James VI. the next heir, but was soon after granted to the king's | was born in 1720. She went to London in 1735, and, being left unprovided for at her father's death, she began to earn her | Two years later the title was granted to Robert Stewart, the king's grand-uncle, second son of John, the 3rd earl, but he in 1580 exchanged it for that of earl of March. On the same day the earldom of Lennox was given to Esme Stewart, first cousin of the king and grandson of the 3rd earl, he being son of John Stewart (adopted heir of the maréchal d'Aubigny) and his French wife, Anne de la Queulle. In the following year Esme was created duke of Lennox, earl of Darnley, Lord Aubigny, Tarboulton and Dalkeith, and other favours were heaped upon him, but the earl of Ruthven sent him back to France where he died soon after. His elder son, Ludovic, was thereupon summoned to Scotland by James, who invested him with all his father's honours and estates, and after his accession to the English throne created him Lord Settrington and earl of Richmond (1613), and earl of Newcastle-upon-Tyne and duke of Richmond (1623), all these titles being in the peerage of England. After holding many appointments the 2nd duke died without issue in 1624, being succeeded in his Scottish titles by his brother Esme, who had already been created earl of March and Lord Clifton of Leighton Bromswold in the peerage of England (1619) and was seigneur d'Aubigny in France. Of his sons, Henry succeeded Aubigny and died young at Venice; Ludovic, seigneur d'Aubigny, entered the Roman Catholic Church and received a cardinal's hat just before his death; while the three other younger sons, George, seigneur d'Aubigny, John and Bernard, were all distinguished as royalists in the Civil War. Each met a soldier's death, George at Edgehill, John at Alresford and Bernard at Rowton Heath. James, the eldest son and 4th duke of Lennox, was created duke of Richmond in 1641, being like his brother a devoted adherent of Charles I. With the death of his little son Esme, the 5th duke, in 1660, the titles, including that of Richmond, passed to his first cousin Charles, who had already been created Lord Stuart of Newbury and earl of Lichfield, being likewise now seigneur d'Aubigny, Disliked by Charles II., principally because of his marriage with "la belle Stuart "-" the noblest romance and example of a brave lady that ever I read in my life," writes Pepys-he was sent into exile as ambassador to Denmark, where he was drowned in 1672. His wife had had the Lennox estates granted to her for life, but his only sister Katharine, wife of Henry O'Brien, heir apparent of the 7th earl of Thomond, was served heir to him. Her only daughter, the countess of Clarendon, was mother of Theodosia Hyde, ancestress of the present earls of Darnley. The Lennox dukedom, being to heirs male, now devolved upon Charles II., who bestowed it with the titles of earl of Darnley and Lord Tarbolton upon one of his bastards, Charles Lennox, son of the celebrated duchess of Portsmouth, he having previously been created duke of Richmond, earl of March and Lord Settrington in the peerage of England The ancient lands of the Lennox title were also granted to him, but these he sold to the duke of Montrose. His son Charles, who inherited his grandmother's French dukedom of Aubigny, was a soldier of distinction, as were the 3rd and 4th dukes. The wife of the last, Lady Charlotte Gordon, as heir of her brother brought the ancient estates of her family to the Lennoxes; the additional name of Gordon being taken by the 5th duke of Richmond and of Lennox on the death of his uncle, the 5th duke of Gordon. In the next generation further honours were granted to the family in the person of the 6th duke, who was rewarded for his great public services with the titles of duke of Gordon and earl of Kinrara in the peerage of the United Kingdom (1876). LENNOX, MARGARET, COUNTESS of (1515-1578), daughter See Scots Peerage, vol. v., for excellent accounts of these peerages by the Rev. John Anderson, curator Historical Dept. H.M. Register House; A. Francis Steuart and Francis J. Grant, Rothesay Herald. See also The Lennox by William Fraser. LENNOX, CHARLOTTE (1720-1804), British writer, daughter of Colonel James Ramsay, lieutenant-governor of New York, I The famous Lennox jewel, made for Lady Lennox as a memento of her husband, was bought by Queen Victoria in 1842. LENO, DAN, the stage-name of George Galvin (1861-1904), English