 7 = + H A m r m A m r m r t — + (m—\)t+r + r mrt A = mrt r + /•' + (m — I ) t If the thickness of the lens be supposed inconsiderable, QRST may be taken as a straight line, and T, O, as one point. It appears from this, that when a pencil of rays enters a thin lens obliquely, that ray which passes through the centre is not re- fracted at all : it serves as an axis to the pencil, and the focus of refracted rays lies on it at the same distance from the centre of the lens as when the axis of the pencil coincides with that of the lens, though the refraction is not quite accurate. 94. To return to the simple approximate formula of the lens. Let illi V = F + F + A- “ = m — 1 1 7 ^ 7h ’ P 0 m — 1 m — 1 1 4 7 — + T P A F ZA p And in like manner, if there be any number n of lenses acting together, we shall have 1 11 = 1 h . ■.(«) p' F l l m — 1 m — 1 m (n) — 1 1 or = d — + . . . • d b — ; I op p A so that their joint effect is the same as that of a single lens, having, for its principal focal length unity divided by 111 J_ ~p + ~p> ' p" ' ' ' ’ ' + ~pF ) J vi — 1 in — 1 m — 1 or 1- 1 — d- • • • • d“ P P P m M - 1 p {n) . 95 . Mr. Herschel calls the reciprocal quantity — the power of b a lens, and enounces the last result thus: “ The power of any system of lenses is the sum of the powers of the component lenses.” Of course, regard must be had to the signs : the power of a concaveAens must be considered as positive, that of a convex one, negative. 96. The same method by which we found the focal length of a lens may be easily applied to any number of surfaces, having a common axis. 69 Let r, r , r " .... be the successive radii, each having its own proper sign as well as magnitude. m , m, m // ....the indices of refraction at the several surfaces. A the original focal distance. A', A ", A'" . . . .those after one, two, three, refractions. Then, if only we neglect the distances between the surfaces along the axis, we shall have 1 _ m — 1 1 A' mr m A 3 1 171 — 1 ^ 1 771 — J 1 _j_ 1 A 1 ' mr' m A' mr mnir mm! A ’ m r m ■ i 1 n tt ’■ m r m A m — 1 , m — 1 t m— 1 ( 1 // n ”1“ 7 // / T } ~ T // . t, m r mm r mm 771 r mmrn A and so on. , 1 1 When the surfaces are those of lenses, m = — , m = — m m and the equations are reducible to those we have already seen. 70 CHAP. IX. ABERRATION IN REFRACTION AT SPHERICAL SURFACES. 97- The question here is precisely similar to those we have met before, namely, to determine the difference between the ulti- mate value of the focal distance for refracted rays, and the value it has for a ray inclined at a sensible though small angle to the axis. To begin with a single surface. Let v, (Fig. 9o.) be the inter- section of the refracted ray and the axis, every thing else as before. Let A v = A '. Referring to the beginning of last Chapter, we find that in strictness Q E Rv QR Ev * or QE . Rv = m . RQ . Ev. Now Q K 2 = EQ 9 + ER' + 2EQ . ER cos AER = ( A — rf + r~ + 2 ( A — r) r . cos 9 — A ' — 2 A r + 2 + 2 A r cos 9 ~ Q, r 2 cos 9 = A 2 — 2r ( A — r) versin 9. Similarly, Rv' — A — 2r ( A ' — r) versin 9. Then putting v for versin 9, (A — r) A ' — 2 p ( A ' — r) v = m ( A ' — r) \ J A ' — 2 /• ( A — r) v . (1 ). Then proceeding as in Chap, iii, we have To obtain the value of d A ' we must differentiate the equation (1), which gives ?i (A - r) . (A' — rv) d A' — r (A' — r) dv A'” — 2 r (A' — r) v y — r (A — r) dv — m A 2 — 2r (A — r) v . d A' — (A' — ;•) . / == • >/ A* — 2?' (A — r) v then making v — 0, A' = A 1 , A 'd A' — r (A' — r) that is, (A — ?■) \dA y - r (A - . ; -- dx>) 1 A' J , iw (A' — /•) (A — r) — m Ad A — A dv. • { (m — 1) A + r} c?A' = r . (A — r) (A' — r) d v ; y'f/ A'\ ?' . (A — r) (A' — r) sm 1 \ * * V dv ) m _ 1 A + r f \2 F 1,1 ^ ,• ./ ^ * = mum together. v A K t K t K vm = m u . - — — x . — . ~.x = — . x, nearly. A r T k k T k (AT and At are very nearly in a ratio of equality); • T I 7 ' K , K A~ k . . vl — v m + m 1 — — x + x = x k k .( 2 ). Comparing this with the former value of v T, we find K + k K . - Ir ■x = a . k K- K — k a -jr- = r •*<*-*>• CL Hence x is at a maximum when k — X K : then x = ~ , 4 4 K AR oT Since m T — ^ q T, mn = therefore the diameter of the least circle of aberration is equal to half the lateral aberration of the extreme ray. Its distance from the focus q is three-fourths of the extreme aberration. Note. What has been proved here for a lens is equally appli- cable to the cases of reflexion, and refraction at a single' surface, as in both of these, the aberration of a ray inclined to the axis varies as the square of the distance of the point of reflexion or refraction from the axis. CHAP. X. \ REFRACTION AT CURVED SURFACES NOT SPHERICAL. 102. In like manner as in Chap. IV. we found that though a spherical surface is not capable of reflecting light accurately, those belonging to the Conic. Sections have that property, so here we shall find that by means of a spheroidal or consiidal surface, 7? rays may be refracted so as to meet accurately in one point without any aberration. This will easily be seen from simple Geometrical considerations. Let AM (Fig. 99*) be the axis major of an ellipse ARM ; S, H the foci., nRN a normal, QR parallel to AM. Then since the angles HRN, SRN are equal, sin QR n sin RNH sin RNS + sin RNH sin NRS sin NRS sin NRS + sin NRF1 RS + RH ~ NS + NH AM ~ SH ' From this it appears that if a transparent spheroid have the ratio of its axis major to its eccentricity equal to its refracting power, rays entering it in a direction parallel to the axis major will be refracted accurately to the farther focus. Moreover if the surface of the spheroid be cut by a spherical surface, having that focus for its centre, a lens will be formed which will refract parallel rays ac- curately to the focus, as there will be no refraction in their passage through the spherical surface. For the same reason, rays diverging from the focus of this lens will be refracted so as to become parallel to the axis. 103. Again, let AM, (Fig. 100.) be the axis major of a pair of hyperbolas, QR a line parallel to it, S, H the foci, RN a normal, sin QRN _ sin RNS _ sin RNS - sin RNH sin n RS sin NRS sin NRS — sin IS RH RS - RH ~ SN - NH AM ~SH as before. Flence it appears that a plano-convex lens having its coin ex surface hyperboloidal, will refract parallel rays accurately k> a point 78 which is the focus of the opposite hyperboloid, and conversely, that rays diverging from a point may be refracted by such a lens so v as to become parallel. 104. Sir I. Newton has given in the 14th section of the first volume of his Principia, two curious propositions relating to the present subject, which are inserted here, to save the trouble of referring to the book itself. The first is : To find the form of the surface of a medium, which will refract rays diverging from a point without it accurately to a point within itself Let A (Fig. 101.) be the focus of the incident rays ; B that of the refracted ; CD the section of the surface ; AD, DB an incident and refracted ray, DE an infinitely small portion of the curve; EF, EG perpendiculars on AD, DB. DF Now = cos EDF = sin incid. DE DG JXE = cos EDG = sin refr. therefore if m be as usual the refracting power, DF—m . DG; but DF, DG are the corresponding increments of AD, BD, so that if we call these u, v, we have this differential equation to the curve, da — — m . dv, or d u + rndv — 0. When B is removed to an infinite distance, the equation becomes du = mdx, if x — AN, DN being perpendicular to AC; now we shall see that this equation belongs to an hyperbola, or an ellipse, according as m is greater or less than unity, du — mdx gives, by integration, u — mx -f n, that is, squaring .r* -f- y* — (m x + nf ; . • . y 2 = (in' — 1 ) a ~ 4- 2 m n x + if . Now the equation to an hyperbola, when the abscissa is measured from the farther focus, is 79 / = (e 3 — 1) {(x-aef-a} = (e" — ] ) { x 1 — 2 a e x + (e~ — 1 ) a 2 }, which agrees with that above, if m — e, /2s ~ n Q,mn= — (e — 1) . 