LENSES AND SYSTEMS OF LENSES. LENSES AND SYSTEMS OF LENSES, TEEATED AFTER THE MANNEE OF GAUSS. BY CHARLES PENDLEBURY, M.A., F.R.A.S., SENIOR MATHEMATICAL MASTER OF ST. PAUL'S; LATE SCHOLAR OF ST. JOHN'S COLLEGE, CAMBRIDGE. CAMBRIDGE: DEIGHTON, BELL AND CO. LONDON: GEORGE BELL AND SONS. 1884. CAMBRIDGE : PRINTED BY W. METCALFE & SON, TRINITY STREET, 3 IN the following pages I have limited myself strictly to the consideration of that which is involved in the term " Lenses and Systems of Lenses"; and I have treated the matter in accor- dance with the methods of Gauss. But I hope that in course of time I may be able to extend the book so as to cover a wider area in the field of Geometrical Optics. C. P. ST. PAUL'S SCHOOL, LONDON. February, 1884. CONTENTS. CHATTER PAGB I. REFRACTION AT A SINGLE SURFACE 1 II. REFRACTION AT TWO SURFACES IN SUCCESSION . . 28 III. REFRACTION AT ANY NUMBER OF SURFACES . . 52 IV. ACHROMATISM ....... 73 V. THE DETERMINATION OF THE FOCI AND OF THE PRINCIPAL POINTS OF A SYSTEM OF LENSES. THE NODAL POINTS 80 VI. THE DIFFERENT FORMS OF LENSES . . . .86 Appendix. ON CONTINUED FRACTIONS . 91 LENSES AND SYSTEMS OF LENSES. CHAPTER I. REFRACTION AT A SINGLE SURFACE. 1. IF a ray of light, or a pencil of rays traverse a system of coaxial lenses, the lenses being of any thickness, of any focal lengths, and of any refractive indices whatever the relation between the positions of the focus of the incident and the focus of the emergent pencil, and a formula for the magnification produced by the system of lenses, could formerly be determined only by an exceedingly cumbrous calculation. It was necessary, moreover, to repeat the process for each different system. For the sake of simplicity it was often assumed that the lenses were indefinitely thin. The laboriousness of the calcu- lations was thereby considerably reduced; but it is clearly a supposition which it is quite improper to make, except under very special circumstances. In a paper communicated by Gauss to the Koyal Society of Gottingen on the 10th of December, 1840,* it was shown how the solution of the problem could be made to depend upon the determination, for each system and once for all, of four fixed points situated upon the axis of the system. These points having been determined, the complete solution of the problem became a matter of simple algebra or Geometry. * C. F. Gauss Werke. Band. V. Gottingen, 1840. B 2 LENSES AND SYSTEMS OF LENSES. It is true that the calculation of the position of the four points is somewhat laborious, but the formulae obtained are symmetrical, although long, and the formulae for a system of n + 1 lenses can be deduced very easily from that for a system of n lenses. If therefore a table of formulae be calculated for 2, 3, 4 ... lenses, which can easily be done, the application to any par- ticular system is a question of Arithmetic and Algebra only. Gauss' method is applicable to any system of coaxial lenses, whatever be the thicknesses of the lenses, whatever be the refractive indices of the media which occupy the spaces between them, and whether the medium in front of the first lens is the same as that behind the last, or not. The problem however becomes much simpler when these first and last media are the same. One restriction, however, must be made. It is supposed that the angle which any ray makes with the axis, and also the distance from the axis of the point at which it cuts any refracting surface, are so small that their squares may be neglected. This is equivalent to neglecting aberration. 2. Let us consider a number of spherical surfaces, all of which have their centres of curvature upon a certain straight line. This straight line we may call the axis of the system. If a system of this kind be intersected by a plane which contains the axis, the section will be such as is represented in fig. 1. FIG. I. In this figure the straight line A^A^^.A^ upon which all the centres of curvature arc situated, is the axis of the REFRACTION AT A SINGLE SURFACE. 3 system ; and the points A^ A z , *..A n in which the axis meets the successive surfaces may be called the vertices of the surfaces. The surfaces may be of any number, of any degree of curvature, and at any distance apart ; moreover, they may have their convex surfaces turned either way. We will now suppose the spaces between each two con- secutive surfaces to be occupied by homogeneous, not doubly refracting, media, of known refractive indices. The media may be all] different, or two or more may be alike. But two similar media should not be adjacent to one another, for the effect would be the same as if the dividing surface were not there. 3. If we consider a particular case, and suppose the number of surfaces to be four, and the successive media to be air, glass, air, glass, and air, we so have the telescope in its simplest form, with one eyepiece and one object glass. 4. If now a ray of light proceed from a luminous point L, and cross the successive surfaces at the points P l7 P^ P 3 , . . . P ( , its course will be bent at each of these points, but in consequence of our hypothesis concerning the nature of the media, its course between any two of the points will be a straight line. The path of the ray may therefore be repre- sented by the broken line LP^Pf z ... P n L' (fig. 2). FIG, 2. The end we aim at is the determination of a relation between the lines LP and PL', so that when we know the 1 n / initial path of a ray entering the system, we may at once be able to ascertain its final path on leaving it. B2 4 LENSES AND SYSTEMS OF LENSES. 5. The general system includes also the case in which one or more of the surfaces are reflective. Tn order to make our results applicable to it, we have simply to consider the particular surface which is reflective, as if it were the boundary between two media whose refractive indices are fj> r and /j, r respectively ; fi r being the refractive index of the medium immediately preceding the surface which reflects. 6. With regard to the rays themselves, we shall consider those only which are inclined at very small angles to the axis of the system, and which cross the surfaces at points very near the vertices. If P be the point of incidence of a ray, A the vertex, and C the centre of curvature of the surface, it will follow that if the arc PA be small, so also will be the angle PGA (fig. 3). FJG.3. We shall suppose that the angle PCA is so small that all powers of its circular measure, higher than the second, may be neglected. Consequently, to this degree of approximation, we may consider the sine, tangent and circular measure of the angle PCA to be equal to one another. 7. In the first place, we will investigate formula? connect- ing the position of the incident ray, with that of the ray after refraction, at one surface only. 8. To find the angle between the incident ray and the refracted ray, after crossing one surface. Let C be the centre of curvature of any refracting surface, and A the vertex. Suppose XPX' to be the course of a ray which crosses the surface at the point P; and let the incident REFRACTION AT A SINGLE SURFACE. 5 ray and the refracted ray, produced if necessary, meet the axis at the points X, X' respectively (fig. 4). FIG 4. In this, and in all subsequent investigations, we will con- sider distances measured from left to right to be positive, and all distances measured from right to left to be negative. The above figure is so drawn that A & and AX', measured from A are positive, and A X negative. At the point P, the ray is bent through the angle XPR, the point B being in X'P produced. We will call this angle, the Deviation at P, and will denote it by 8. Let the normal CP be produced to (7', and let and then LCPX' But if fj> Q and /^ be the refractive indices of the two media, we have /jb Q sin <> = //.j sin ' = ; 6 LENSES AND SYSTEMS OF LENSES. therefore S = < - 9. We may put the expression for the deviation in another form. Let the arc AP= h, and denote the distances from A of X) X') and (7, by - w, v, and r respectively ; we then get if we neglect 10. If rays of light proceed from a luminous point X on the axiS) they will^ after crossing a refracting surface, meet again in one and the same point X', which also lies upon the axis. To find the relation between the positions of X and X'. With the same notation as before, we have (fig. 4) therefore /K O sin (PXA + PC A) = ^ sin (PC A - PX'A] ; therefore therefore cy _ & = ^~J^ ................... (i) v u r if we neglect h*. In this formula /^ , ytt, and r are quantities which depend only upon the nature of the media, and upon the curvature of REFRACTION AT A SINGLE SURFACE. 7 the separating surface; they are the same whatever be the ray which we may consider. Hence we see, that since the formula gives only one value of v corresponding to a par- ticular value of u, it follows that all the rays, which proceed from any point X on the axis, will after refraction meet again in a point X', also on the axis; the relative positions of these two points being given by equation (1). Conversely, for a particular value of v we get from (1) only one value for u. Hence the rays, which after crossing a refracting surface meet together in a point X 1 on the axis, must before incidence have proceeded from one and the same point X, also lying upon the axis. Again, if we consider X' instead of X as the luminous point, or the origin of the rays, and that the rays travel from right to left, it is clear that after refraction at the surface they will all meet together in the point X. Consequently the point X bears the same relation to X r , when X' is the origin of rays, that X' bears to X } when the origin of rays is at X. Two points such as X and X', which are related to one another in this way, are called Conjugate Points. 11. There are now two particular cases to be considered : (1) If the point X is at an infinite distance from A, we have u GO , and the incident rays are all parallel to the axis. The equation connecting u and v being v u r if we put u = co , we get *--- ........................ (2). fV-A*. (2) If the point X' be at an infinite distance from A, we have v = oo ; and the rays after refraction are parallel to the axis. The corresponding value of u is given by the equation = -^- ........................ (3). /*o-/*i 8 LENSES AND SYSTEMS OP LENSES. The two points determined by (2) and (3) are called the Focal Points or Foci of the given surface with respect to the given media, and are such that all rays whose paths in the tirst medium are parallel to the axis, will after refraction pass through the point given by (2) ; and all rays whose paths after refraction are parallel to the axis, must in the first medium have travelled in directions which, produced if necessary, would have passed through the point given by (3). The Foci are commonly denoted by the letters F' and F. Their distances from the vertex A are called the Focal Distances of the surface, and are denoted by the symbols /' and/. Hence r _ J ~~ and therefore /a /' -f /&,/= 0. N ^ If we introduce these symbols into equation (1), it becomes + = 1 (4). f f' 12. The equation - + J - = 1 connects the distances of two conjugate points from the vertex of the refracting surface. It is sometimes convenient to have a relation between their distances from the centre of curvature of the surface, and sometimes between their distances from the foci- These relations we will now determine. 