comedian, who was born at Somers Town, London, in February 1861. His parents were actors, known as Mr and Mrs Johnny Wilde. Dan Leno was trained to be an acrobat, but soon became a dancer, travelling with his brother as "the brothers Leno," and winning the world's championship in clogdancing at Leeds in 1880. Shortly afterwards he appeared in London at the Oxford, and in 1886-1887 at the Surrey Theatre. In 1888-1889 he was engaged by Sir Augustus Harris to play the Baroness in the Babes in the Wood, and from that time he was a principal figure in the Drury Lane pantomimes. He was the wittiest and most popular comedian of his day, and delighted London music-hall audiences by his shop-walker, stores-proprietor, waiter, doctor, beef-eater, bathing attendant, "Mrs Kelly," and other impersonations. In 1900 he engaged to give his entire services to the Pavilion Music Hall, where he received £100 per week. In November 1901 he was summoned to Sand. ringham to do a "turn" before the king, and was proud from that time to call himself the "king's jester." Dan Leno's generosity endeared him to his profession, and he was the object of much sympathy during the brain failure which recurred during the last eighteen months of his life. He died on the 31st of October 1904. a summer resident from 1836-1853; and with Henry Ward Beecher (see his Star Papers). Elizabeth (Mrs Charles) Sedgwick, the sister-in-law of Catherine Sedgwick, maintained here from 1828 to 1864 a school for girls, in which Harriet Hosmer, the sculptor, and Maria S. Cummins (1827-1866), the novelist, were educated, and in Lenox academy (1803), a famous classical school (now a public high school) were educated W. L. Yancey, A. H. Stephens, Mark Hopkins and David Davis (1815-1886), a circuit judge of Illinois from 1848 to 1862, a justice (1862-1877) of the United States Supreme Court, a Republican member of the United States Senate from Illinois in 1877-1883, and president of the Senate from the 31st of October 1881, when he succeeded Chester A. Arthur, until the 3rd of March 1883. There is a statue commemorating General John Paterson (17441808) a soldier from Lenox in the War of Independence. See R. de W. Mallary, Lenox and the Berkshire Highlands (1902); J. C. Adams, Nature Studies in Berkshire; C. F. Warner, Picturesque Berkshire (1890); and Katherine M. Abbott, Old Paths and Legends of the New England Border (1907). LENORMANT, FRANÇOIS (1837-1883), French Assyriologist | of the Seven Gables and the Wonder Book; with Fanny Kemble, and archaeologist, was born in Paris on the 17th of January 1837. His father, Charles Lenormant, distinguished as an archaeologist, numismatist and Egyptologist, was anxious that his son should follow in his steps. He made him begin Greek at the age of six, and the child responded so well to this precocious scheme of instruction, that when he was only fourteen an essay of his, on the Greek tablets found at Memphis, appeared in the Revue archéologique. In 1856 he won the numismatic prize of the Académie des Inscriptions with an essay entitled Classification des monnaies des Lagides. In 1862 he became sub-librarian of the Institute. In 1859 he accompanied his father on a journey of exploration to Greece, during which Charles Lenormant succumbed to fever at Athens (24th November). Lenormant returned to Greece three times during the next six years, and gave up all the time he could spare from his official work to archaeological research. These peaceful labours were rudely interrupted by the war of 1870, when Lenormant served with the army and was wounded in the siege of Paris. In 1874 he was appointed professor of archaeology at the National Library, and in the following year he collaborated with Baron de Witte in founding the Gazette archéologique. As early as 1867 he had turned his attention to Assyrian studies; he was among the first to recognize in the cuneiform inscriptions the existence of a non-Semitic language, now known as Accadian. Lenormant's knowledge was of encyclopaedic extent, ranging over an immense number of subjects, and at the same time thorough, though somewhat lacking perhaps in the strict accuracy of the modern school. Most of his varied studies were directed towards tracing the origins of the two great civilizations of the ancient world, which were to be sought in Mesopotamia and on the shores of the Mediterranean. He had a perfect passion for exploration. Besides his early expeditions to Greece, he