2a e ; a — — , m~ — 1 — « = : (e — 1) a , or a — —5 -5 , as before. OT — 1 (W — 1) If m be less than unity, the integrated equation becomes y 2 = n~ + 2 m n x — ( 1 — in 2 ) x~, which agrees with the equation to the ellipse, y 2 = (l—e') {d : - (x-aef }, or y 1 — ( 1 — e 2 ) { ( 1 — e") a~ + 2 a ex — x % } . The general equation integrated gives u + mv = n, that is, if AB = c, y/ x~ + y 2 + m (c — or) 2 + y" = n. The curve to which this belongs is a certain oval, which Descartes has described. 105 . Newton’s second proposition is: To find the form of a convex lens , that shall refract light ac- curately from one point to another. He supposes the first surface given, and determines the second thus, (Fig. 102.) Let A be the focus of incident, B of refracted rays ; A DFB the course of a ray ; CP, ER, circular arcs with centres A, B ; CQ, ES, orthogonal trajectories to DF. Let AB, AD, DF, be produced so that BG=(m — l) CE, AH = AG ; DK = — DH. Join KB, and let a circle with centre m D, and radius DI 1 cut it in L. Draw BF parallel to DL. In the first place PD = m . DQ ; FR — Fn. FS. For let AD, AE 80 (Fig. 103.) be two incident rays inclined to each other at an in- finitely small angle, DK, EL, the refracted rays. Let EF, EG be perpendicular to AD, DK. Then PF=Ep , QG = Eq\ therefore DF—d . DP, DG — d . DQ. Now it was proved in the former proposition that DF—m. DG, and DP, DQ are evidently integrals of these differentials beginning together from nothing. To return then to our problem, PD m ~ Uq ’ DL FB DL-FB PH-PD-FB but m — = — — = = DK FK DK-FK FQ~ QD ’ PH- FB _ CE -f EG - BR - FR " FQ ~ CE — FS CE + BG-FR CE-FS CE + (m — l) . CE — FR = CE - FS ; FR m ~ FS 5 and therefore the ray ADF is refracted to B. CHAP. XI CAUSTICS PRODUCED BY REFRACTION. lOG. These caustics are exactly analogous to those before treated of, being formed by the successive intersections of refracted rays, as those were by reflected ones. Prop. Required the caustic to a plane refracting surface. Let QR (Fig. 104.) represent an incident ray, q RS the re- fracted one; AM, MP are the rectangular co-ordinates of a point P on that line; PN is parallel to AM. 81 Let AQ = A, z AQR = 9, £ A qR = 9', AM=x, MP —y. Then MP — AR- NR, that is, y — A tan 9 — x tan 9'. This is the equation to any point P on the refracted ray. If this point be on the caustic, it must be common to two successive re- fracted rays infinitely near each other, that is, x and y must be the same for the refracted rays answering to 9 and 9-\~d9. We may therefore equate to nothing the differential of our equation with respect to 9 and 9', considering x and y as invariable. This gives us d9 _ dff cos 9' cos 9 2 We have, moreover, between 9 and 9', d9 and d9‘, the equations sin 9 — m. sin 9' ; cos 9 . d9 = m cos 9' . d& . From these, by the elimination of the functions of 9 and 9', we shall obtain an equation containing only x , y, m, and A, which will be that of the caustic. 10/- Prop. Required the focus of a thin pencil of rays after being refracted obliquely at a curved surface. Let QR, QR', (Fig. 105.) represent two rays inclined to each other at an infinitely small angle, incident obliquely on a curved surface at R, R ; Rq, R' q the refracted rays ; RE, R E, normals. Let EQ = q, Eq — t, ER — r, QR = u, Rq — v, z QRZ — 0, z ERq = cp . L 82 Then we have these equations : sin (f) = m . sin V (0 dn + mdv = 0 (Art. 104.) (2) q~ = r 2 -(-2?/ r cos

' v from ^ = — X' md

') dv + vr sin ip' dip', . u + r cos (p du that is, u = : — — x — , r sm

. , . — j and — for — — , , we obtain nu cos ip mdip v v — r cos ip' sin 0 cos (p ' m w + r cos (p sin ip cos (p ’ ir/ 1 cos ) tan u = y- — = r cos (p . — . tan

3 . AE. The caustic here extends further both in length and breadth than in the last case. It begins of course at the point m, Em Q being the angle whose sine is -§-. (2) When AQ, = 3 AE, A q is infinite, so that the branches of the caustic becomes asymptotic to the axis, as in Fig. 117. (3) . When AQ is less than three times AE, the curve opens, in a form something similar to that in F'ig. 30. 86 For instance, when Q is at the extremity of the diameter of a sphere, Fig. 118.) AQ = 2JE, A q = 4AE, EmQ = 41°. 49, QEm= 96 °. 22 '; Rv, R'v are the asymptotes*; Ev = 3.949 AE, EvR=\ 1 °. 25'. (4) Let now Q come within the sphere, (Fig. 11 9.). 2 Provided EQ be greater than ^ AE, a segment of a circle on EQ capable of containing an angle of 41° 49', will cut the section of the sphere in two points m, n, at which rays incident from Q will be refracted parallel to the surface. Between the points m, n, there will be no refraction: those rays which fall on Am will, after refrac- tion, form a caustic of the same kind as that of the last case : those which fall on a n will form another caustic nq, q being the focus for rays refracted at a. * The place of the asymptote is thus calculated : Since v is to be infinite, and « = 2 r cos cos . ; — - . tan

cos d>' = 2 . - , or - — o -f- = 2 . sin (p cos

T V Ar — A v v — r Ar + A v v + r ’ and therefore if vve put a for the aperture Bb, v ~ r no = v-\- r Suppose, for instance, the lens be of crown-glass, v — . 56 r = . 54, v — r .02 1 v -|- r 1.1 55 The diameter of the least circle of aberration is therefore — of 55 the aperture. 1 28. With regard to the distribution of the light over the surface of the circle of least aberration, it will be sufficient to observe that the vertices of all the cones of coloured light being on the axis of the lens, the centre of the circle is one of them, so that it must be strongly illuminated, having the whole of the light of one sort thrown on it, besides portions of the others; the circumference on the con- trary is enlightened only by the extreme rays of the red and blue cones, so that it is the least bright part, and it will be easily seen that the light diminishes gradually from the centre of the circle to the edge. On this account the effect of this aberration on images produced by lenses, is not so great as one might imagine from the great magni- tude of the least circle of aberration: it certainly substitutes fo^ single foci, so many interconfased circles, but as these are bright only at their centres, and above all, as the yellow light, which is the brightest in the spectrum, converges nearly to those centres, the haziness is not very considerable except in cases where the light is very much condensed by a lens of short focal length. 1 29- The chromatic aberration is, however, a much more serious bar to the perfection of optical instruments depending on the lens, than that owing to the spherical figure, for this latter imperfection can be made quite insensible in most cases, by diminishing the aperture of the lens, since it varies as the square of this line, whereas the former varying as the simple power of the aperture, will be diminished certainly, but very considerably less than the other. It has therefore been a great desideratum to find some way of constructing a lens, so as to be achromatic, and this has been tole- rably well effected, by joining together two or more lenses, made of substances having different dispersive powers, so that the disper- sions may be equal and opposite, though the refraction be not wholly destroyed. 130. The expression for the principal focal length* of a com- bination of lenses placed close together was found (p. 68.) to be 1 m — 1 rn — 1 m" — 1 ~ = I ; 1 77— + • • • • 9 P P P If therefore 1 +r and 1 + v represent the values of m for red and violet rays, we shall have, taking only two lenses, 1 r i' — lor the red ray, 9 P P 1 v v . . — = - -1 . , tor the violet. 9 p p Now it is clear that if we chuse to leave p and p indeterminate, we may equate these two values of

