Chapter XI: Act 1867: every British ship going to other countries where lemon or (2)

LENNOX, CHARLOTTE (1720-1804), British writer, daughter of Colonel James Ramsay, lieutenant-governor of New York, 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 living by writing. She made some unsuccessful appearances on the stage and married in 1748. Samuel Johnson had an exaggerated admiration for her. "Three such women," he said, speaking of Elizabeth Carter, Hannah More and Fanny Burney, "are not to be found; I know not where to find a fourth, except Mrs Lennox, who is superior to them all." Her chief works are: _The Female Quixote; or the Adventures of Arabella_ (1752), a novel; _Shakespear illustrated; or the novels and histories on which the plays ... are founded_ (1753-1754), in which she argued that Shakespeare had spoiled the stories he borrowed for his plots by interpolating unnecessary intrigues and incidents; _The Life of Harriot Stuart_ (1751), a novel; and _The Sister_, a comedy produced at Covent Garden (18th February 1769). This last was withdrawn after the first night, after a stormy reception, due, said Goldsmith, to the fact that its author had abused Shakespeare.

LENNOX, MARGARET, COUNTESS OF (1515-1578), daughter of Archibald Douglas, 6th earl of Angus, and Margaret Tudor, daughter of Henry VII. of England and widow of James IV. of Scotland, was born at Harbottle Castle, Northumberland, on the 8th of October 1515. On account of her nearness to the English crown, Lady Margaret Douglas was brought up chiefly at the English court in close association with the Princess Mary, who remained her fast friend throughout life. She was high in Henry VIII.'s favour, but was twice disgraced; first for an attachment to Lord Thomas Howard, who died in the Tower in 1537, and again in 1541 for a similar affair with Sir Charles Howard, brother of Queen Catherine Howard. In 1544 she married a Scottish exile, Matthew Stewart, 4th earl of Lennox (1516-1571), who was regent of Scotland in 1570-1571. During Mary's reign the countess of Lennox had rooms in Westminster Palace; but on Elizabeth's accession she removed to Yorkshire, where her home at Temple Newsam became a centre for Catholic intrigue. By a series of successful manoeuvres she married her son Henry Stewart, Lord Darnley, to Mary, queen of Scots. In 1566 she was sent to the Tower, but after the murder of Darnley in 1567 she was released. She was at first loud in her denunciations of Mary, but was eventually reconciled with her daughter-in-law. In 1574 she again aroused Elizabeth's anger by the marriage of her son Charles, earl of Lennox, with Elizabeth Cavendish, daughter of the earl of Shrewsbury. She was sent to the Tower with Lady Shrewsbury, and was only pardoned after her son's death in 1577. Her diplomacy largely contributed to the future succession of her grandson James to the English throne. She died on the 7th of March 1578.

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 clog-dancing 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 Sandringham 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.

LENORMANT, FRANÇOIS (1837-1883), French Assyriologist 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 Lenox-on-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 county-seat 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 of the Seven Gables_ and the _Wonder Book_; with Fanny Kemble, 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 (1744-1808) 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).

LENS, a town of Northern France, in the department of Pas-de-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 lighthouse 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" or "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.

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, C1, C2 (fig. 1), of the bounding surfaces is termed the _axis_; the points S1, S2, at 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, i.e. 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 rays, 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
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 O1, O2 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´1, O´2, are at the intersections of the two
image-planes with the axis. We denote the four planes by the letters
O1, O2, and O´1, O´2. If two points A, C be taken in the plane O1,
their images are A´, C´ in the plane O´1, and since the planes are
represented similarly, we have O´1A´:O1A = O´1C´1:O1C = [beta]1 (say),
in which [beta]1 is easily seen to be the _linear magnification_ of
the plane-pair O1, O´1. Similarly, if two points B, D be taken in the
plane O2 and their images B´, D´ in the plane O´2, we have O´2B´:O2B =
O´2D´:O2D = [beta]2 (say), [beta]2 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 O1, O2
in the points E, F, and by means of the magnifications [beta]1,
[beta]2 determine the conjugate points E´, F´ in the planes O´1, O´2.
Since the planes O1, O2 are parallel, then AC/AE = BD/BF; and since
these planes are represented similarly in O´1, O´2, 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 P1, vertically under P and defined by the
two rays CD1, and EF1, the conjugate point P´1 will be determined by
the intersection of the conjugate rays C´D´1 and E´F´1, the points
D´1, F´1, being readily found from the magnifications [beta]1,
[beta]2. Since PP1 is parallel to CE and also to DF, then DF = D1F1.
Since the plane O2 is similarly represented in O´2, D´F´ = D´1F´1;
this is impossible unless P´P´1 be parallel to C´E´. Therefore every
perpendicular object-plane is represented by a perpendicular
image-plane.