visited the south of Italy three times with this object, and it was while exploring in Calabria that he met with an accident which ended fatally in Paris on the 9th of December 1883, after a long illness. The amount and variety of Lenormant's work is truly amazing when it is remembered that he died at the early age of forty-six. Probably the best known of his books are Les Origines de l'histoire d'après la Bible, and his ancient history of the East and account of Chaldean magic. For breadth of view, combined with extraordinary subtlety of intuition, he was probably unrivalled. LENOX, a township of Berkshire county, Massachusetts, U.S.A. Pop. (1900) 2942, (1905) 3058; (1910) 3060. Area, 19.2 sq. m. The principal village, also named Lenox (or Lenoxon-the-Heights), lies about 2 m. W. of the Housatonic river, at an altitude of about 1000 ft,, and about it are high hills Yokun Seat (2080 ft.), South Mountain (1200 ft.), Bald Head (1583 ft.), and Rattlesnake Hill (1540 ft.). New Lenox and Lenoxdale are other villages in the township. Lenox is a fashionable summer and autumn resort, much frequented by wealthy people from Washington, Newport and New York. There are innumerable lovely walks and drives in the surrounding region, which contains some of the most beautiful country of the Berkshires-hills, lakes, charming intervales and woods. As early as 1835 Lenox began to attract summer residents. In the next decade began the creation of large estates, although the great holdings of the present day, and the villas scattered over the hills, are comparatively recent features. The height of the season is in the autumn, when there are horse-shows, golf, tennis, hunts and other outdoor amusements. The Lenox library (1855) contained about 20,000 volumes in 1908. Lenox was settled about 1750, was included in Richmond township in 1765, and became an independent township in 1767. The names were those of Sir Charles Lennox, third duke of Richmond and of Lennox (1735-1806), one of the staunch friends of the American colonies during the War of Independence. Lenox was the countyseat from 1787 to 1868. It has literary associations with Catherine M. Sedgwick (1789-1867), who passed here the second half of her life; with Nathaniel Hawthorne, whose brief residence here (1850-1851) was marked by the production of the House | LENS, a town of Northern France, in the department of Pasde-Calais, 13 m. N.N.E. of Arras by rail on the Déûle and on the Lens canal. Pop. (1906) 27,692. Lens has important iron and steel foundries, and engineering works and manufactories of steel cables, and occupies a central position in the coalfields of the department. Two and a half miles W.S.W. lies Liévin (pop. 22,070), likewise a centre of the coalfield. In 1648 the neighbourhood of Lens was the scene of a celebrated victory gained by Louis II. of Bourbon, prince of Condé, over the Spaniards. LENS (from Lat. lens, lentil, on account of the similarity of the form of a lens to that of a lentil seed), in optics, an instrument which refracts the luminous rays proceeding from an object in such a manner as to produce an image of the object. It may be regarded as having four principal functions: (1) to produce an image larger than the object, as in the magnifying glass, microscope, &c.; (2) to produce an image smaller than the object, as in the ordinary photographic camera; (3) to convert rays proceeding from a point or other luminous source into a definite pencil, as in light-house lenses, the engraver's globe, &c.; (4) to collect luminous and heating rays into a smaller area, as in the burning glass. A lens made up of two or more lenses cemented together or very close to each other is termed " composite " "compound "; several lenses arranged in succession at a distance from each other form a system of lenses," and if the axes be collinear a "centred system." This article is concerned with the general theory of lenses, and more particularly with spherical lenses. For a special part of the theory of lenses see ABERRATION; the instruments in which the lenses occur are treated under their own headings. ог The most important type of lens is the spherical lens, which is a piece of transparent material bounded by two spherical surfaces, the boundary at the edge being usually cylindrical or conical. The line joining the centres, C, C2 (fig. 1), of the bounding surfaces is termed the axis; the points S1, S2, at FIG. 1. which the axis intersects the surfaces, are