aud so obtain proper values for p and p , * It is quite sufficient to consider the principal focal length, as it will easily be seen, that in the case of rays not parallel, one has only to add -i to the expression. 96 - + -/ — + ~/> P P P P / , / / . / P v—r rp +r p = vp + vp, or — = , p v — r which shows that p and p must be of different signs, or one lens concave and the other convex; and that they are as the respective dispersions of the lenses. In order therefore to bring the most unequally refrangible rays, namely, the red and violet, to one focus, we have only to put together a convex and a concave-lens, and to make the quantities represented by p and p (or the principal focal lengths, for any one kind of simple light, which are in the same ratio) proportional to the dis- persive powers of the substances, of which the lenses are made. I 131. The common practice of opticians is to use flint-glass, and crown-glass, the dispersive powers of which are in the ratio of 50 to 33; and therefore a compound-lens such as that represented in Fig. 141. in which the separate focal lengths, for the same kind of homogeneous light, are as 50: 33, will make the red and violet rays of the solar, or any similar light, converge accurately to one point. To illustrate this, let v, g, r, (Fig. 142.) be the points to which the convex-lens by itself would throw the violet, green, and red rays. The addition of a concave-lens diminishes the convergency, and therefore throws the foci farther off ; but it affects the violet and green light more than the red, so that they are all brought closer together, and if the lenses be so matched that the dispersive power of the first is just balanced in all parts by the counter-dispersive power of the second, the rays will all be brought to one single point f. If, however, the substances of which the two lenses are made, do not act with equal inequality , on the different coloured rays, the object will not be attained. If for instance, (Fig. 143.) the first lens disperses the rays so that the foci v, g, r are equidistant, but the second lens acts very nearly as strongly on the green rays as on the violet, it may throw the red focus from r to r, the violet from c to v close to it, vv being greater than r r , but the green will go from g to g , making gg nearly equal to vv, so that the three foci will not coincide. 97 Now this is in some degree the case with respect to flint and crown-glass : they do not disperse the different coloured rays pro- portionally, and in consequence, if two lenses be matched so that their dispersions are equal and opposite for the extreme rays, there will still be some aberration of green and blue rays uucorrected. 132. Dr. Brewster in his excellent “ Treatise on new Philo- sophical Instruments ” details some experiments tending to shew that prisms and lenses of the same substance might be combined so as to correct each other’s dispersion, without destroying all the re- fraction. He found that when a beam of light passed through a flint-glass prism, so that the deviation was a minimum, (the angles of incidence and emergence being equal,) and the dispersion was corrected by a smaller prism of the same substance, inclined to increase its refrac- tion, the colourless pencil was still considerably refracted from its original direction, by the prism with the greater refracting angle. This combination, represented in Fig. 144. he proposes to imitate with a pair of lenses, by making them of the form shewn in Fig. 145. The reason why the preceding theory did not lead to any such conclusions as these, appears to be as follows: It was taken for granted, that a given substance has always the same dispersive power into whatever form it be put, or however its surface be inclined to the light, that is, that the dispersion bears a constant ratio to the mean refraction. Thence it was argued that the dispersiou of a lens was as the dispersive power of the substance, and the power of the lens, jointly, or as the dispersive power directly, and focal length inversely, and that therefore the dispersions of a convex and concave-lens might be made equal and opposite, if the dispersive power . fraction — — was the same m each : but it appears focal length from Dr. Brewster’s experiments, that our premises are not true, for that when the angle of incidence is changed, the ratio of refrac- tion is not constant for each kind of primitive light. 133. Dollond, who first constructed achromatic compound- lenses, made them of three different parts, as represented in Fig. 14(j, N 98 two convex lenses of crown-glass, with a concave one of white flint- glass between them. In this case we may consider the two outer lenses as producing one single refraction, and the inner one as correcting it. CHAP. XIV. THE EYE. 134. The organ by which we are most usually, and most easily informed of the presence of external objects, and without which we should often be ignorant of their form, and always of their colour, is the eye, a most curious combination of parts so ad- mirably contrived to answer all the purposes required, that nothing short of divine intelligence could have been capable of constructing it, and the mere imitation of it is far beyond the reach of human skill. 135. The eye is, in form, nearly spherical, as will be seen by referring to Fig. 147, which represents a horizontal section of the right eye*. > Its several parts are as follows : The cornea A is a transparent membrane which covers the con- vexity in front of the eye. It is formed like a meniscus, being thickest in the middle. The sclerotica HIK is a thick tough coat, which covers the re- mainder of the eye, and is intimately united with the cornea round the edge of the convexity. The choroid - coat EFG lines the sclerotica; (but not the cornea): these two integuments are united rather loosely in general, except round the edge of the cornea, where they are firmly fastened toge- ther by a circular band, called the ciliary ligament. This figure is copied from Dr. Young. 99 The choroid is continued, so as to form, by a sort of doubling or fold, the ciliary processes, and is again continued in front of these. This continuation is called the uvea, and is like a circular basin thinned away towards the centre, where there is an aperture B. This aperture is denominated the pupil. It is of various forms in different animals, and is capable of being contracted, or enlarged. In man it is always circular, but in animals of the feline kind, its ver- tical diameter is invariable, so that its figure varies from a circle to a straight line (Fig. 148.). In ruminating animals on the contrary, it is transversely oblong, and when contracted to the utmost becomes a horizontal straight line (Fig. 149-). The change in the size or form of the pupil, is effected by certain muscles, which in general perform their office spontaneously, being of the kind called involuntary . The cat however is said to have a great command over this mechanism. The iris is a coloured membrane coating the exterior surface of the uvea. It is of different hues in men, varying through many shades of blue, gray, brown, and green. The interior surface of the choroid is covered with a dark mucus, in which is imbedded a fine net-work, called the retina. This pro- ceeds from the optic nerve, which enters obliquely at the back of the eye through a tube (part of which is shown in the figure) which connects the eye with the brain, the coats of which, called the dura mater and pia mater, are by some writers said to be identical with the sclerotica and choroid. M. Cuvier says on this subject: 1 ‘ Le F posing q to be as near as possible. On the whole then, the magnifying power is b-F + F F (?