13. To find a relation between the distances of two con" jugate points from the centre of curvature of the refracting surface. Let C be the centre of curvature and A the vertex of the surface, X the origin of light, XP the path of a ray which meets the surface at P, and after refraction passes in the REFRACTION AT A SINGLE SURFACE. direction X'P. The refracted ray produced backwards meets the axis at the point X'. CPE is the normal at P (fig. 5). and Let CX' = q GA =r arc AP h Then we have p Q sin XPE = /*, sin X'PE therefore /* sin (PGA + PXA) = ft sin (PCM + PX'A) ; /^ A \ /A h \ , T , 3 therefore a. - 4- - = A*, I - + H- MLh . \r ^? - rj l \r q-rj Hence, to the required degree of approximation, we have r(p-r) r(q-r)> therefore therefore or q p P (5), (6), The figure to which this article applies has been so drawn, that the distances from (7 of the various points might be all positive. Other cases may easily be deduced from this one. 10 LENSES AND SYSTEMS OF LENSES. 14. To find a relation between the distances of two con- jugate points from the foci. Let XF =d and X'F' = d' Then we have d = u /, therefore dd' = (u-f)(v-f) = uv- uf - vf+ff (7) This relation, dd =ff\ is called Newton's formula. 15. To our degree, of approximation, the tangent plane at the vertex may be considered as coincident with the surface itself. Let the incident and refracted rays meet the tangent plane at the points Q, Q' respectively ; and let L PXA = a | LPX'A = *\ (fig. 6). LPGA =0J X F1G.6. Then we have AQ A Q = AX.' tan a, , AQ A X tan a therefore ~ = r^-, , AQ AX tana -4^Ysina 2^'^7 approximately PX' AX_ AX' since PX\ PX differ from AX', AX only by quantities of the REFRACTION AT A SINGLE SURFACE. second order ; therefore 11 therefore the points Q, Q' and P coincide to our degree of approximation. Hence the course of the ray may be represented by the broken line XQX' instead of by XPX'. 16. If, through the foci F and F\ straight lines be drawn perpendicular to the axis, to meet the incident ray and the refracted ray in the points D and D' respectively , then Y being the point where the ray crosses the tangent plane at the vertex.* Let FD =z-| F'D' = z'\(&%. 7), AY = h) FIG. 7. Then we have from (4) u v or therefore, by similar triangles, y + = 1, or h ' h z + z =L * Die Haupt~ und Brenn- Puncte eines Linsen-systems, yon Carl Neumann. Leipzig. 1866. 12 LENSES AND SYSTEMS OF LENSES. 17. If the ray XYX' goes through the vertex A } we have h 0; therefore or z = z. 18. From the formula z -\- z = h we get a simple geome- trical construction, whereby the path of the refracted ray may be determined when that of the incident ray is given ; or, conversely. If the incident ray be given, so are the distances FD and A Y. Hence we can find AY- FD, which is equal to F'D'. If, therefore, through the focus F' we draw the line F'D' perpendicular to the axis and equal in length to A Y FD, we shall so determine the point 7>', which is a point on the refracted ray. The straight line YD is then the path of the ray after refraction. 19. DEFINITION. If any number of rays proceed from the same luminous point, and be refracted in crossing a spherical surface which divides the first medium from another of different refractive index, the rays will after refraction meet again in one and the same point. Two such points, namely the point whence the rays proceed, and the point at which they meet again after refraction, are said to be conjugate to one another with respect to the media and the surface considered. They are called briefly conjugate points. 20. This definition involves a general theorem, of which we have not yet proved more than a particular case ; that case, namely, in which the luminous point lies upon the axis. We will next prove the theorem to hold for a luminous point which does not lie in the axis, but only for rays from it which lie in a plane passing through the axis. Finally, we will consider the general case, and prove the theorem to be true for any position of the luminous point, and for rays from it in any directions whatever. REFRACTION AT A SINGLE SURFACE. 13 21. Wherever the luminous point may be, all rays pro- ceeding from it in the plane which contains the luminous point and the axis, will after refraction meet together again in one and the same point. We will suppose the plane of the paper to contain the luminous point and the axis. Let (?, chosen arbitrarily, be the position of a luminous point, and let QP^Q', QP^Q be the paths of any two rays which proceed from it in the plane of the paper, and which meet the tangent plane at the vertex at P, and P 2 respec- tively; Q' being the point where they meet together after refraction. Also let QP 3 Q" be a third ray which crosses the tangent plane at P 3 and meets the ray QP l Q' at the point Q". We will show that the points Q' and Q" coincide. From Q, Q' and Q" draw QN, Q'N' and Q"N" perpendi- cular to the axis, and meeting it at the points N } N', N" respectively (fig. 8). FIG.8, Through F and F' draw straight lines perpendicular to the axis, and meeting the rays at the points D^ D^ D 3 and D/, D^ DS respectively. We have then and 14 LENSES AND SYSTEMS OF LENSES. therefore, by subtraction, D,D^ D^D therefore -^f + r \ r * therefore, by similar triangles, , -i. ' ~ therefore therefore therefore therefore NF N'F'_ lt NA-NF N'A-N'F' FA F'A NA ' N'A~ J- + -,-7 = 1. NA N A In an exactly similar way, by considering the two rays ^" and QP 3 Q", we shall get JL f--i NA + N"A " Comparing these two results, we see at once that N'A=N"A. Hence the points N' and N", and consequently the points Q and ^", coincide. Wherefore all rays which proceed from any luminous point Q, will after refraction meet again in one and the same point Q'.* 22. From the result of the preceding article we obtain a simple geometrical construction, whereby we may determine the position of a point conjugate to a given one. Let Q be the given point. Then we know that the point conjugate to Q is the point of concurrence of all the rays * Lie Haupt- und Brenn- Puncte eines Linsen Systems, von Carl Neumann. Leipzig. 1866. REFRACTION AT A SINGLE SURFACE. 15 which proceed from Q. Hence it will be enough to deter- mine the point of concurrence of any two. It happens that there are two rays, whose paths we know from beginning to end ; namely, the ray which proceeds from Q in a direction parallel to the axis, and the ray which passes through F. The former of these passes through F' after refraction, and the latter after refraction travels in a direction parallel to the axis. Hence we have the following geometrical construction for the determination of the point conjugate to Q (fig. 9). FIG.S- From Q draw QP parallel to the axis and meeting the tangent plane at the vertex at P. Join PF', and produce it. Again, join QF and produce it to meet the tangent plane at P'. Through P' draw P' Q' parallel to the axis, and let it meet PF' produced at Q'. Then, since the rays QP and QF meet after refraction at the point $', it follows that Q' is the point conjugate to Q. 23. In Arts. 10 and 21 have been proved two particular cases of our fundamental theorem. We will now consider the general case, in which the position of the luminous point, and the directions of the rays, are both unrestricted. For this purpose it will be necessary to use the algebraic equa- tions to the straight line in Three Dimensions.* 24. We will take the axis of the surface for the axis of #, and any plane perpendicular to it for the plane of yz. * Verdet (Euvres, tome iv. part ii. Conferences de Physique. 16 LENSES AND SYSTEMS OF LENSES. The equations of the incident ray may then be written in the form y = mx + b] z = nx + c) ' or, more conveniently, in the equally general form = <^( X -a)+\ ' Mo ) where /* is the index of refraction of the first medium ; ~ , the tangents of the angles of inclination to the axis Mo Mo of x of the projections of the incident ray on the planes of xy and xz respectively ; a the abscissa of the vertex of the refracting surface ; a, > , c the coordinates of the point at which the ray meets the tangent plane at the vertex. In a similar manner the refracted ray may be represented by the equations W, , v z = -L (x - a) -f Cj /i, being the index of refraction of the second medium. It follows from our hypothesis, that - , , ! , , , M Mo M, M, c , b { and c, are all small quantities of the first order. 25. We will first investigate the relation between \ and , and between c, and c . Let P be the point of incidence, C the centre and r the radius of curvature of the refracting surface, A the vertex ; and let a plane drawn through C parallel to the plane zy be met by the incident and refracted rays at T and T' respectively. From the fact that the incident ray, the REFRACTION AT A SINGLE SURFACE. 17 refracted ray, and PC the normal at P, are all in the same plane, it follows that CT'T is a straight line (fig. 10)* FIG.IO. L PC A = 0, = a and let #, y, and z be the coordinates of P. Draw PNj NM perpendicular to the planes xy, xz re- spectively. Then = a + r (I cos0). Now, at the point P which is common to the incident and refracted rays, the expressions for y given by their respective equations must be the same, hence in, where x - Hence, substituting for x the value found for it above, we get (1-COS0) m.r 18 LENSES AND SYSTEMS OF LEXSEg. therefore b } = b fi + r (1 - cos 6) ^ - ^1 . In accordance with our hypotheses - and l are small /*0 /*! quantities of the ^r^ order, and 1 cos# is a small quantity of the second order, consequently is a small quantity of the third order and may be neglected. Therefore, to this degree of approximation, we get *, = *o5 and in a similar way it may be shown that C , = c o5 therefore the equations to the refracted ray may be written in the form 26. Next, we will determine the relations between and wi , . We have from the figure CP _ ~Cf~'"CP'~CT _ sincf), sin a sin a' ' sin< /u, sn a CT' But the ratio is also equal to the ratio of the y coor- dinates of the points T' and 7 1 , i.e. of the points on the two portions of the ray whose abscissa is a + r ; therefore _.!ii SinOC ' m o r i ^ REFRACTION AT A SINGLE SURFACE. !9 mr 7 fj> n sin a (mr \ therefore -f b = -. , ( + 6 J . But a and a' are both very nearly right angles ; therefore -. since , differs from unity by a small quantity of the second 771 order. Also -I- b Q is a small quantity of the first order ; Mo therefore, if we neglect small quantities of the third order, we get therefore therefore W?,f f*i mr In a similar way it may be shown that subject to the condition, that we may neglect small quan- tities of the third order. 27. We have now found expressions for the constants involved in the equations to the refracted ray, in terms of the constants involved in the equations to the incident ray. It follows that if the equations to the incident ray be then the equations to the refracted ray will be C2 20 LENSES AND SYSTEMS OF LENSES. 