termed the " vertices " of the lens; and the distance between the vertices is termed the "thickness." If the edge be everywhere equidistant from the vertex, the lens is "centred." Although light is really a wave motion in the aether, it is only necessary, in the investigation of the optical properties of systems of lenses, to trace the rectilinear path of the waves, i.e. the direction of the normal to the wave front, and this can be done by purely geometrical methods. It will be assumed that light, so long as it traverses the same medium, always travels in a straight line; and in following out the geometrical theory it will always be assumed that the light travels from left to right; accordingly all distances measured in this direction are positive, while those measured in the opposite direction are negative. Theory of Optical Representation.-If a pencil of rays, ie. the totality of the rays proceeding from a luminous point, falls on a lens or lens system, a section of the pencil, determined by the dimensions of the system, will be transmitted. The emergent rays will have directions differing from those of the incident ra the alteration, however, being such that the transmitted rays are convergent in the "image-point," just as the incident rays diverge from the "object-point." With each incident ray is associated an emergent ray; such pairs are termed conjugate ray pairs." Similarly we define an object-point and its image-point as conjugate points"; all object-points lie in the "object-space," and all image-points lie in the " image-space." .. The laws of optical representations were first deduced in their most general form by E. Abbe, who assumed (1) that an optical representation always exists, and (2) that to every point in the the two rays CD1, and EF1; the conjugate point P will be deter points D', F', being readily found from the magnifications B, B. mined by the intersection of the conjugate rays C'D', and E'F', the Since PP, is parallel to CE and also to DF, then DFDF. Since the plane O is similarly represented in O', D'F' D'F'; this is impossible unless P'P' be parallel to C'E'. Therefore every perpendicular object-plane is represented by a perpendicular imageplane. Let O be the intersection of the line PP, with the axis, and let O' be its conjugate; then it may be shown that a fixed magnification 8, exists for the planes O and O'. For PP/FF=001/0,02 P'P/F'F', =O'O'O'O'1⁄2, and F'F' =2FF. Eliminating FF, and F'F' between these ratios, we have P'P'./PP1ß2 = O'O',·0,01/001. O'O'2, or ẞ3=ß1⁄2‚O′O′,.O102/001.0'0'1⁄2, i.e. B1 =BXa product of the axial distances. The determination of the image-point of a given object-point is facilitated by means of the so-called "cardinal points "of the optical system. To determine the image-point O', (fig. 3) correspond ing to the object-point O1, we begin by choosing from the ray pencil proceeding from O, the ray parallel with the axis, i.e. intersecting the axis at infinity. Since the axis is its own conjugate, the parallel ray through O, must intersect the axis after refraction (say at F'). Then F' is the image-point of an object-point situated at infinity on the axis, and is termed the "second principal focus" (German der bildseitige Brennpunkt, the image-side focus). Similarly if O' be on the parallel through O, but in the image-space, then the conjugate ray must intersect the axis at a point (say F), which is conjugate with the point at infinity on the axis in the image-space. This point is termed the "first principal focus" (German der objekt seilige Brennpunkt, the object-side focus). Let H, H' be the intersections of the focal rays through F and F with the line 0,0'. These two points are in the position of object and image, since they are each determined by two pairs of conjugate rays (OH being conjugate with H'F', and O'H', with HF). It has already been shown that object-planes perpendicular to the axis are represented by image-planes also perpendicular to the axis, Two vertical planes through H, and H', are related as object- and image-planes; and if these planes intersect the axis in two points H and H', these points are named the " principal," or "Gauss points" of the system, H being the " object-side and H' the image-side principal point." The vertical planes containing H and H' are the "principal planes." It is obvious that conjugate points in these planes are equidistant from the axis; in other words, the magnification 8 of the pair of