+')■ The final image is inverted with respect to the object, as may be seen by the figure. From the known equation — = — — £ = A"F A"+F ’ and therefore — , we deduce r A" _ A "+F A ~ F Amici’s reflecting Microscope. Fig-. 172. 161. In the 18 th volume of the Transactions ot the Italian Society, there is a Memoir giving a detailed account of a catoptrical microscope, invented by Professor Amici, of Modena*. At one end A of a tube 12 inches long, and 1 ~ in diameter, is placed a concave metallic speculum of a spheroidal form, having its foci Q,q, at the distances 2 ~ and 12 inches, respectively. The object to be examined is placed on a little shelf C projecting from the stand, half an inch below the tube, and reduced to the nearer focus Q by a plane mirror F, placed at half a right angle to the axis of the speculum, at the distance of 1^ inch from it. The image, formed in the farther focus q, is viewed through a lens B, which may be changed at pleasure, so as to increase or diminish the magnifying power, the object remaining unmoved, which gives this instrument a great advantage, in point of convenience, over the common compound refracting microscope. In order to make the object sufficiently bright, there are attached to the instrument two concave mirrors D, F, one of 3 inches diameter, and 2-^ focus, at the foot of the stand, and the other directly over the object, having an aperture in the centre like that in the tube, to admit the rays to the plane mirror. This instrument has a magnifying power of near a million, and is found extremely convenient from the horizontal position of the tube, which enables the observer to examine an object more at his ease, and for a longer time, than when stooping over a microscope of the common construction. The Professor has contrived, by a very ingenious arrangement, to convert his instrument into a species of camera lucicla , which enables him to draw any object on a magnified scale. Dr. Smith’s Reflecting Microscope. Fig-. 1/3. 162. Here the office of the object-glass is executed by a pair of mirrors A, B, the former concave, the latter convex, having aper- * Tins account is borrowed from the Edinburgh Philosophical Journal, Vol. II, 126 lures pierced through their centres to give a passage to the light. By this means, a small object being placed at Q, a first reflexion at A would produce an image at q, which being farther from the centre of the surface, would be larger than the object, but a second reflexion at the surface B sends back the light which is proceeding to form this, and thus throws the image to q 2 , still farther magnified ; there an eye glass D receives the rays, and trans- mits them with the proper divergence for distinct vision. A small screen is placed at C to prevent the rays from coming directly from the object to the lens. The Solar Microscope. Fig - . 1/4. 163. This is merely a sort of magic lantern, in which the light of the Sun, reflected by two plane mirrors A, B, and condensed by lenses C, D, is thrown on a minute transparent object jE, of which a magnified image is formed by means of a lens F. In the figure, the object to be exhibited is placed near the focus of the first combination of lenses, so as to be entirely en- lightened by the rays coming through them, and not to be burnt, which would be the case, were it exactly at the focus. The Heliostat. Fig. 175. 164. This instrument consists of a plane mirror M, moveable about a horizontal axis AA and a vertical one CP, which is made to revolve by clock-work about an axis parallel to that of the heavens, so as to reflect the Sun’s light constantly in one same direction, during the course of the day. DR is the hand of the clock, connected with the mirror by means of the apparatus F into which is inserted the rod QM. It is found very useful in many optical experiments where a small pencil of solar light admitted into a darkened room, is to be subjected to reflexion, refraction, dispersion, &c. It is a very convenient appendage to the solar microscope, which is the reason of its description being inserted here. The Astronomical Telescope. Fig. 176. 165. Generally speaking, the Telescope is in construction analogous to the compound microscope : the only difference is, that 127 the latter instrument is used to examine an object placed near it, which is not distinctly visible, on account of its minuteness, whereas the former is calculated to assist the eye in viewing objects, which look small or indistinct, only on account of their distance. The telescope substitutes for a distant object an image at any distance required, which subtends at the eye a much larger angle than the object, and from which more light proceeds, than the eye could receive from that object. The manner in which this is effected in this instance, is as follows. By means of a convex lens A, called the object glass, there is produced an inverted image PFR of a distant object, which image is, of course, at F the principal focus of the glass, and has collected on it nearly all the light which, proceeding from the object, falls on the surface of the object glass. This image is viewed and magnified by means of an eye glass B, another convex lens, which is placed so that the image F is at its focus f, or a little within it, in which latter case, a second image pqr (though not a real one,) is produced at the most convenient distance for correct vision. It is evident then, that if the eye glass B be a powerful lens, which it always is in practice, the angle pBr which the final image subtends at the eye, placed close to the eye glass, is much larger than PAR , or P AR', that subtended at A by the object, and it is brighter, for the eye receives in the one case ail the light which falls on the object glass, in the other, only as much of that coming from the object as can pass through the pupil of the eye. l66. If the foci F, f of the two lenses coincide, since the angles RBP, RAP, are inversely as BF, AF, the magnifying power of AF F this instrument is expressed by or ~p, 5 if F, F represent the focal lengths of the object and eye glass. If the final image be brought to the nearest distance of distinct vision (c), it is greater in c c the proportion of — + 1 to — , (see compound microscope,) and r r therefore its value is 128 It appears from this, that to make a telescope of great mag- nifying power, the object glass should be of considerable focal length or small power, and the eye glass, on the contrary, very powerful *. In consequence, before the discovery of reflecting telescopes, and achromatic combinations of lenses, astronomers were obliged to use instruments of so great a length as to be hardly manageable, till an ingenious mechanic, named Hartsocker, divided the object glass from the eye piece, fixing the former with its frame on the roof of a house, or ou a high pole. Huyghens, improving upon Hartsocker’s plans, made use of the apparatus represented in Fig. 177. When merely common lenses are used in this telescope, the eye glass is limited in power by the necessary confusion in the first image from the aberrations, which become very sensible when that image is much magnified, particularly those arising from the unequal refraction of the light. For this reason, all good telescopes of this kind are made with an achromatic compound lens for an object glass, and instead of a single eye glass, we also find an eye piece composed of two or three lenses, also achromatic. The spherical aberration may be lessened, preserving the mag- nifying power, by using a weak object glass, assisted by another, called a Jield-gfass, as in Fig. 178. 