28. With the help of these equations may be proved the most general case of the proposition, that All rays which proceed from one and the same point will, after refraction, meet again in one and the same point. Let f , 77, f be any point on the incident ray, and f ', 77', ?' a point on the refracted ray ; then we have Eliminating 6 from these two equations, we get fi therefore In a similar manner it may be shown, that Each of these relations between ', 77', f and f, 97, f involves an unknown parameter in the first degree, m n in the first relation and in the second. These quantities depend upon the particular ray considered. Each is constant so long as we consider the same ray, but assumes a different value if we change to another ray. Now it will be noticed that the coefficient of m Q is the same as the coefficient of n . If we equate this coefficient to zero, we get REFRACTION AT A SINGLE SURFACE. 21 and consequently and r-ft^-aj+i ............ (iii). The constants which are involved in these three equations, namely, a, /K O , /& r, depend entirely upon the nature of the media, and on the position and curvature of the refracting surface. They are therefore absolute constants. It follows that these equations hold for any ray whatever that before incidence passed through the point From (i) we get '-a f t _ Mo "Mi I -a] ^ g- Mi 1 r Mo I" Mo therefore f = a + - *f~ . , Mo** - (M a -M,) (?-)* which gives us f ' in terms of f . If we substitute this value of f ' in equations (ii) and (iii), we get iff in terms of f, 77, and f ' in terms of , . Hence we see that we have found a point fV?' on tne refracted ray, whose position depends solely upon that of f^f, and is independent of the particular ray considered. Consequently, every ray which before incidence passes through the point whose coordinates are , ?, f, will after refraction pass through the point whose coordinates f, rj\ ' are de- termined by the equations (i), (ii) and (iii), whether the inci- dent ray meets the axis or not. This proves the theorem completely. 29. Equation (i) in the preceding article leads to a very important geometrical property of conjugate points. It shows us that (' depends upon f alone, and is independent of 77 and f. Hence, to all points which have the same f, correspond conjugate points, all of which have the same f. Conse- quently, if any number of points lie upon a plane perpen- dicular to the axis, the points conjugate to them will also all lie upon a plane perpendicular to the axis. 22 LENSES AND SYSTEMS OF LENSES. Two such planes are said to be conjugate to one another, and are called briefly Conjugate Planes. The property of conjugate planes may however be proved in an elementary manner in the case in which the plane of incidence contains the axis of the refracting surface. We will assume that the plane of incidence is coincident with the plane of the paper. 30. To investigate for mulce for the Geometrical determina- tion of the position of the point conjugate to a given one. Let P, P be two points conjugate to one another. The position of P being given, it is required to find that of P' FIG II. , \3 ? We know that PAP ' will be the course of the ray which passes through the vertex A. Draw PN) P'N' perpendicular to the axis and meeting it in JV, N' respectively; and through F and F' let FD, F'D' be drawn perpendicular to the axis to meet the incident and refracted rays respectively in D and D'. From the formula z + ' = h (Art. 16) we have, since h = 0, therefore FD and F'D' are equal in length, but on opposite sides of the axis ; therefore, by similar triangles, FD AF _ _ PN == AN ~ AN 9 and also _ __ P'N' " AN'" AN" REFRACTION AT A SINGLE SURFACE. 23 f.PN f.P'N' ~AN~ AN' ' numericall 7> and we have previously shown (Art. 21) that / f These equations are sufficient to determine AN' and P'N' when AN and PN are known. If we put M, y, v, y for A N, PN, AN', P'N' respectively, the equations may be written in the form fy = fj/_ u v (i), u . 31. Equation (ii) of the preceding article leads at once to the property of conjugate planes, which, in Art. 29, we deduced from the formula in the general theorem. The equation shows that the value of v depends only upon that of M, and is independent of y. Hence, if any number of points be taken, lying upon a plane perpendicular to the axis of the refracting surface, the points conjugate to them will also lie upon a plane per- pendicular to the axis. The distances M, v of two conjugate planes from the vertex are connected by the equation U 32. In Art. 22 it was shown how the conjugate point might be found graphically by following the paths of two known rays to their point of intersection. Now, however, by means of the equation u we may determine the plane conjugate to that which passes through the luminous point; and for the determination of 24 LENSES AND SYSTEMS OF LENSES. the point conjugate to the luminous point, we need follow only one ray to its intersection with the plane. The ray most suitable for the purpose is that which passes through the centre of curvature of the refracting surface. This ray crosses the surface without undergoing refraction, and its path from beginning to end is in one straight line (fig. 12). FIG,I2, Let Q be the luminous point, C the centre of curvature, and QN the plane perpendicular to the axis which contains Qj and let Q'N' be the plane conjugate to QN. Join QC, and produce it to meet the plane Q'N' at Q'. Then Q' is the point which is conjugate to Q. Hence we have the general theorem that all straight lines which join points to their conjugates pass through the centre of curvature of the refracting surface, 33. Definition. The point $', through which pass after refraction all the rays which proceed from $, is called the image of Q, 34. Let us suppose that there are a number of points such as Q, all of which lie on the plane QN. To each of these will correspond an image lying on the plane Q'N'. And if the points Q form in the aggregate an object of finite size and definite shape, the points Q' will form an image exactly similar in shape to the original object formed by the points Q, The size of the image will not, however, be the same REFRACTION AT A SINGLE SURFACE. 25 as that of the object. It is necessary therefore to determine the relation between them. 35. Definition. The ratio of the linear dimensions of the image to the linear dimensions of the object is called the magnification, or the magnifying power of the surface. 36. To find an expression for the magnification caused by refraction at a surface. Let Q be a point on the boundary of the object, then the conjugate point Q' will be a point on the boundary of the image (fig. 12). Consequently the magnification is represented by the ratio of Q'N' to QN. This ratio we will denote by m, and we will find an expression for m in terms of u and t?,' the distances respec- tively of N and N' from the vertex A. Q'N' We have m QN ON' , = y^ by similar triangles, GJ.V v r u r' But a._^ = /Vi^o v u r therefore fi^u (rv) fi Q v (r u) ; v r Lu n v , , r therefore u r therefore m = ^- (I). ft* 37. We will now introduce a notation which will sub- sequently be found very useful. The symbols u and v denote what may be called the absolute distances of JVand N' from A. 26 LENSES AND SYSTEMS OF LENSES. Let us denote and by u and v respectively, then the formula for the magnification becomes v m = u .(a). We may call u and v' the reduced distances of N and .AT from A. The same notation may be adopted for all linear magni- tudes, any reduced distance being obtained by dividing the corresponding absolute distance by the refractive index of the medium in which it is measured. 38. Helmholtz* formula for magnification** Let a and a' be the angles at which an incident ray XY and the refracted ray YX' are respectively inclined to the axis (fig. 13). FIG. 13. Then therefore u tan a = A Y = v tan a' ; m = ?8- tan a' .(3). This formula is an important one, for it connects the magnification with the angle of divergence of the rays, and is independent of the curvature of the refracting surface. * Helmholtz, Optiqne Fhysiologiqne. REFRACTION AT A SINGLE SURFACE. 27 39. It may be noticed that an expression for the mag- nification may also be obtained from equations (ii) and (Hi) of Article 28. We get =*- where Hence W (I - q) + 1 - (/* - A*) (f - a ) ,(4). CHAPTER II. REFRACTION AT TWO SURFACES IN SUCCESSION. 40. Many of the properties which belong to a ray of light when refracted at one surface only, may be extended almost directly to the case in which the ray is bent a second time in traversing a second surface. We assume that the second surface is related in position to the first in the manner described in fig. 1 ; that it is wholly independent of it as far as curvature is concerned ; and that the two surfaces are at any distance whatever apart. Two such surfaces combined form an ordinary thick lens, the character of which depends upon the curvatures of the two surfaces, the directions in which their concavities are turned, and the refractive index of the medium between them. The form of what we may call our standard lens is given in fig. 14 ; for, in accordance with our convention, both the radii of curvature are there positive. It is therefore the simplest case to demonstrate, as well as the one from which particular cases can most easily be deduced. FIG. 14- . We will also assume, as a rule, that the first and last media are similar air, for example and we will denote REFRACTION AT TWO SURFACES IN SUCCESSION. 29 the refractive indices of the successive media by /^ , /,& /u respectively. If, then, r 1 and r 2 be the radii of curvature of the two surfaces, the properties of any particular lens may be deduced from this general case by assigning to r t and r 2 their proper values, and to /^ the value of the refractive index of the particular material. 41. If rays proceed from a luminous point and traverse two refracting surfaces in succession, they will, after emergence, meet again in one and the same point. Suppose Pto be the luminous point. It has been proved in Chap. I. that the rays from JP* will, after refraction at the fast surface, . meet again in a certain point P,. The point P t , or the image of P with respect to the first surface, may be considered as a source of light from which rays proceed across the second surface. All these rays, after refraction at the second surface, will, in consequence of the same law, meet again in a certain point P', the point P' being the image of P l with respect to the second surface. Hence, all rays which proceed from a luminous point P and traverse two refracting surfaces in succession will, on emergence, meet again in one and the same point P'. This result holds whatever be the position of P, and whether the plane of incidence contain the axis of the lens or not. 42. It is obvious that, just as in the case of one surface only, P and P' are reciprocally related to one another, and that if we were to consider P' as the source of light, all rays from it which traverse the two surfaces would on emergence meet together at the point P. Hence P and P' are conjugate to one another with respect to the lens considered ; or, in other words, P' is the image of P. 43. The definition of conjugate points leads directly to the two following propositions : 30 LENSES AND SYSTEMS OF LENSES. (i) If 5 and a- denote any two incident rays, and s and a' the corresponding emergent rays, the point of concurrence of s and a is conjugate to the point of concurrence of 8 and a. (ii) If P and Q be a pair of conjugate points, and^? and q another pair, a ray which before incidence passes through P and p will after emergence pass through Q and q. 44. If a number of points P lie upon a plane perpen* dicular to the axis, all the points P' conjugate to them will also lie upon a plane perpendicular to the axis. For the points which are conjugate to the system P with respect to the first surface lie upon a certain plane perpen- dicular to the axis. This was proved in the former chapter. We will call this system of points P t , and we may consider the points as sources of light from which rays traverse the second surface. Again, we know that all the points P which are conjugate to the points P t with respect to the second surface also lie upon a certain plane perpendicular to the axis. But the points P are conjugate to the points P with respect to the lens. Whence the proposition follows. 