planes is unity. An additional characteristic of the principal planes is that the object and image are direct and not inverted. The distances between F and H, and between F' and H' are termed the focal lengths; the former may be called the " object-side focal length and the latter the "image-side focal length." The two focal points and the two principal points constitute the so-called four cardinal points of the system, and with their aid the image of any object can be readily determined. D. FIG. 2. object-space there corresponds a point in the image-space, these points being mutually convertible by straight rays; in other words, with each object-point is associated one, and only one, image-point, and if the object-point be placed at the image-point, the conjugate point is the original object-point. Such a transformation is termed a "collineation," since it transforms points into points and straight lines into straight lines. Prior to Abbe, however, James Clerk Maxwell published, in 1856, a geometrical theory of optical representation, but his methods were unknown to Abbe and to his pupils until O. Eppenstein drew attention to them. Although Maxwell's theory is not so general as Abbe's, it is used here since its methods permit a simple and convenient deduction of the laws. "" Maxwell assumed that two object-planes perpendicular to the axis are represented sharply and similarly in two image-planes also perpendicular to the axis (by "sharply is meant that the assumed ideal instrument unites all the rays proceeding from an object-point in one of the two planes in its image-point, the rays being generally transmitted by the system). The symmetry of the axis being premised, it is sufficient to deduce laws for a plane containing the axis. In fig. 2 let O, O be the two points in which the perpendicular object-planes meet the axis; and since the axis corresponds to itself, the two conjugate points O', O's, are at the intersections of the two image-planes with the axis. We denote the four planes by the letters O, O2, and O'1, O'2. If two points A, C be taken in the plane O, their images are A', C' in the plane O, and since the planes are represented similarly, we have O'A': O1A =0'1C'1 :O1C =B (say), in which B is easily seen to be the linear magnification of the plane-pair O, O'. Similarly, if two points B. D be taken in the plane O, and their images B', D' in the plane O's, we have O'B': O,B=O'D': O,D=82 (say), B2 being the linear magnification of the plane-pair O2, O'2. The joins of A and B and of C and D intersect in a point P, and the joins of the conjugate points similarly determine the point P'. | If P' is the only possible image-point of the object-point P, then the conjugate of every ray passing through P must pass through P. To prove this, take a third line through P intersecting the planes O, O, in the points E, F, and by means of the magnifications B. B determine the conjugate points E', F' in the planes O'1, O's: Since the planes O, O are parallel, then AC/AE=BD/BF; and since these planes are represented similarly in O, O's, then A'C'/A'E' B'D'/B'F'. This proportion is only possible when the straight line E'F' contains the point P. Since P was any point whatever, it follows that every point of the object-space is represented in one and only one point in the image-space. Take a second object-point P, vertically under P and defined by H2 FIG. 3. Choose from the pencil a second ray which contains F and intersects the principal plane H in H; then the conjugate ray must contain points corresponding to F and H2. The conjugate of F is the point at infinity on the axis, i.e. on the ray parallel to the axis. The image of H, must be in the plane H' at the same distance from, and on the same side of, the axis, as in H'. The straight line passing through H' parallel to the axis intersects the ray H'F in the point O, which must be the image of O. If O be the foot of the perpendicular from O, to the axis, then OO, is represented by the line O'O', also perpendicular to the axis. This construction is not applicable if the object or image be infinitely distant. For example, if the object 00, be at infinity (O being assumed to be on the axis for the sake of simplicity), so that the object appears under a constant angle w, we know that the second principal focus is conjugate with the infinitely distant axis-point. If the object is at infinity in a plane perpendicular to the axis, the image must be in the perpendicular