167 . The field of view of a telescope is the lateral extent of prospect it affords in one position, the greater or less portion of the heavens it makes visible at once. To determine this we have only to find the extreme direction in which a ray can pass from one lens through the other : for this we must join the corresponding extremi- ties of the object and eye glass. Fig. 179; the lines Mm, Nn will thus bound the visible image rqp, and the field of view is measured be the angle r A p\. * The brightness of the image varies as the surface or aperture of the object glass directly, and as the magnitude of the image, (that is, as the focal length,) inversely. On this account, the lens should be large in proportion to its focal length, which it may safely, to a certain extent, which is deter- mined by the aberrations becoming sensible. t This is nearly equal to the angle subtended by the object-glass at the e pe-glass, which is the common measure of the field of view. 129 In point of fact, the field of view is not so large as represented here, for the point p receives but one single ray that can fall on the eye glass, and therefore will not be visible. If the opposite extremities of the glasses be joined by lines Mn, Nm, they will bound apart of the image tqs, which transmits all its light. It will be seen from the figure, that the eye must be placed at a little distance from the eye glass to receive the rays proceeding from the extreme verge of the field of view. 1 68 . The image is inverted as in the compound microscope. It may be set upright by an additional pair of lenses C, D, (Fig. 180), which are placed so as to have a common focus, and, usually, have no effect on the magnifying power. This construction is used in what are called Day Telescopes , which are chiefly employed for viewing distant terrestrial objects; for observations on the heavenly bodies, the additional glasses are dispensed with, in order to save the light that is lost by the two additional refractions. The lenses in the figures are drawn of their full diameter, but in practice it is usual to limit their apertures in order to diminish the aberrations, (see Fig. 181). In this case we must consider the lenses as extending no farther than these apertures. 1 69 . In telescopes to be used for astronomical observations it is usual to put a net-work of fine wire or sometimes of spider’s web at the focus of the object glass, in order to determine the precise position of a star as it passes by them. This apparatus is called a Micrometer, and its simplest form is represented in Fig. 182. having five parallel wires dividing the diameter of a circular dia- phragm into equal parts, and a sixth bisecting them all perpendi- cularly. Another kind of wire micrometer represented in Fig. 183, con- sists of two parallel wires, ah, cd 3 the one ( cd ) fixed, and the other moveable by means of a fine screw, with a third ef, perpendicular to them. This is used for measuring the apparent diameters of the heavenly bodies. K 130 170 . There is another kind of micrometer, which may as well be described here while we are on the subject: it is called the divided object glass micrometer , and is, in fact, an object glass divided into two by a plane passing through its axis, and of which the two parts, when placed with their centres not coinciding, act as two separate lenses, (Figs. 184, and 185.) The use of this construction is to make the images of two stars, not far distant from each other, coincide on the axis of the telescope, and to determine their angular distance by observing how much the centres of the half lenses have been displaced, by graduated scales on the edges. Sometimes the eye glass of a telescope is divided in this manner instead of the object glass, and this is thought by many persons a preferable arrangement. Galileo’s Telescope. Fig - . 186 . 170. In this the rays refracted through the object glass, and proceeding to form an image at its focus, as in the Astronomical Telescope, are stopped by a concave lens, which, if it have a common focus with the object glass, makes them emerge parallel, or if placed nearer, gives them any degree of divergency suited to the eye. By this means the first image is made imaginary , and the second, likewise imaginary, is thrown on the opposite side of the eye glass, and is therefore erect. This is the principle of the opera glass, which has the advan- tage of being much shorter than the astronomical telescope, in which the eye glass is beyond the first image, besides representing objects in their natural erect position. This was the first telescope invented. The magnifying power, which in this instrument is commonly very low, is represented as before by the fraction T P ' This telescope might be used as a microscope, but it would require to be lengthened, as the first image would be thrown farther from the object glass than its principal focus; the magnifying power would in this case be increased. The field of view is here found by joining the opposite extremi- ties of the glasses by the lines Mn, N m, (Fig. 187.) which mark on the first image the extreme points p, r, to which a ray belonging can fall on the eye glass. The angle r Ap thus measures the field of view, which is much larger than in the astronomical tele- scope. The lines Mm, Nn define the bright part of the field. Dr. ITerschel’s Telescope. Fig. 188 . l7l. The construction of this instrument is better seen in Fig. 189. where A represents a concave metallic speculum giving an image of a distant object at its focus q, where it is viewed through an eye glass B. In practice, the image is thrown to one side as in Fig. 190. as otherwise the head of the observer would intercept the best part of the incident light. The magnifying power is plainly / f being the focal length of the speculum, and F' that of the eye glass. Dr. Herschel has constructed telescopes of this kind that magnify several thousand times*, but he generally used powers of only 500 or 600 which gave more brightness to the image. The visible part of the image is bounded by the lines Mm, Nn, determining the points r, p, from which, single lays are sent to the eye glass. The field of view is measured by the angle rEp, which is nearly the apparent magnitude of the eye glass seen from the spe- culum, and is of course'vei'y small, for which reason there is often attached to telescopes of this kind a small refracting one, of low * magnifying power but considerable field, which has its axis parallel to that of the other, and is called a finder, as it serves to direct the large telescope to any desired point, as a particular star. 172. Reflecting telescopes in general have these advantages over refractors : They are almost entirely free from chromatic aberration, being subject only to that of the eye glass, which is very inconsiderable. * His great telescope is 39 feet 4 inches in length, 4 feet in diameter, and magnifies 6000 times. 132 They are shorter, cateris paribus, for the foca! length of a con- cave speculum is only half its radius, whereas that of a glass lens with equal surfaces is the w'hole radius. They give brighter images, for there is less light lost in their re- flexions, than in the refraction through an object glass. Notwithstanding this, they are not so much used, because they are less manageable from their weight, more expensive, more apt to get out of order, and more troublesome to use with any nicety, as the least shaking of the instrument or its stand causes great con- fusion in the image, which is not the case in refracting telescopes. Sir Isaac Newton’s Telescope. Fig-. 