45. If P and P' be conjugate points, and planes pass through them perpendicular to the axis, it follows that any point on one of the planes has its conjugate on the other. Two such planes are said to be conjugate to one another with respect to the lens, and are called briefly Conjugate Planes. Also the points where the planes meet the axis of the lens are called Conjugate Foci. 46. If any two conjugate planes be taken, and any number of points on one plane be joined to their conjugate points on the other, all these straight lines will meet the axis in the same point. REFRACTION AT TWO SURFACES IN SUCCESSION. 31 Let PN, P,JV;, P'N' be planes such that PN and are conjugate to one another with respect to the first surface, and Pj^Vj and P'N' conjugate with respect to the second surface; and let (7,, C7 a be the centres of curvature of the two surfaces respectively (fig. 15). FIG. 15. Let P' be conjugate to P with respect to the Jens, and let the straight line PP' meet the axis at the point C'. We will show that C' is a fixed point, for different positions of P in the plane PN. If Pj be conjugate to P with respect to the first surface, and therefore conjugate to P' with respect to the second surface, it has been proved in Art. 32, that the straight lines PP, and Pf pass through G* and (7 2 respectively. PN NC. We have constant, and therefore therefore therefore therefore PK NO PN r = constant ; NO' r, = constant ; NN' , = constant ; N'C' = constant ; LENSES AND SYSTEMS OF LENSES. therefore C 1 is a fixed point for all positions of P in the plane PN. 47. If the point N move along the axis to an infinite distance from the lens, the rays which proceed from it, in the limiting position, will before incidence be parallel to the axis, and after emergence will meet at a certain point N' on the axis. Again, if the point N' move along the axis to an infinite distance from the lens, the emergent rays which converge to N 1 will, in the limiting position, be parallel to the axis, and must before incidence have issued from an origin of light at a point N situated upon the axis. The limiting position of N' as N moves off to an infinite distance, and the limiting position of N as N' moves off to an infinite distance, are called the Principal Foci of the lens. They are commonly referred to simply as the Foci, and are denoted by the letters F' and F respectively. Hence all rays which before incidence are parallel to the axis will after refraction pass through the point F', and all rays which after emergence are parallel to the axis must have proceeded before incidence from the point F. 48. If we denote by / and /' respectively the infinitely distant points towards which N and N' move, it follows that / and F') and F and. /' are pairs of conjugate points. 49. The planes through the foci F and F' perpendicular to the axis are called the Focal Planes. 50. The planes conjugate to the Focal Planes are at an infinite distance. Hence, if the luminous point be on a Focal Plane, it follows that all the rays which proceed from it will, on emergence, be parallel to one another. Also, if an image fall on a Focal Plane, it follows in the same way that the incident rays must all have been parallel to one another. REFRACTION AT TWO SURFACES IN SUCCESSION. 33 51. In the case of refraction at one surface only it was shown that the distances u and v of conjugate foci from the vertex of the surface are connected by the equation or by '- + - = 1. u v We will now investigate the corresponding formula for the case of two surfaces. 52. To find the relation between the positions of conjugate foci when a ray is refracted through a lens. Let QPP'X^ be the path of a ray, which crosses the surfaces at the points P, P respectively, and let X^ JT 2 be the points at which the portions $P, PP', produced if necessary, meet the axis: X 3 being the point at which the axis is met by the emergent ray. FIG. 16. X/ Then JSTj, -5T 2 are conjugate to one another with respect to the first surface; and JT 2 , JT 3 are conjugate with respect to the second surface. Let A and A be the two vertices. The thickness of the lens, AA, is a positive quantity, but we may represent it by _ ^ if we consider t to be itself negative. With this notation we shall make the formulae more symmetrical. D 34 LENSES AND SYSTEMS OF LENSES. Let M, v t , be the distances of X^ X z respectively from the vertex A ; v l 4 t, v, the distances of X^ X 3 respectively from the vertex A; r, 5, the radii of the first and second surfaces respectively. Then we have ^ Mi ~ Mo ' Bnda lso V, + e wil by <7. Then we get For simplicity we will denote by p and v v 4 t These equations may be simplified still further by using reduced distances instead of absolute distances ; as explained in Art. 37. We will write u' for , v,' for - , t' for , and so on. The equations then become _!_ !_ 1 v+ It will however be unnecessary to use the accents if we remember that in future the symbols represent reduced and not absolute distances. So we get I i v i u 1 1 = are also connected by the equation AD - BC= ( , -j , and are therefore also equal to one another, if the extreme media are the same ; but they are not necessarily equal, if the extreme media are different. 67. Definition. The distance H'F' is called the Focal Length of the Lens. We may however take the reduced distance for the Focal Length, if it be distinctly understood that we do so. The two are the same when * = 1. 42 LENSES AND SYSTEMS OF LENSES. If we denote it by /, we have /= -r j and the formula of Art. 65 becomes = -% . v u f 68. Definition. The quantity A is called the Power of the Lens. 69. The property of Principal Planes may be viewed in a slightly different way. If we consider a pair of conjugate planes, and join any number of points on one plane to their conjugates on the other, we know that all these straight lines will meet the axis in one and the same point (Art. 46). But if one of the straight lines be parallel to the axis, it may be supposed to meet it at an infinite distance. Hence they will all meet the axis at an infinite distance, and consequently they will all be parallel to the axis. But if two conjugate planes be so situated that the line joining any point on one of them to its conjugate on the other be parallel to the axis, it is obvious that to an object on one plane will correspond an image of exactly the same size on the other. This is the property which belongs by definition to Principal Planes. Consequently we may define Principal Planes and Prin- cipal Points as follows : If two conjugate planes be such that the lines joining pairs of conjugate points on them are all parallel to the axis, these planes are called Principal Planes, and the points where they meet the axis are called Principal Points. 70. It may be noticed that conversely if any straight line parallel to the axis, meet the Principal Planes at the points P and P' respectively, then P and P' are conjugate to one another. 71. To determine geometrically the position of a point P 1 which is conjugate to a given point P. REFRACTION AT TWO SURFACES IN SUCCESSION. 43 We will assume tbat the Foci F and F', and the Prin- cipal Points H and H' have been determined, and that both the Principal Points lie between the Foci (fig. 17). FIG. n. \ We know from the definition that all the rays from P will, after refraction, pass through P'; hence it will be sufficient to find the ultimate intersection of any two of them. We will select that which proceeds from P in the direc- tion Pa, parallel to the axis, and which meets the first Principal Plane at a point a ; and also the ray PF@ which passes through the focus F and meets the Principal Plane at a point fi. The incident ray Pa passes through 7, the point at infinity, and a. Hence, (Art. 43), the corresponding emergent ray will pass through the points conjugate respectively to / and a. Let Pa be produced to meet the other Principal Plane in a'. Then we know that a is conjugate to a, and F' to I. Consequently a'F' will be the direction of the ray on emergence. Again, the incident ray PF/3 passes through F and /. If a straight line be drawn through (3 parallel to the axis } and meeting the other principal plane at /3', then /3' is con- jugate to /3. Also I' is conjugate to F. Hence /3'P' drawn parallel to the axis, will be the path of this ray on emergence. The point P', which is conjugate to P, is the intersection of a'F' and /3/3' produced. 44 LENSES AND SYSTEMS OF LENSES. We have consequently the following Geometrical con- struction for the determination of P'; through P draw POLO! parallel to the axis to meet the second Principal Plane at a', and draw also PF(B to meet the first Principal Plane at /3. Draw 0ff parallel to the axis. Then a!F' meets /3/3', pro- duced if necessary, at the point P' required. 72. It should be noticed that if the first and last media be the same, the figure Pa'P'fi will be a parallelo- gram. 73. To investigate algebraical formula connecting the positions of two conjugate points which do not lie upon the axis. The letters in the accompanying figure have their cus- tomary signification. We will assume, for simplicity, that H and H' lie between F and F', and we will consider numerical values only. The signs can be readily determined by inspection. FIG. 18. let Let PN t P'N' be drawn perpendicular to the axis ; and PN = y, NH = x HF =// H'F'=f REFRACTION AT TWO SURFACES IN SUCCESSION. 45 Then, from the similar triangles jBHF and ySaP, we get HF ^P'' / = x Again, from the triangles a'H'F' and a'/3'P', we get in a similar way x' y + y' ' 7 -/ therefore / + % = !. xx and ^ = -'-4 u; ic If ic and y be given, these two equations are sufficient to determine x and y '. Hence, if P be given, the position of P' can be determined. 