plane through the focal point F' (fig. 4). The size y of the image is readily deduced. Of the parallel rays from the object subtending the angle w, there is one which passes through the first principal focus F, and intersects the principal plane H in H1. Its conjugate ray passes through H' parallel to, and at the same distance from the axis, and intersects the image-side focal plane in O'; this point is the image of O, and y' is its magnitude. From the figure we have tan w= HH1/FH=yf, or f=y/tanw; this equation was used by Gauss to define the focal length. Referring to fig. 3, we have from the similarity of the triangles OOF and HH2F, HH2/OO1 = FH/FO, or O'O'1/001 = FH/FO. Let y be the magnitude of the object 001, y' that of the image O'O', the focal distance FO of the object, and ƒ the object-side focal distance FH; then the above equation may be written | yly fix. From the similar triangles H'H'F' and O'Ŏ'F', we obtain O'O'1/00 F'O'/F'H'. Let x' be the focal distance of the image F'O', and f the image side focal length F'H'; then yly= x'f'. The ratio of the size of the image to the size of the Denoting this by 8, we (1) (2) F H TH FIG. 4. object is termed the lateral magnification. and also H H, xx' =ƒƒ'. By differentiating equation (2) we obtain (3) The ratio of the displacement of the image dx' to the displacement of the object dx is the axial magnification, and is denoted by a Equation (3) gives important information on the displacement of the image when the object is moved. Since ƒ and f' always have contrary signs (as is proved below), the product - is invariably positive, and since x2 is positive for all values of x, it follows that dx and de' have the same sign, i.e. the object and image always move in the same direction, either both in the direction of the light, or both in the opposite direction. This is shown in fig. 3 by the object 0,0, and the image O'O'2. H ** If two conjugate rays be drawn from two conjugate points on the axis, making angles u and u' with the axis, as for example the rays OH, O'H', in fig. 3, u is termed the "angular aperture for the object," and the angular aperture for the image.' The ratio of the tangents of these angles is termed the "convergence" and is denoted by y, thus y=tan utan u. Now tan = H'H'1/O'H' =H'H',/(O'F'+F'H') = H'H'1/(F'H'-F'O'). Also tan u=HH/OH = HH1/(OF+FH)=HH1/(FH-FO). Consequently y=(FH-FO) /(F'H' F'0'), or, in our previous notation, y= {ƒ −x)/(f' —x'). | From equation (1) f/x=x'f', we obtain by subtracting unity from both sides (f-x)/x = (x' —ƒ')}f', and consequently ----- =1. H corresponding image subtend the same angle at the principal points. A=HO HF+FO-FO-FH=x-f, whence I+5+1=0; this becomes in the special case when f= -f', I I A ̃ ̄‡ To express the linear magnification in terms of the principal point distances, we start with equation (4) (ƒ—x)/(f' —x') = −x/f'. From this we obtain A/A'=-x/f', or x= -f'A/A'; and by using equation (1) we have ẞ —ƒA'/ƒ'A. In the special case of f-f', this becomes ẞ= A'/A=y'/y, from which it follows that the ratio of the dimensions of the object and image is equal to the ratio of the distances of the object and image from the principal points. The convergence can be determined in terms of A and A' by substituting x-f'A/A' in equation (4), when we obtain y=A/A'. Compound Systems.—In discussing the laws relating to compound systems, we assume that the cardinal points of the component systems are known, and also that the combinations are centred, i.e. that the axes of the component lenses coincide. If some object be represented by two systems arranged one behind the other, we can regard the systems as co-operating in the formation of the final image. Let such a system be represented in fig. 6. The two single systems are denoted by the suffixes 1 and 2; for example, Fi is the first (4) From equations (1), (3) and (4), it is seen that a simple relation exists between the lateral magnification, the axial magnification and the convergence, viz. ay=B. KK In addition to the four cardinal points F, H, F', H', J. B. Listing,y H2 FIG. 5. yf'? fy f'fa = (5) tan w A tan w's Δ the values given in equation (2), and we obtain y7y=B, it follows that 1/(x-X)=6/(x-X). Replace x' and X' by By taking a ray proceeding from the image-side we obtain for the or 1=-ptX first principal focal distance of the combination ===-x=3/(1-1) f=-fifala. In the particular case in which A=0, the two focal planes F1, F2 coincide, and the