191. 173. This differs from Dr. Herschel’s only in having a plane mirror C placed at an angle of 45° to the axis, which throws the image q to the side of the instrument q , where the eye glass B is placed. Newton sometimes used a rectangular prism of glass for a plane reflector. The magnifying power is of course the same as in Herschel’s telescope, as is likewise the field of view, provided the plane re- flector be large enough to convey all the rays to the eye glass. The Gregorian Telescope. Fig. 192. 174. In this the image formed, as in the' last two instruments, at the focus q of a concave speculum A, is reflected by a second small concave mirror B having its focus f a little beyond q, so that there is a second image at q', which is erect , and is viewed through an eye-piece fixed in an aperture in the centre of the principal speculum. This appears at first sight to be a very disadvantageous con- struction, as the central rays are all stopped by the smaller mirror, and the best part of the great speculum is lost. It is, however, to be observed, that the small reflector is usually of very confined dimensions, and when the object speculum is well ground, it is found that the lateral rays converge quite sufficiently well to make a distinct image. To find the magnifying power, we must compare the angle subtended by the second image at the eye glass with that of the 133 object or first image at the centre of the great mirror : but since the focal length of the mirror is half its radius, this is the same thing as the angle of the first image seen from the mirror. The magnifying power may then be measured by ?' (p, we must put for m the value that it has for any desired sort of 'homo- geneous light refracted between air and water, and by the help of a table of natural sines and cosines, we shall obtain the angles

+!) 0 ~ ; 0 = 7T — (7T — 0) — (/> + l) tp' + (p — 1) “ 2 = tp - (p + 1) 0' + (p- 1) - . d0 TT dtp' Hence, - t— = 1 — (p + 1)— - ; and dtp cos tp or dtp vr 'dtp’ dtp ’ m. cos tp' p -f 1 We have then m cos tp' — (p -f ]) cos tp ; m" cos 0' 2 = (p + l) 3 cos tp' , and ni~ sin tp ' 2 = sin tp', .'. w 2 = (p+ 1 f cos tp : + sin tp 2 = (p‘ -f- 2jp) cos 0 2 + 1 ; cos tp = v/4 nr — 1 p + 2 p 0 is of course found as before, 1 , 2, 3. . . .being put for p, according as the question relates to the primary, secondary, or tertiary bow. In this manner the radius * of the innermost arc of the lower bow, is found to be 40° 17 , that of the outermost 42° 2'. And the extreme values for the second bow, are 50° 57', and 54° 7'. 182. It is easy to verify these results by observation, for as the centre of the bows is in the line joining the centre of the Sun * These arcs are considered as parts of small circles of the celestial sphere, and the radius is the distance of each from its pole. 142 and the eye of the spectator, (Fig. 210.) the radius of any arc of which A is the highest point, is equal to the sum of its altitude AOh, and that of the Sun SOH, or h OS. We have therefore only to take with a sextant, or other equivalent instrument, the greatest height of any arc above the horizon, and add that of the Sun, to obtain the radius of the arc. 183. It is sometimes required to determine, from observations on the rainbow, the ratios of refraction, for the different kinds of coloured light, between air and water. Suppose that we have found the value of 0, or 2 = 2 tan (p , or tan

all respects to the laws deduced by M. Biot from observation. These interferences of the rays may be produced without the assistance of crystalline laminae; w'e may equally employ thick plates, provided the rays pass through them at very small incli- 185 nations to their crystalline axes. If the experiment be made with a conical pencil of light, large enough to give the various rays com- posing its inclinations sensibly different to the axes, so that they experience double refractions sensibly unequal, these rays, analyzed after they emerge, offer different colours united in the same system of polarization ; and the union of these colours forms round the axes coloured zones, the configuration of which indicates the system of polarizing action exerted by the substance under consideration. This kind of experiment is therefore very proper to exhibit the axes and to indicate the mode of polarization with which any given substance affects the rays. Upon the whole, the interferences of polarized rays offer very remarkable properties, many of which have been discovered and analyzed by Messrs. Arago and Fresnel with great ingenuity and considerable success, but as the limits of this Work do not allow of a full exposition of them, I will only cite one, which is, that rays polarized at right angles do not affect each other when they are made to interfere, whereas they preserve that power when they are polarized in the same direction. It is not only crystalline bodies that modify polarization impressed on the rays of light : Messrs. Malus and Biot found by different experiments made about the same time, that if a ray be refracted successively by several glass plates placed parallel to each other, it will at length be polarized in one single direction perpendicular to the plane of refraction. Malus, by a very ingenious analysis of this phaeuomenon, has more- over shown that it is progressive, the first glass polarizing a small portion of the incident light, the second a part of that which had escaped the action of the first, and so on. M. Arago, measuring the successive intensities by a method of his own invention has shown that they are exactly equal to the quantity of light polarized in contrary directions at each reflexion. A phamomenon analogous to this is produced naturally in prisms of tourmaline, which appear to be composed of a multitude of smaller prisms, united together, but without any immediate contact. All light passing through one of these prisms perpendicularly is found to be polarized in a direc- tion perpendicular to the edges, so that if two such prisms be placed at right angles, on looking through them a dark spot is seen where they cross. This property of the tourmaline affords a very convenient method to impress on a pencil of rays a polarization in A A 18 G any required direction, or to discover such polarization when it exists. Moreover, M. Biot has discovered that certain solid bodies, and even certain fluids, possess the faculty of changing progressively polarization previously impressed on rays passing through them ; and by an analysis of the phaenomena produced by those substances he has shown that the same faculty resides in their smallest mole- cules, so that they preserve it in all states solid, liquid, and aeri- form, and even in all combinations into which they may happen to enter. M. Fresnel has found certain analogies between these phae- nomena and those of double refraction, which seem to connect the two together most intimately through the intermediation of total jeflexion. Since reflexion and refraction, even of the ordinary kind, modify the polarization of light, we may expect to find this effect produced when rays of light are made to pass through media of regularly varying density. It is accordingly found that all transparent bodies which are sufficiently elastic to admit of different positions of their particles round a given state of equilibrium, as glass, crystals, animal jellies, horn, &c. produce phaenomena of polarization when they are com- pressed or expanded, or made unequally dense by being considerably heated and then cooled suddenly and unequally. These