74. If the extreme media be the same, we have /and /' equal to one another numerically, but of opposite signs. In this case the formulae of the preceding article reduce to and * = * x x ) The first of these is similar to the formula obtained in the case of refraction at one surface only ; the distances u and v in that case being measured from the vertex. The second formula, y - = y - , shows that if Pand P (fig. 19) be any two conjugate points, and if they be joined to H and H' respectively, the straight lines PH and P'H' are parallel to one another ; that is to say, an incident ray through the 46 LENSES AND SYSTEMS OP LENSES, FIG. Id. Principal Point H will produce a parallel emergent ray through the other Principal Point H'. Hence for a lens, when the extreme media are the same, the Principal Points possess the property which belongs in the general case to what are called the Nodal Points. The general case will be considered in a subsequent chapter. 75. The results of Art. 73 may be put in another form. We have g _ y + y ~~ therefore similarly therefore (x-f)(x'-f)=ff (iii). 76. If straight lines be drawn through F and F' per- pendicular to the axis, to meet the incident and emergent rays respectively in D and D' then Let the incident and emergent rays, produced if necessary, meet the axis at the points X, X' respectively (fig. 20). REFRACTION AT TWO SURFACES IN SUCCESSION. 47 FIG, 20. X F H Then we know that X and X' are conjugate to one another; therefore / /' therefore therefore therefore therefore HX * H'X' HX-f H'X'-f HX H'X' FX F'X' JJV 77"' V ) JHj\. JjL A. FD F'D' Ha. ' H'al ' FD+F'D = Hc for 0.0! is parallel to the axis, and therefore H 'a! = Ha.. If we denote FD, F'D\ Ha. by z, z', h respectively, we get which is similar to the corresponding formula obtained in Art. 16 in the case of a ray refracted at one surface only.* 77. The image of a plane luminous object formed by a thick lens. We assume that the object lies on a plane P perpen- dicular to the axis ; and that it may be regarded as a cluster of luminous points, each of which has its image on the plane * Carl Neumann : Ueber die Haupt- und Brenn- Puncte. 48 LENSES AND SYSTEMS OF LENSES. P conjugate to P. These point-images, in the aggregate form the image required. The determination of the position of the plane P' when the Principal Planes have been found, has been explained in Arts. 71, 73. It will be shown in Chapter V. how the Prin- cipal Planes themselves may be determined experimentally. Again, it has been proved that all the lines joining cor- responding points of object and image, meet one another at a point C' on the axis. Hence the image is similar in form to the object, and will be inverted or upright according as G ' lies between them or not. 78. To determine the image graphically* Let us take any point of the object and join it to> its conjugate point. This line, produced if necessary, will meet the axis at a point C '. If we now describe a cone which has its vertex at 0', and the object P for its base, then the section of this cone made by the conjugate plane P' will be the image required. 79. Definition. When a ray parallel to the axis is refracted by a lens, it receives a certain deviation. A thin lens, which, when placed coaxally with the lens, would produce the same deviation in the same ray, is said to be equivalent to the given lens. It is called, briefly, the Equivalent Lens. 80. To find the deviation produced by a thick lens. Let QPP'Q' be the course of a ray, which before inci- dence is parallel to the axis, and which crosses the refracting surfaces at the points Pand P' respectively (fig. 20 a). Let Af = h, and let the radii of the surfaces be r and s. We will consider all deviations positive when they are towards the axis, and we will denote the deviation at P by - & ; then from Art. 8 we have REFRACTION AT TWO SURFACES IN SUCCESSION, HG.ZOM P' 49 We have, from the Geometry of the figure, h + 1 , if - t = absolute thickness of lens, = h (1 -f pi], if t reduced thickness. Again, deviation at P' -M. [*(*+pQ ..J Mo I s ) - (1 + pt) /* -/* t But total deviation = deviation at P-f deviation at P' therefore total deviation = + 3 + ^ ( (1 50 LENSES AND SYSTEMS OF LENSES. 81. The focal length, of the equivalent Lens. In Art. 9 we have shewn that a single refracting sur- face, which may be considered as an indefinitely thin lens, produces in a ray parallel to the axis a deviation f being the absolute focal length of the surface. Also (Art. 80) a thick lens produces a deviation _Ah ~ /*0 * Hence the absolute focal length of the thin lens which would produce the same deviation as a given thick lens, or, in other words, the absolute focal length of the equivalent lens, is given by the formula h _ Ah J~ ^ /-2- Consequently, the reduced focal length of the equivalent lens 1 ~ A' 82. If the results of Arts. 67 and 81 be compared, it will be seen that the focal length of the equivalent lens is equal to what we have defined as the focal length of the thick lens itself, for each is equal to -r . This equality is evident geometrically. For if we con- sider a ray which before incidence is parallel to the axis, we know that on emergence it will pass through F' (see fig. 21) ; the deviation being represented in the figure by the angle between PO.OL produced and a!F'. And if the ray pass through the equivalent lens, the deviation produced being the same, REFRACTION AT TWO SURFACES IN SUCCESSION. 51 FIG.2L it is clear that the emergent ray will cut the axis at SL distance from the lens equal to H'F'^ the same as before. Consequently the two focal lengths are equal to one another. 83. In this chapter we have considered the passage of a ray of light across two surfaces only, and we have proved (Arts. 5"2, 58j 59) that the positions of conjugate points are 1 connected by the relation _ Cu + D " and that the magnification is given by the formula m= G - Av, 1 or m In the next chapter we shall prove that however mainy surfaces be crossed by the ray, the formulae which correspond to those given above are exactly analogous to them; and that in the general case, for any number of surfaces, the constants A, B, (7, D are connected with one another in exactly the same manner as in the case of an ordinary thick lens (Art. 53). E2 ( 52 CHAPTER III. REFRACTION AT ANY NUMBER OF SURFACES. 84. The refracting surfaces to be considered are spherical, and have all their centres of curvature upon the same straight line, which is the axis of the system. The spaces between the surfaces are supposed to be occupied by homogeneous media, such that the medium on one side of any surface and that on the other side have different refractive indices. By assigning suitable values to the indices of refraction, to the distances between the vertices, and to the radii of curvature, this system of refracting surfaces may be adapted to the case of any system whatever of any number of co-axai lenses, simple or compound. 85. Several of the properties that have been proved for a lens can be extended at once to the general case. It will be sufficient if we merely state the propositions, and then leave the reader to prove them by generalising the corresponding propositions in Chapter II. In many cases the requisite alterations will be but verbal. 86. (i) // any number of rays proceed from a luminous point P, and traverse a system of any number of refracting surfaces in succession, they will, after emergence from the system, pass through one and the same point P' (Art. 41). Two points such as P and P are said to be conjugate to one another with respect to the system of surfaces. (ii) The point of concurrence of any two incident rays is conjugate to the point of concurrence of the corresponding emergent rays. REFRACTION AT ANY NUMBER OF SURFACES. 53 (Hi) If P, P' be a pair of conjugate points, and p, p another pair, then the incident ray which passes through P and p will produce an emergent ray passing through P' and p. (iv) To a system of points P lying on a plane perpendicular to the axis, corresponds a system of points P , which are conjugate respectively to the points P, and also lie on a plane perpendicular to the axis. Two such planes are called Conjugate Planes, that is to say, planes conjugate to one another with respect to the system of surfaces considered. The points where they meet the axis are called Conjugate Foci. (v) If P and P' be two conjugate points, a plane through P perpendicular to the axis will be conjugate to a plane through P perpendicular to the axis. (vi) If two conjugate planes be taken, and any number of points on one be joined to their conjugates on the other, all these straight lines will meet the axis at the same point. 87. If the incident rays are all parallel to the axis, they may be considered as proceeding from a point / on the axis at an infinite distance from the vertex of the first surface. After emergence from the system they will meet together at a point F' on the axis. Again, the rays which after emergence from the system are all parallel to the axis may be supposed to meet the axis at an infinitely distant point I', and must before incidence on the first surface have proceeded from a certain point F on the axis. The points F and F 1 are called the Principal Foci, or briefly, the Foci of the system. The planes through them perpendicular to the axis are the Focal Planes of the system. Also F, 1' and /, F' are pairs of conjugate points. The properties (Art. 50) of rays which before incidence or after emergence, meet at a point on a Focal Plane, are true also whatever be the number of surfaces crossed by the rays. 88. We will now suppose that the source of light is at a point X on the axis of the system, and we will denote by u 54 LENSES AND SYSTEMS OF LENSES. its reduced distance from the vertex of the refracting surface nearest to it. The reduced distance of X^ the point con- jugate to X with respect to the first surface, will be denoted by tv If X 2 be conjugate to X^ with respect to the second surface, their distances from the second vertex may be denoted by v l + fj and t? a , where, with the notation explained in Art. 52, ^ is the reduced distance between the two vertices. Again, if 2 be the distance between the second and third vertices, the distances from the third vertex of X y and the point conjugate to it with respect to the third surface may be denoted by v a + 2 and v a ; and so on. In this way, if we suppose that there are n surfaces, we Jiave for the last pair of distances V, + '-, and (abcY (alcd)'* the successive numerators and denominators being connected by the relations (be) = c (b) +(1), (bed) = d (be) + (b), &c. The functional sign is purposely the same throughout, for we know that (bed) in the fourth numerator is exactly the same function of 6, c, df, that (abc) in the third deno- minator is of a, b, c. Among the known properties of the numerators and denominators of these successive convergents we have the following ; (abcd...hk) = (kh...dcba) ................. I, that is to say, any function (abcd...hk) is unaltered if we reverse the order of the letters. Also -y (abc) =-7- (cba) da v ' da v II. &c. These theorems are very important, but we have not been able to find them in any English book on the subject. It may therefore be useful to the reader if we give proofs of them in an Appendix at the end of this volume. 