focal lengths f, f' are infinite. Such a system is called a telescopic system, and this condition is realized in a telescope focused for a normal eye. ́ff" Since ẞ=flx=x'f', we have ƒ' = -−X, f= −X'. These equations show that to determine the nodal points, it is only necessary to measure the focal distance of the second principal focus from the first principal focus, and vice versa. In the special case when the initial and final medium is the same, as for example, a lens in air, we have ƒ= -f', and the nodal points coincide with the principal points of the system; we then speak of the "nodal point property of the principal points," meaning that the object and So far we have assumed that all the rays proceeding from an objectpoint are exactly united in an image-point after transmission through the ideal system. The question now arises so to how far this assumption is justified for spherical lenses. To investigate this it is simplest to trace the path of a ray through one spherical H FIG. 6. principal focus of the first, and F', the second principal focus of the second system. A ray parallel to the axis at a distance y passes through the second principal focus F' of the first system, intersecting the axis at an angle w'. The point F' will be represented in the second system by the point F', which is therefore conjugate to the point at infinity for the entire system, i.e. it is the second principal focus of the compound system. The representation of Fi in F' by the second system leads to the relations F2F'1 = x2, and F'F'x', whence x2x2=f2f2. Denoting the distance between the adjacent focal planes F', F2 by A, we have A=FF2-F2F', so that x=-faf'2/A. A similar ray parallel to the axis at a distance proceeding from the image-side will intersect the axis at the focal point F1; and by finding the image of this point in the first system, we determine the first principal focus of the compound system. Equation (2) gives xix'=fif', and since x'=FIFA, we have x=ff/A as the distance of the first principal focus F of the compound system from the first principal focus F1 of the first system. To determine the focal lengths f and f' of the compound system and the principal points H and H', we employ the equations defining the focal lengths, viz. f=y'/tan w, and f'=y/tan w'. From the construction (fig. 6) tan w'i=y/f'. The variation of the angle w's by the second system is deduced from the equation to the convergence, viz. y=tan w/tan w/f'2=A/f', and since w=w', we have tan w' (A/f') tan w'. Since w'w' in our system of notation, we have refracting surface. Let such a surface divide media of refractive | f having the opposite sign to fi. Denoting the distance F, F, by 4. indices n and n', the former being to the left. The point where the we have AF'‚F2 = F',S,+S,S,+S»F2 =F}}S; +S1Sz−F2S2 = ƒ's +d+f. axis intersects the surface is the vertex S (fig. 7). Denote the Substituting for f', and f1⁄2 we obtain distance of the axial object-point O from S by s; the distance from t_n' (s'-1) FIG. 7. O to the point of incidence P by p; the radius of the 'spherical surface by r; and the distance OC by c, C being the centre of the sphere. Let u be the angle made by the ray with the axis, and i the angle of incidence, i.e. the angle between the ray and the normal to the sphere at the point of incidence. The corresponding quantities in the image-space are denoted by the same letters with a dash. By the same method we obtain for the second principal focal length From the triangle O'PC we have sin u=(7/c) sin i, and from the triangle O'PC we have sin u' = (r/c') sin i'. By Snell's law we have n'/sin i/sin i', and also u'ti'. Consequently' and the position of the image may be found. To determine whether all the rays proceeding from O are refracted through O', we investigate the triangle OPO'. We have p/p' sin u'/sin u. Substituting for sin u and sin u' the values found above, we obtain p'/p=c' sin /c sin i'n'c'/nc. Also c=OC=CS+ SO=-SC+SO=s-r, and similarly cs-r. Substituting these values we obtain n(5-7)' or n(s-r) _n' (s'-r) p2 = (s—r)2+r2+2r(s− r) { 1 −2+28 + ... }. 4! and therefore for such values of for which the second and higher powers may be neglected, we have p(s—r)2+r2+2r(s-r), i.e. p=s, and similarly p'=s'. Equation (6) then becomes n(s-r)/s= n'(s'-r)/s' or f'=-n'r/(n'-n) By joining this simple refracting system with a similar one, so that the second spherical surface limits the medium of refractive index n', we derive the spherical lens. Generally the two spherical surfaces enclose a glass lens, and are bounded on the outside by air of refractive index 1. 