phaeno- mena, discovered originally by M. Seebeck, have been since studied and considerably extended by Dr. Brewster, who has moreover re- marked, that successive reflexions of light on metallic plates pro- duced phaenomena of colours in which both M. Biot and he have recognized all the characters of alternate polarization. Knowing, by what has preceded, the experimental laws, accord- ing to which light is decomposed in crystals endued with double refraction, we may consider these effects as proofs proper to cha- racterise the mode of intimate aggregation of the particles of such bodies, and to give some insight into the nature of their crystalline structure. Light becomes thus, as it were, a delicate sounding in- strument w'ith which we probe the substance of matter, and which, insinuating itself between their minutest parts, permits us to study their arrangement, at which Mineralogists previously guessed only by inspection of their external forms. M. Biot has shown the use of this method, applying it to a numerous class of minerals desig- 187 nated by the general name of Mica, and he thinks he has decisive reasons to believe that several substances of natures so extremely different as to their composition and structure have been impro- perly comprised under that name. He has also made use of the phenomena of alternate polarization, to construct an instrument which he calls a colorigrade, which, producing in all cases the same series of colours in exactly the same order, merely by the nature of its construction, affords a mode of designation just as convenient for comparison as that furnished by the thermometer for temperatures. Many other experiments have been made, and are daily making; many other properties have been discovered in polarized light; but the limits of this Work do not allow us to give any detailed account of them, so that we have been obliged to confine ourselves to the results, which are, perhaps not the most important part of the sub- ject, but the easiest to explain ; our aim in this rapid sketch being rather to stimulate than satisfy the desire of knowledge on this branch of science which presents so vast a field for research both in theory and experiment, and which, though so lately discovered, has already furnished some useful applications to Physics and Mineralogy. 188 TABLE.' Of the Refractive and Dispersive Powers of different Substances, with their Densities compared with that of Water, which is taken as the Unit. The substances marked (*) are combustible. The refraction is supposed to take place between the given substance and a vacuum. t Substance. Ratio of, refraction. Dispersive power. Density. Chromate of lead (strongest). . . 2.974 0.4 5.8 Realgar. 2. o 49 0.267 3.4 Chromate of lead (weakest). . . . 2.503 0.262 5.8 * Diamond 2.45 0.038 3.521 * Sulphur (native). 2.115 2.033 Carbonate of lead (strongest). . . 2.084 \ n on i 6.071 “——weakest 1.813 c u.uy i 4.000 Garnet 1.815 0.033 3.213 Axinite. 1.735 0.030 Calcareous Spar (strongest)... . ——weakest 1.665 1.519 0.04 j 2.715 *Oil of Cassia 1.641 0.139 Flint glass 1.616 0.048 3.329 another kind ... . 1.590 Rock crystal 1.562 0.026 2.653 Rock salt 1 .557 0.053 2.130 Canada balsam 1.549 0.045 Crown glass 1.544 0.036 2.642 Selenite 1.536 0.037 2.322 Plate glass 1.527 0.032 2.4 S 8 Gum arabic 1.512 0.036 1.452 *Oil of almonds 1.483 , 0.917 *Oil of turpentine 1.475 0.042 0.869 189 Table of the Refractive and Dispersive Powers oj different Substances, continued. Substance. Ratio of refraction. Dispersive power. Density. Borax 1.475 0.030 1.718 Sulphuric acid 1.440 0.031 1.850 Fluor spar 1.436 0.022 ’ 3.168 Nitric acid 1.406 0.045 - 1.217 Muriatic acid 1.374 0.043 1.194 * Alcohol 1.374 0.029 0.825 White of egg 1.361 0.037 1.090 Salt water 1.343 1.026 Water 1.336 0.035 1.000 Ice 1.307 0.930 Air 1 .00029 0.0013 Oxygen 1 .00028 0.0014 * Hydrogen 1.00014 0.000 1 Nitrogen 1.00029 0.0012 Carbonic acid gas 1.00045 0.0018 190 ADDENDA The following Articles should have been inserted at the end of Chap. IX. Ramsden’s Dynamometer. The magnifying power of a telescope of any construction, ex- cept Galileo’s, is easily found practically by means of an instrument invented by Ramsden the Optician, the nature and use of which will probably be understood from the following account. If an astronomical telescope, for instance, be directed toward the Heavens, and a sheet of paper held behind the eye-glass at a certain small distance which will be easily discovered, a bright and clearly defined circle will be observed, which is, in fact, an image of the object-glass, or, more properly, the edge of which is an image of the rim of the object-glass, produced by the eye-glass. The diameter of this circle compared with that of the object-glass gives at once the magnifying pow'er of the instrument: for these diameters are as the distances from the eye-glass, one of which, namely, that of the object-glass, is F + F' (Alt. 166.), and the other is easily de- termined bv the common formula for the lens applied to this case. Calling it S, and observing to affix to each quantity its peculiar sign, we have 11 1 _ F $ ~ F' ~ F+F' ~ F\F+F') ' F+F' _ F Hence — s — — — . d T X ow F+F r is the same as the ratio of the diameters of the 191 object-glass, and its image on the paper; and— , is by Art. 166, the magnifying power of the instrument. Ramsden’s invention consists of a thin plate of horn, a substance preferable to paper, on which the diameter of the bright circle is measured at once by means of an accurate scale of equal parts drawn on it. The observer has only to place the instrument so that the diameter of the circle shall coincide with the line on which this scale is marked. The reason why Galileo’s telescope alone does not admit of the use of the dynamometer is sufficiently obvious : the eye-glass being a concave lens, no image of the rim of the object-glass is formed by it. The field of view of a telescope is determined in practice by a very simple observation. Any person at all acquainted with Astronomy is aware that the apparent motion of a star situated on the equator is at the rate of a minute of a degree in four seconds of time. If then the telescope be directed to a star on the equator, or very near it, which will answer quite w r ell enough for all usual purposes, we have only to observe the number of seconds occupied in its passage across the field of view', and multiply this number by 4 to obtain in degrees, a measure of that field. m MISCELLANEOUS QUESTIONS. 1. A luminous point is placed at the distance of 5 feet from a concave mirror of one foot radius ; to what point will the rays be reflected ? 2. In the above instance, determine the initial velocity of the focus, supposing the luminous point to advance towards the mirror at the ratio of 2 feet per second. 3. Supposing two rays inclined to each other at an angle of one degree to fall nearly perpendicularly on a convex mirror of which the radius is 5 feet, placed at 3j> feet from the point of in- tersection of the rays; what will be their mutual inclination after reflexion ? 4. A light is placed behind a screen, at the distance of two inches from a concave reflector of 9 inches principal focal length ; whence do the rays appear to proceed ? 5. What is the extreme aberration when rays diverging from a point fall perpendicularly on a convex mirror of 3 inches diameter, and 10 inches focal length, at the distance of 30 feet? 6. Supposing the focal length and aperture to be the same, which gives the greater aberration, a concave mirror or a convex one ? 7- A concave mirror being formed by the revolution of an ellipse (whose axes are 3 and 2 feet), about the axis major, to what point will those rays be reflected which diverging from the centre of the figure, fall on the circle whose diameter is the latus rectum ? 