95. The Quantity A is the denominator of the penulti- mate convergent to the continued fraction JL l 1 i 11 ~+ + """ + + u ' 58 LENSES AND SYSTEMS OF LENSES, and is consequently a function of Hence, we may write o Pi- and therefore by Theorem I of the preceding article, we have also and, by Theorem II, we have dA B = dp, OJ D Hence, whatever be the number of surfaces, Z?, C and D can always be expressed in terms of A. These results are precisely analogous to those obtained in Art. 53. 96. Let us represent the denominators of the successive convergents to the continued fraction JL 1 1_ _1_ ^ 0*4"'' ~\~ D ~\~ t ~\~ O rn H 1 r n 1 ' i ' r i then we know that each of these quantities is connected with the two preceding it by the relations &c., of which the last are REFRACTION AT ANY NUMBER OF SURFACES. 59 D7. If, for example, we suppose that there are four surfaces, we have 2n I 7, and we get - A - A A - A. = o = o -0 = Hence, solving these equations for A^ we get A. 1, -p,J -1, o, o, o, o, 1, -*i, -1, o, o, o, o, 1, ~P 2 1 -1, o, o, o, o, 1, -*, -1, o, o, o, o, 1, -Pa -I o, o, o, o, o, 1, -'3 o, o, o, o, o, 1 o, -p,> -1, o, o, o, o, 1, -*,1 -1, 0, o, o, o, 1, -Pa -1, o, o, o, o, 1, - -1, o, o, o, o, 1, -p.) -1 1, o, o, o, o, *, -, P41 o, o, o, o, o, 1 The coefficient of A n in the above result is a determinant which has all the terms on one side of its diagonal zero, .and all the terms in the diagonal unity. The value of the .determinant reduces therefore to unity ; and we get -Pi, -1, > o, o, ) If -,* 1, > J J o, 1, -P.J -], J ? 7 ) 1, -4 -1, J J o, o, 1, -p> -1, 7 > o, o, li -tt -1 o, o, > > o, 1, ~P4 00 LENSES AND SYSTEMS OF LENSES. 98. If, again, we suppose that there are two refracting surfaces, we have 2n - 1 = 3, and therefore -/> - therefore o, i, -P. = ftft, + ft + ft. which agrees with the result obtained for a thick lens in the preceding chapter. 99. For the case of one surface only, we have which, too, agrees with previous results. 100. For the general case of n surfaces, we have o, o, o, 0, 0, 0, 1, -p z , -1, 0, o, o, i, -V-i, i, -t 101. We may now investigate a relation whereby the value of A for n surfaces may be determined from that for 9il surfaces 5 that is to say, we will investigate an equation connecting A^ and ^4 2 ,,_ 3 . We know that A^ is the denominator of the last con- vergent to the continued fraction L 1 1 1 1 REFRACTION AT ANY NUMBER OF SURFACES. 61 that is, to the continued fraction _]_J_JL 1_ 1 1 p^*T+ p7+ * 2 + '"U+p? obtained by reversing the order of the letters in the pre- ceding one. If we denote the denominators of the successive con- vergents to the latter fraction by A A A A -*1J -^t? -^31 " ^2-l> we have A^ = Pn A^ + A^ . therefore 102. It is clear that A=/>, ........................... From this, by means of the formula just proved, we get Hence -4 B = (1 + P 3 * 2 ) ^ 3 + Ps ^ = (1 + PA) (Pt + P 2 + P.PA) + P (1 + M)-("i), and so on. The formula of Art. 101 consequently enables us to de- termine the value of A for three surfaces from its value for two, then its value for four surfaces from its value for three, and thus by successive steps to its value for any number of surfaces whatever. Moreover, from the value of A in any particular case, we can determine the corresponding values of B, C and D. Thus we see that .the values of -4, B, C, D for any system of surfaces may be deduced from the value of A for one surface only. 62 LENSES AND SYSTEMS OF LENSES. 103. Magnification. It has been proved in the particular case of a lens, that m or the magnification can be expressed as a linear function of v. and as a linear function of u. m We will now show that the magnification can be so ex- pressed, whatever be the number of refracting surfaces ; that the formulae in the general case are precisely analogous to those obtained in the particular case ; and that they involve the constants A, B, and C in exactly the same way. 104. The magnification produced by a system of n re- 1 fracting surfaces can be expressed as a linear function of v. Let us consider the system formed by the first n 1 surfaces, and let (w) n _ t denote the magnification produced by it. Also let t> n _, be the reduced distance of the (n l) tb image from the (n l) th vertex. We will assume that (^) w _ t can be expressed as a linear function of t^, and thence show that on this supposition, the magnification can again be expressed as a linear function of V M , if we cause the rays to pass through an additional or n th surface ; v n being the reduced distance of the n tb image from the vertex of the n th surface. We will assume that and we will suppose that t n ^ is the reduced distance between the (n - l) th and the n ih vertices. It has been proved, that the magnification produced by a single surface Consequently the magnification caused by the n th surface alone REFRACTION AT ANY NUMBER OF SURFACES. 63 Hence, if we denote by m the whole magnification pro- duced by the n surfaces, we have 1YI = (ry #15 ) But I- L 41 therefore therefore m = (7 + *_, - a (Vt + '-,] Vi Hence the magnification can be expressed as a linear function of v n . 105. Again, if P and P' be two points that are conjugate to one another with respect to the system of surfaces, we know that which ever we consider as the source of light, the rays from it will produce an image at the other point. The same is true of objects and images of a definite size. Consequently, if we transpose the terms object and image in the preceding article, and consider as an origin of light what is there treated as an image, the image of it produced by the system of surfaces will be what we originally considered as the object. The magnification produced by the system thus trans- posed will clearly m Also the quantity corresponding to v n is obviously u. Hence, we obtain as a result, that for a system of any 64 LENSES AND SYSTEMS OF LENSES. number of surfaces the inverse magnification, or , can be expressed as a linear function of w, or - = /3'+aV m 106. We have proved so far that the magnification produced by a system of n refracting surfaces may be ex- pressed in either of the forms and - 7/1 If we suppress the subscript letter n, the former becomes m = y a'u. These formulae are similar in form to those obtained for a lens. We have now to determine the yet unknown constants /3', a ",7>'. 107. The formulae for the magnification, namely, m = y av 1 i = /3'+a"u ' m ) must clearly be equivalent^ so that if in the former we were to put for v its known value in terms of w, we should certainly get the latter. It follows, therefore, that if we eliminate m from the two expressions, the result of elimination will be an expression involving only w, v, and constants, which must be identicallv the same as the known relation Eliminating wz, we get l = ( 7 '-a't;)&8' + a"M), or 1 = 7 '/9' + a'V u - j3'av - a!a."uv, or OLOL'UV + ffa'v - a."y'u + 1 - y'fl' = (i) REFRACTION AT ANT NUMBER OF SURFACES. 65 Cu+D Also from v = -; - ^ . we get Au + B ' * * \ / * The equations (i) and (ii) must be identically the same; hence, comparing coefficients, we have oca" _ ffa' _ tt'V _ 1-7'ff. J. " : 6' -/> ' therefore r = -^ = X -4 (7 therefore, substituting for a', a", 7', y8', in the equation a'q" 1 - A -D we get \\'AD = - 1 + therefore XV (AD - 5(7} - 1 ; therefore XV = 1 .......................... (iv). Again, substituting in the formulae for m the values of a', a", /:?', 7' given by (iii), we get m = A, <7 The resemblance between the formulae for n surfaces and the formulae for a lens has now become more distinct ; but the method described here does not lead to the determination of X in an elementary way. The method is in itself impor- tant, but for our special purpose it will be better to consider the matter directly. 108. To investigate a formula for the magnification pro- duced by a system of n refracting surfaces. Let us consider the last n 1 surfaces and let A', B' y , C'j D 1 be the corresponding values of A y B, C, D. F 66 LENSES AND SYSTEMS OF LENSES. If m denote the magnification produced by these n - 1 surfaces, we will assume that *- AV+B>> . where u is the distance of the second image from the second vertex. The magnification produced by the first surface is given by i = - m \ v i * Hence wi, or the resultant magnification produced by the whole system, is given by -=(^V+')- m J v But 1 1 u Vj w H ' therefore u - =l-f w^; therefore 1 _ A>u ' + A't,)(l+up,) = u (A 1 -f ^> t + ^Xp,) -f 5' + A\. Now -4' is the denominator of the last convergent to the continued fraction _J_ JL _i_ I P 2 + ^+P+;-P ; we have also B' = -= . dp, ' and we have proved that successive denominators are con- nected with one another by the equation . + P., (Art. 101). Hence A = ( Pl t t + 1 ) A' + p"#', and B = ^=A' dp, REFRACTION AT ANY NUMBER OF SURFACES. 67 therefore Au + B .............................. (i) ; m and if in this formula we substitute for u its value in terms of v found from the equation _Au + B = Cu~VD> we get m = - Av + G ..................... (ii). Hence the magnification produced by a system of n refracting surfaces can be expressed in exactly the same form as that produced by a lens. 109. If the foci be taken as the origins of distances, the formula for the magnification become (Art. 63) - = m u and v being measured from F and F' respectively. 110. Helmholt^ formula for the magnification produced by a system of n surfaces. If /u. , yu-j, /i a , ...fJt> n be the refractive indices of the suc- cessive media; and a, a 1? a 2 , ... a n be the angles at which the succes- sive portions of the ray are inclined to the axis of the system ; and m^ m^ m^...m n the magnifications produced by the surfaces; we have u> n tan a m i = * - ? /*, tana, a, tana. m, = C1 - - 1 . /a 2 tana s 7 ^ r! tani i1 /x n tana H r /x tana therefore wi = ^ . F2 68 LENSES AND SYSTEMS OF LENSES. 111. Points of Unit Magnification. If in the formulae for the magnification \ve put w = we get 0-1 The points so determined were called by Gauss the Principal Points of the system of surfaces; the planes through them perpendicular to the axis are the Principal Planes. The Principal Planes are conjugate to one another with respect to the system of surfaces, and are such that to an object on one plane will correspond an image on the other, the image and the object being of exactly the same size. 112. From the definition of Principal planes, that they are Conjugate Planes and also Planes of Unit Magnification, it follows that a straight line parallel to the axis will meet them in points that are conjugate to one another. 113. All that we have said with respect to the Principal Points of a lens is equally true of the Principal Points of a system of n surfaces. It will be enough perhaps if we simply state the facts, the proofs of them being already given in the corresponding articles of Chapter II. (i) If the Principal Points be taken as origins from which distances are measured, we get m = - Av + 1 I 1 , (Art. 64), = Au + 1 m J I- 1 -- 4 (Art. 65). REFRACTION AT ANY NUMBER OF SURFACES. 