7271 71-1 n-1 Writing RA(n-1), this relation becomes 717272 (10) (8) The deduction of the cardinal points of a spherical glass lens in air from the relations already proved is readily effected if we regard the lens as a combination of two systems each having one refracting surface, the light passing in the first system from air to glass, and in the second from glass to air. If we know the refractive index of the glass n, the radii, 7 of the spherical surfaces, and the distances of the two lens-vertices (or the thickness of the lens d) we can determine all the properties of the lens. A biconvex lens is shown in fig. 8. Let F be the first principal focus of the first system of 71, and F the second principal and let S be its vertex. Denote the distance F S (the first principal focal length) by fi, and the corresponding distance F', S, by f'. Let the corre sponding quantities in the second system be denoted by the same letters with the suffix 2. ra By equations (8) and (9) we have 717972 f = (n−1)R ̄(n−1){n(12−1)+d(n−1)}' which is equivalent to *=3+"=" (7) This relation shows that in a very small central aperture in which the equation ps holds, all rays proceeding from an object-point are exactly united in an image-point, and therefore the equations previously deduced are valid for this aperture. K. F. Gaussi.e. derived the equations for thin pencils in his Dioptrische Untersuchungen (1840) by very elegant methods. More recently the laws relating to systems with finite aperture have been approximately realized, as for example, in well-corrected photographic objectives. Position of the Cardinal Points of a Lens.-Taking the case of a single spherical refracting surface, and limiting ourselves to the small central aperture, it is seen that the second principal focus F' is obtained when s is infinitely great. Consequently s'--f; the difference of sign is obvious, since s' is measured from S, while f' so that is measured from F'. The focal lengths are directly deducible from equation (7): (n-1)'d 71727 If the lens be infinitely thin, i.e. if d be zero, we have for the first principal focal length, (6) FIG. 8. To obtain and p' we use the triangles OPC and O'PC; we have (sr)+2+27 (sr) cos, p'2 = (s'−7)2+r2+2r(s'-r) cos. denote the reciprocal of the radii by the symbol p; we thus have Hence if s, r, and n' be constant, s' must vary as varies. The=1/f, p=1/r. Equation (10) thus becomes refracted rays therefore do not reunite in a point, and the deflection is termed the spherical aberration (see ABERRATION). Developing cos in powers of ø, we obtain (n − 1)2dpɩîî ̧ } = (n = 1) { // SH 72 nrir2 -f. The reciprocal of the focal length is termed the power of the lens and is denoted by . In formulae involving it is customary to H (n-1)R' where x, F.F. and x'1 =F',F2 =A. The distance of the first principal focus from the vertex S, i.e. SF, which we denote by s is given by sp=S,F=SF+FF = − FS+FF. Now FS, is the distance from the vertex of the first principal focus of the first system, f1, and FF = x. Substituting these values, we obtain 71 nr2 r1(nr1+R). Sp=21-1 (n-1)R (n-1)R The distance F'F' or x' is similarly determined by considering F', to be represented by the second system in F'. We have The two focal lengths and the distances of the foci from the vertices being known, the positions of the remaining cardinal points, i.e. the principal points H and H', are readily determined. Let =SH, ie, the distance of the object-side principal point from the vertex of the first surface, and s,,SH', i.e. the distance of the image-side principal point from the vertex of the second surface, then f=FHFS+SH=~S,F+S1H = − Sp+SH; hence su=5+1 --dr/R. Similarly Susp1+f' = -dr2/R. It is readily seen that the distances s, and s,, are in the ratio of the radii, and . The distance between the two principal planes (the interstitium) The interstitium becomes zero, or the two principal planes coincide, if d-r-12. We have now derived all the properties of the lens in terms of its elements, viz. the refractive index, the radii of the surfaces, and the thickness. Forms of Lenses.-By varying the signs and relative magnitude of the radii, lenses may be divided into two groups according to their action, and into four groups according to their form. According to their action, lenses are either collecting, convergent |