8. A small object is placed between two plane mirrors inclined to each other, so that a perpendicular drawn from the object to their intersection makes an angle of 5° with one, and of 12° with the other ; what is the number of images formed ? 193 9« A straight line is placed at the distance of 3 inches from a concave mirror of 9 inches radius : find the dimensions of the image. 10. Define the image of a portion of a parabola plac ed before a concave mirror, and having the centre of the mirror for its locus. 11. There are three transparent plates A, B, C, bounded by plane surfaces. The ratio of refraction between A and B placed in contact is 4, and between A and C 4 ; what is it between B and C? 12. A ray of light falls at an angle of 45° on the plane surface of a refracting medium ; the ratio of refraction is : what is the deviation ? 13. What must be the refracting power of a transparent sphere in order that it may just collect parallel rays to a point on its surface ? 14. At what distance from a luminous point should a convex lens of two feet focal length be placed, in order that the focus of refracted rays may be at the same distance on the other side of it ? 15. What equiradial lens is equivalent to a meniscus, the radii of which are 6 and 10 inches ? l6\ A double convex lens whose thickness is 3 inches and radii 30, and 20, is placed in air : what is its focal length ? 17* What is the focal length of a lens composed of water con- tained between two meniscus-shaped watch-glasses, the radii of the surfaces being 5 and 7 inches, and the thickness supposed incon- siderable. 18. Supposing a diamond sphere to be just inclosed in a cube of glass, what would be the focus of parallel rays incident perpen- dicularly at one of the points where the surfaces touch ? 19. The caustic by refraction of a plane surface, for rays diverging from a point, is the evolute of an ellipse or an hyperbola, according as the passage is into a denser or a rarer medium. 20. What is the form of the caustic produced by a parabolic conoid of glass, the incident rays being all parallel to the axis. B B 194 21. A small rectilinear object is placed before a double convex lens, of inconsiderable thickness, inclined to the axis at an angle of 30°, and the distance of its intersection with the axis from the lens, is four times the focal length; shew that the image is an arc of an ellipse, and find the axis major of it. 22. If an object be placed in the principal focus of a convex lens, the visual angle is the same, whatever be the place of the eye on the axis, 23. In a convex lens with surfaces of equal curvature, the spherical aberration exceeds the chromatic, if the semi-aperture be greater than ± of the radius. 24. If the dispersive powers of tw'O prims be inversely as their refracting angles, they will form an achromatic compound prism, when placed against each other in opposite directions. 25. It is required to achromatize a double concave lens of rock crystal, by means of a meniscus of iceland spar, which is just to fit into it. What must be the radius of the inner surface of the me- niscus, supposing those of the concave lens to be each 5 inches, the refracting powers of the substances being 1.547, and 1.657? 26. How many times will the surface of a minute object be magnified by a globule of spirit of wine ~th of an inch in diameter, supposing the least distance of correct vision to be 5 inches? 27. What must be the limit of the angular distance of two stars, that they may be both seen at once through an Astronomical telescope 4 feet long which magnifies 47 times, the apertures of the glasses being 3 inches and ^ of an inch. 28. Compare the fields of view of an Astronomical and a Galilean telescope, supposing the object glasses to be each 4 inches in diameter, and of 3 feet focal length, and the eye glasses of ^ inch aperture, and 1 inch focal length. 29- How much must the Galilean telescope, mentioned in the last question, be lengthened, to be used as a microscope, sup- posing that the object to be examined would be placed at 3ft. 2 in. from the object glass ? 30. What is the focal length of the eye-glass of Sir W.Herschel’s great telescope ? Seepage 131, Note. 195 31. Whereabouts should a plane reflector be placed in that telescope, according to Newton’s construction, so that the eye glass, of 1 inch focal length, may be set in the side of it? 32. What must be the ratio of refraction for yellow light, be- tween air and water, supposing the radius of the arc of that colour in the secondary rainbow to be 52° 10'. 33. A plane mirror two feet in height is placed against a ver- tical wall, so that its lower edge is 4ft. Sin. from the ground. What part of his figure can a man 6 feet high see in it, when standing upright on the ground, supposing the vertical distance of the eyes from the crown of the head to be - of the whole height? 34. If parallel rays be incident on a sphere of a given refract- ing power, find that ray of which, when produced, the part included within the sphere, is to the analogous part of the refracted ray in a given ratio. 35. Given the distance of the points of incidence of two parallel rays on a transparent sphere, one of which passes through the centre; required the distance between the points of emergence. 36. Given the apparent perpendicular depth of a fish under water; find the direction in which an arrow should be shot to hit it. 37. Let the surface of a plane reflector be always perpen- dicular to a line which revolves about one of its extremities, and cuts two other lines given in position; it is required to determine the focus of the reflector, so that an object moving , in the inter- section of the revolving line, with one of the given lines, the image shall move in its intersection with the other. 38. If a ray of light refracted into a sphere emerge from it after any given number of reflexions; determine the distance of the inci- dent ray from the axis, when the arc of the circle intercepted be- tween the axis and the point of emergence, is a minimum. 39. Supposing that a person can see distinctly in air, what must be the nature and power of a lens which will enable him to see distinctly under water. 40. If rays be incident, parallel to the axis, 011 the plane 196 surface of a plano-convex lens, whose thickness is t, and radius r: and emerge after two refractions at the plane surface, and one re- flexion from the spherical ; prove that the distance of the geometrical focus of the reflecto-refracted rays from the plane surface r — It sin inc. = ; where m — — — . 2 m sin refr. 41. If two parallel rays be incident on a transparent sphere, one perpendicularly, and the other at an angle of incidence 9 ; the arc included between the emergent rays is one whose sine sin 2 9 vr? — sin O' — sin 9 (m 2 — 2 sin 9 2 ) _ J m \ i m being the ratio of refraction. 42. Determine the apparent magnitude of a straight rectilinear object, placed at a given depth parallel to a surface of water, the eye being situated at a given point in the plane , which passes through the object perpendicular to the surface. 43. If the radius of the anterior surface of a concave glass speculum of inconsiderable thickness (c) = a; then if the radius 13 of the second surface = a + — c, the image of a distant object formed by reflexion at the first su rface will coincide with that pro- duced by reflexion at the second a ltd refraction at the first. /•/ A TK m. Ncsle.t Sou-fc 14 ’ Strand < ^LCt*vQ_ 91-^b 34,00 THE GETTY CENTER LIBRARY ^Kn<4ii>55^nS5SSSSr55S^rjaHHSHI®3?il^L«'W>i*W>Tr l >W?i?W!i?«>;