69 I (Art. 66), HF - i these being reduced distances. In absolute measurements HF =-? If the two extreme media be the same, these absolute distances become numerically equal. The absolute distance H'F' is called the Focal Length of the system. If we denote the reduced distance H'F' by /, we get /= -= , and therefore 1 f Also the Foci lie either both between or both beyond the Principal Points. (iii) To determine the position of the point P' conjugate to a given point P, we have the following construction : Through P draw Pan! parallel to the axis to meet the second Principal Plane at a', and draw also PFft to meet the first Principal Plane at ft. Draw ft ft' parallel to the axis. Then a'F' produced will meet ftft' at the point P' required. (Art. 71). (iv) If (xy}i (#V) be the coordinates of two conjugate points P and P' respectively, as explained in Art. 73, we have X X /and/' being the absolute distances HF, H'F respectively. Also (*-/)(*'-/')=//' 70 LENSES AND SYSTEMS OF LENSES. (v) If straight lines be drawn through F and F' perpen- dicular to the axis to meet the incident ray and the emergent ray respectively in D and D', then a being the point at which the incident ray meets the Prin- cipal Plane. (Art. 76). 114. The image produced by a system of surfaces can be determined in the same manner as in the case of a lens. (Arts. 77, 78). 115. Definition. When a ray parallel to the axis is refracted by a system of surfaces, the ray receives a certain deviation. A thin lens, which when placed eoaxally with the system would produce the same deviation in the same ray, is said to be equivalent to the given system of surfaces. It is called, briefly, the Equivalent Lens. 116. To find the deviation produced by a system of surf aces. If a ray parallel to the axis cross a refracting surface at a distance h from the axis, it has been shown (Art. 80) that the deviation produced ph = ^' where //, is the refractive index of the medium following the surface, and p the power of the surface. Let the deviations after crossing the successive surfaces be * **- 8 nl and the distances from the axis of the points of incidence be A n h h 3 , ... h n - the powers of the surfaces being as before Pit P P* '" Pn' REFRACTION AT ANY NUMBER OF SURFACES. 71 Assuming as before that the deviations are measured towards the axis we get easily (as in Art. 80) that, and so on, t tj &c. being the absolute thicknesses. Now if we write S\ for /^S,, 8' 2 for /* a S 4 , &c., thess equations become, using reduced thicknesses, If we form the continued fraction . 1 1 it is easy to see that the denominator of the last convergent will be y tt , i.e. HHTj and the same denominator has also "i "i been represented by A. Hence 117. If /be the absolute focal length of the equivalent lens, we have therefore therefore the reduced focal length = . 72 LENSE8 AND SYSTEMS OF LENSES. 118. Again, since the denominators of the last conver- gents to the fractions and are the same, it follows that the two reduced focal lengths of the system are equal to one another. Hence if we denote the absolute lengths by / and /', we have /= /*o A*.' If the first and last media are the same, we have 1*0 = A*. 5 bence / = /' numerically, or 'f~f algebraically. 119. It should be noticed that the so-called equivalent lens merely produces the same deviation as the system. It does not bring the rays to the same focus, nor therefore does it produce an image in the same position. It may be shown however that both these conditions may be satisfied by using two thin lenses properly situated. ( 73 CHAPTER IV. . ACHROMATISM. 120. When an object is viewed through a lens or through a system of lenses it commonly appears to have a sort of coloured border. This is due to the fact that ordinary sun- light is composite and not simple. It is a combination of an indefinitely large number of different kinds of light, which have different degrees of refrangibility and all varieties of colour. The various component rays which make up a resultant ray of* ordinary sun-light can be separated from one another by allowing the ray to pass through a glass prism. If the light as it emerges be cast upon a screen, there will appear an elongated continuous coloured band. The component rays being of different refrangibilities must necessarily meet the screen at different points. Now, if P be a luminous point and P be its conjugate point, or image, we know that the position of P' is a function of the refractive indices of a ray with respect to the various media. Consequently each component of the light that proceeds from P will produce a separate image, in a separate position, and of its own proper colour. Moreover, the magnification also is a function of the refractive indices, and therefore the images formed by the component lights will all be of different sizes. These separate images are all formed on planes perpen- dicular to the axis, and one behind another. The image actually seen by the eye is the resultant of them, that is to say, the image obtained by their superposition. The central 74 LENSES AND SYSTEMS OF LENSES. portion of it is colourless as ordinary light, for there every component plays its part. But towards the edge, in conse- quence of the difference in the sizes and positions of the component images, one colour after another ceases to appear and the resultant image is seen surrounded with a rainbow-like border. So long as the image has this coloured edge, it is indefinite and unsatisfactory. We will endeavour to show whether it can be got rid of, and if it can, by what means and under what conditions. A lens, or system of lenses, which produces an image with a distinct border free from colour is said to be achromatic. 121. The problem before us is simply this : If an object be viewed through a lens or system of lenses, what conditions must the lens or lenses satisfy in order that the images formed by rays of two or more colours may be coincident. It is clear from what has been said that (i) the focus- conjugate to a given point on the axis must be the same for two or more colours, and (ii) the magnification must be the same for two or more colours. It should be remarked that the refractive indices differ but slightly from one another and that they all lie within certain limits. 122. We will now express the conditions in an algebraical form, by means of the equations - = .4w+, m _ Cu + D Au + B' The distances w, v being reduced depend on the distances refractive indices of the first and last media. In most Astro- nomical instruments these are both the same, and vary very little (under ordinary circumstances) for different colours. We shall therefore suppose u to be unaltered by the variation of the retractive index of the first medium, and v to vary only by reason of the indices contained in A, B^ (7, D. ACHROMATISM. 75 v 123. Considering the second requirement, we have m therefore B (-} = uSA + S , \mj u being constant, and the symbol 8 expressing variation due to variations in the values of the /A'S. If the magnification be the same for all, then S(!)=0; \mj therefore uA + S#;= 0. This must be true for all values of w, therefore SA = 0] =0 1 = 0} and SB. - r TJiese are the conditions that the different coloured images may be all of the same size. 124. We will now consider the other requirement, that the images be formed all in the same place. We have Cu + D ~Au + B> and we have to find the condition that v may remain the same while A, B, (7, and D undergo small variations. This condition gives us B) But, from Art. 123, Au + B const.; therefore therefore This must be true for all values of u ; therefore BC=Q These are the conditions that the component images may be all in the same plane. 76 LENSES AND SYSTEMS OF LENSES. 125. If we combine the results of Arts. 123 and 124 we see that both requirements will be satisfied if SA = SB=SC=BD=0 ............... (Hi). The quantities A, 5, (7, D however are connected by the relation and the four conditions contained in (iii) are not all independent. The necessary and sufficient conditions for achromatism are SA = S = 5(7 = 0, or or 126. If it be required that the system shall be achromatic only for a particular value of u y we must eliminate u between the two equations therefore the sufficient condition is BD BB 127. In Art. 177 and in subsequent Articles of Parkinson's Optics the problem of achromatism is considered in two ways r according as the pencil passes centrically or excentrically. These methods are important in such an instrument, for example, as the telescope. 128. The former case may be considered by the methods which we have just explained. When the pencil passes centrically through the system, all the conjugate foci lie upon the axis, and our object is gained if we make as many as possible of these coincide. ACHROMATISM. The condition is Cw-f D But t? = therefore (uSA + SB) (Cu + D) = (Au + J5) (wS (7 + therefore w 8 (A.SC-C.$A)+u(A.$D+RSC-I>.SA- C.SB) +RSD-D.SB =0. 129. The case of excentrical refraction can best be considered by the method given in Parkinson's Optics, Art. 180. We may however obtain a similar result by making &A' = 0, where A' is the power of the quasi-equivalent lens, that is, the lens which would produce the same deviation as the system does in a ray initially inclined and not parallel to the axis. An expression for A' may be obtained in a way similar to that in Art. 117. 130. The quantities 8 A, SJ?, &c., may be determined as follows. We have /(/>!> note t vertices of the surfaces ; therefore where t^ ... denote the reduced distances between the Also &c., where t'^ 1\, ... are the absolute distances between the vertices. 78 LENSES AND SYSTEMS OF LENSES. Therefore Sp p = - (p p - /x p _,) & p = 8f^ In this way we may get Sp,, & 1? &c. ; hence, by substi- tution, we get S A in terms of fy*,, ..., &c. 131. Considering a single thick lens of refractive index p and thickness a, and supposing the external air to be of constant index 1, we have /Lt- 1 1 - p a where Pl = -^- , Pf = , *=-. Hence 5^ = (1 + p t) B Pl + (1 + p t ) 3/9, + Pl p&, x dp ~ S/A ., a8/L6 and op. = , op = - , 6f = - ^ . r, r g ' yu. The condition S^l = gives us A /A-l \ S/A _ / ^-1 a\ 8/A \ ?* 2 p) ^ \ r i p) r 2 ( A -i or Again jB = 1 + p./. Calculating the variation of B in the same manner, and making it equal to zero, we get either a = 0, or - = 0> T 2 or -=0. ACHROMATISM. 79 In the first case the two radii must be equal in order to satisfy (i), and the lens ceases to be of any optical value. The second case = needs to be zero also, r * r v with the same result as before. Lastly, - = is an im- V> possibility. Our conclusion therefore is that a single lens cannot be achromatic. ( 80 CHAPTER V. THE DETERMINATION OF THE FOCI AND OF THE PRINCIPAL POINTS OF A SYSTEM OF LENSES. THE NODAL POINTS. 132. To determine theoretically the positions of the Foci and of the Principal Points of a lens or system of lenses. Conjugate Foci lying upon the axis are connected with one another by the relation where x, x are the distances of the conjugate points from the Principal Points 77, H' respectively. Let us take any point on the axis of the system, and use it as an origin from which all our distances may be reckoned ; and let F, F', H, H r be the distances from it of the Principal Foci and the Principal Points respec- tively. The above relation may now be written in the form H-F _ E'-F' -H= + W=t' =1 .................. M' where f , f ' are the distances from of any pair of con- jugate points. We have shown, in Art. 113, how to determine the position of the point conjugate to a given one. If therefore we take any four points on the axis, and determine the positions of the four points respectively conjugate to them, we get four simultaneous values of f and ^'. These, when substituted successively in (ii), give us four independent equations for the determination of the four unknown quan- tities Fj F 1 , H, H'. Thus are found the positions of the Principal Foci and also of the Principal Points. DETERMINATION OF THE FOCI. 81 133. To determine experimentally the positions of the Foci of a lens or system of lenses. We will suppose that, in the following figure, A is a micrometer or a frame holding two spider lines crossing one another, G a stand supporting a telescope, and that B supports a cylinder enclosing the system of lenses. Moreover Aj J5, and G are supposed to be moveable by the hand or by means of screws to and fro along the graduated bar MN, and also to be so adjusted that the micrometer, .the lenses, and the telescope have the same axis (fig. 22). FIG. 22. n M N Let the telescope be turned first towards a distant object, and then accurately focused. The rays from the distant object are approximately parallel, and the image will be formed at the Principal Focus of the telescope. When this has been done, the telescope must be placed on the stand G so that the micrometer A may be viewed through the system of lenses; the micrometer must then be moved to and fro along the graduated bar until the image of it, seen through the system of lenses and the telescope, becomes clear and distinct. Now this image is seen through a telescope which has been focused upon a distant object, hence we know that the image of the micrometer can be distinct only when the rays that fall upon the object glass of the telescope are parallel to the axis. Consequently the rays that emerge from the cylinder B must be parallel to the axis. Therefore the micrometer A must be at one of the Principal Foci of the lens-system. The Focus is thus determined in position. G 82 LENSES AND SYSTEMS OF LENSES. We have still to measure its distance from the nearest surface of the system. This might be done by moving the micrometer along the graduated bar until it came into contact with the surface, and then taking the difference of the readings in the two positions given by the scale. There is however a practical difficulty in ascertaining the exact moment of contact, and this method consequently leads to an unsatisfactory result. The distance between the micro- meter and the nearest surface may be measured more accurately by a simple optical contrivance. For this purpose let the telescope be focused upon some near object whose distance is greater than that which we have to measure; and let the telescope be removed to the other end of the bar so that the micrometer may be Between it and the lens-system. If when this has been done the telescope be moved along the graduated bar until first the micrometer^ and then the dust on the face of the lens be in focus successively ; and if the scale be read for these two positions of the telescope, the difference between the readings will give us the distance between the micro- meter and the face of the lens-system with tolerable accuracy. The micrometer being at a focus of the system, we thus get the distance of the focus from the surface nearest to it. 134. To determine the positions of the Principal Points when the Foci are known. If d and d" be the distances from the Foci of any two conjugate points on the axis, and if f be the distance of a Focus from the corresponding Principal Point, we have dd' =f. It has been shown how d and d' may be determined; hence the above equation enables us to determine f, and consequently the positions of H and H'. 135. In order that the method described above may be a directly practical one, it is necessary that the lens-system should be a convex lens or equivalent to a THE NODAL POINTS. 83 convex lens. Otherwise no real images will be formed by it. If the lens-system itself gives a real image, the method can be applied at once. But if it does not, it can be made to do so by combining with it a known convex lens of sufficient power. The method may then be applied to the joint-system, and by making allowance for the effect of the known convex lens, the proper result for the original system can be deduced from the one thus obtained. 136. The Principal Points, introduced by Gauss, have been supplemented by two other points which Listing intro- duced and called Nodal Points. They are principally of importance when the extreme media are not the same. This is found to be the case in the .human eye. The Nodal Points are situated upon the axis of the lens-system, and are conjugate to one another. Their dis- tinguishing property is that an incident ray through one will produce an emergent ray in a parallel direction passing through the other. When the extreme media are the same, we have seen that this is a property of the Principal Points. Hence in this case the Principal Points and the Nodal Points coincide. When there is only one refracting surface we might call its centre of curvature the Nodal Point, for we know that an incident ray which passes through the centre of curvature crosses the surface without deviation. 137. To determine the positions of the Nodal Points. Let H, H', F, F' be the Principal Points and Foci of a lens-system, and let T be any point on the Focal Plane through F (fig. 23). We know that a ray through T parallel to the axis, and meeting the Principal Planes at a, a' respectively, will on emergence pass through F'. Moreover, since T is a point in a focal plane, its conjugate is on a plane at an infinite distance, and therefore all rays from it will on G2 84 LENSES AND SYSTEMS OF LENSES. FIG. 23. j < N' emergence be parallel to one another. They will therefore all be parallel to a.'F'. Hence if T(B meet the first Principal Plane at /8, and if $/3' drawn parallel to the axis meet the other Principal Plane at /3', then 0'N' drawn parallel to a!F' will be the emergent ray produced by the incident ray 7)3. Let Tj3 produced and the corresponding emergent ray meet the axis of the system at the points N, N' respectively. Then we see from the figure that the triangles TFN and a'H'F' are equal in all respects; therefore FN=H'F' = constant. Hence the position of N is independent of the position of Tj therefore N is a fixed point, and its distance from F is equal to the second focal distance. In a similar way it may be shown that F'N' = FH; therefore N' also is a fixed point independent of T, and is at a distance from F' equal to the first focal distance. The points N and N' are clearly the Nodal Points referred to in Art. 136. 138. If the extreme media be the same, the two focal distances are equal ; hence, as we have already noticed, the points N and N' coincide with H and H'. 139. From the figure we have also NN' = /3ff = HE'. THE NODAL POINTS. 85 W Hence the distance between the Nodal Points is equal to the distance between the Principal Points. 140. We have also HN= H'N' = H'F' - HF. 141. When the Nodal Points have been determined, we may with their help very readily determine the direc- tion of the emergent ray produced by a given incident ray, and also the position of a point conjugate to a given one (fig. 24). FIG.24. N H With the usual notation we will suppose XTa. to be an incident ray meeting the Focal Plane through F at T. Join TN. Let aa' parallel to the axis meet the second Principal Plane at a'. Through a' draw cdT'X' parallel to TN. Then a'T'X' is the direction of the emergent ray produced by the incident ray XTa, and X, X' are a pair of conjugate points. Otherwise: Draw N'T' parallel to XTa and meeting the Focal plane through F' at the point T ; then a!T' is the direction of the emergent ray. If an object lens be situated on the Nodal Plane through N its image will be situated on the Nodal Plane through N'' t and it may easily be proved that the linear dimensions of object and image are to one another inversely as the indices of refraction of the first and last media. 86 CHAPTER VI. THE DIFFERENT FORMS OF LENSES. 142. WE will now apply some of the results of Chapter II to determine the positions of the Principal Points and Foci for the five most important forms of the simple lens. We will consider, (1) a double convex lens, (2) a plano- convex lens, (3) a double concave lens, (4) a plano-concave lens, (5) a meniscus; and finally we will consider (6) the case of two mirrors placed upon the same axis, and facing one another, in such manner as we find in Gregory's and Cassegrain 7 3 telescopes. 143. In Arts. 55 and 64, it was shown that the distances of the Foci and of the Principal Points from the vertices of the lens are given by the formulae AF- * A pat + p -|- cr ' G_ tp + l A PC AH = l ~ B A pat + p 4 a ' A-H'- ?=**, A pat + p + 8,8 > ...X > /. l v)6 (3). If we were to consider only the quantities &, c, c?, ... n in the one series, and 8, e, ... X, /A, v in the other, we should in a similar way obtain the relation n = 6r'5 4- (8, s, ... X, /u, v) c. Substituting now (8, e, ... X, /*, /) If the expressions for n in (3) and (4) be compared, it will be obvious that , e, ...X, and therefore, by analogy, ff-*k ... \ From (3) and (4) we also get v. Hence, substituting in this equation the expressions already obtained for H and ', we get PROPERTIES OF CONTINUED FRACTIONS. 93 Again, if we put a = 0, 6 = 1, in the last three of the relations (1), we get w = <(7, 8, s, ...X, /*, v), m = <(7, 8,e, ...X, /*), and if we substitute these expressions for n, ?w, and Z, in the relation n = ym -f 7, we get But from (5) we have Hence, it is clear by a comparison of (6) and (7) that if we can prove that (7, 8, e, ... X, fi) = (f) (p, X, ... e, 8, 7), and that (7, 8, e, ... X) = (X, ... e, 8, 7), it will follow at once that the same property holds if we consider an additional quantity v ; that is to say, we shall thus prove that (7, S, e, - \ /*, ") = (v, H, \ ... e, 8, 7). But we have already shown that and that (7, 8, s) = 87 + s + 7 = $ (e, 8, 7). Hence, it follows by induction that the general theorem is true ; namely, that (7, S, > MI *, A*j ") = * (", A*> *> - > 8 , 7)j however many there may be of the quantities 7, 8, e, ... X, /*, v. It will be seen that the relations (1) are those which hold between the denominators of successive convergents to a continued fraction, and that the theorem here proved is the first of those quoted in Art. 94. 94 LENSES AND SYSTEMS OF LENSES. 3. We have also - $ (7, 8, e, ... X, * c (v, /*, X, ... e, 8, 7) = ; {'*&*, X,...,8,7)-HKX,...s,S,7)} , x,...s, 8, 7) In a similar manner we get J- (7, 8, e, ... JS : f) 5 Therefore dpdv = <(X, ...e, 8, 7) = 0(7, 8, e, ...X). (7, 8, s, ... X, /A, v) = < (7, 8, e, ... X)., These are the other theorems quoted in Art. 94. 4. In the two preceding articles the theorems have been considered in a general form, but they may be proved very readily by considering the value of A expressed as a determinant. 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