Let O be the intersection of the line PP1 with the axis, and let O´ be
its conjugate; then it may be shown that a fixed magnification [beta]3
exists for the planes O and O´. For PP1/FF1 = OO1/O1O2, P´P´1/F´F´1 =
O´O´/O´1O´2, and F´F´1 = [beta]2FF1. Eliminating FF1 and F´F´1 between
these ratios, we have P´P´1/PP1[beta]2 = O´O´1·O1O2/OO1. O´1O´2, or
[beta]3 = [beta]2·O´O´1·O1O2/OO1·O´1O´2, i.e. [beta]3 = [beta]2 × a
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´1 (fig. 3) corresponding to the
object-point O1, we begin by choosing from the ray pencil proceeding
from O1, the ray parallel with the axis, i.e. intersecting the axis at
infinity. Since the axis is its own conjugate, the parallel ray
through O1 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´4 be on
the parallel through O1 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 objektseitige
Brennpunkt_, the object-side focus).

Let H1, H´1 be the intersections of the focal rays through F and F´
with the line O1O´4. These two points are in the position of object
and image, since they are each determined by two pairs of conjugate
rays (O1H1 being conjugate with H´1F´, and O´4H´1 with H1F). 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 H1 and H´1, 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 [beta] 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.

_Equations relating to the Focal Points._--We know that the ray
proceeding from the object point O1, parallel to the axis and
intersecting the principal plane H in H1, passes through H´1 and F´.
Choose from the pencil a second ray which contains F and intersects
the principal plane H in H2; 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 H2 must be in the plane H´ at the same distance from, and on the
same side of, the axis, as in H´2. The straight line passing through
H´2 parallel to the axis intersects the ray H´1F´ in the point O´1,
which must be the image of O1. If O be the foot of the perpendicular
from O1 to the axis, then OO1 is represented by the line O´O´1 also
perpendicular to the axis.

This construction is not applicable if the object or image be
infinitely distant. For example, if the object OO1 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´1; this point is the image of O1, and y´ is its magnitude. From the
figure we have tan w = HH1/FH = y´/f, or f = y´/tan w; this equation
was used by Gauss to define the focal length.

Referring to fig. 3, we have from the similarity of the triangles OO1F
and HH2F, HH2/OO1 = FH/FO, or O´O´1/OO1 = FH/FO. Let y be the
magnitude of the object OO1, y´ that of the image O´O´1, x the focal
distance FO of the object, and f the object-side focal distance FH;
then the above equation may be written y´/y = f/x. From the similar
triangles H´1H´F´ and O´1O´F´, we obtain O´O´1/OO1 = 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 y´/y = x´/f´. The ratio of the size of the image to
the size of the object is termed the _lateral magnification_. Denoting
this by [beta], we have

[beta] = y´/y = f/x = x´/f´, (1)

and also

xx´ = ff´. (2)

By differentiating equation (2) we obtain

dx´= -(ff´/x²)dx or dx´/dx = -ff´/x². (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 [alpha].
Equation (3) gives important information on the displacement of the
image when the object is moved. Since f and f´ always have contrary
signs (as is proved below), the product -ff´ is invariably positive,
and since x² is positive for all values of x, it follows that dx and
dx´ 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 O3O2 and
the image O´3O´2.

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 OH1,
O´H´1, in fig. 3, u is termed the "angular aperture for the object,"
and u´ the "angular aperture for the image." The ratio of the tangents
of these angles is termed the "convergence" and is denoted by [gamma],
thus [gamma] = tan u´/tan u. Now tan u´= H´H´1/O´H´ = H´H´1/(O´F´+
F´H´) = H´H´1/(F´H´- F´O´). Also tan u = HH1/OH = HH1/(OF + FH) =
HH1/(FH-FO). Consequently [gamma] = (FH - FO)/(F´H´-F´O´), or, in our
previous notation, [gamma] = (f - x)/(f´- x´).

From equation (1) f/x = x´/f´, we obtain by subtracting unity from
both sides (f-x)/x = (x´-f´)/f´, and consequently

f - x x f
------- = - -- = - -- = [gamma]. (4)
f´ - x´ f´ x´

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. [alpha][gamma] = [beta].

In addition to the four cardinal points F, H, F´, H´, J. B. Listing,
"Beiträge aus physiologischen Optik," _Göttinger Studien_ (1845)
introduced the so-called "nodal points" (_Knotenpunkte_) of the
system, which are the two conjugate points from which the object and
image appear under the same angle. In fig. 5 let K be the nodal point
from which the object y appears under the same angle as the image y´
from the other nodal point K´. Then OO1/KO = O´O´1/K´O´, or OO1/(KF +
FO) = O´O´1/(K´F´+ F´O´), or OO1/(FO - FK) = O´O´1/(F´O´- F´K´).
Calling the focal distances FK and F´K´, X and X´, we have y/(x - X) =
y´/(x´- X´), and since y´/y = [beta], it follows that 1/(x - X) =
[beta]/(x´- X´). Replace x´ and X´ by the values given in equation
(2), and we obtain

1 /ff´ ff´\ xX
----- = [beta]/( --- - --- ) or 1 = -[beta]---.
x - X \ x X / ff´

Since [beta] = f/x = x´/f´, we have f´ = -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 f = -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
corresponding image subtend the same angle at the principal points.

_Equations Relating to the Principal Points._--It is sometimes
desirable to determine the distances of an object and its image, not
from the focal points, but from the principal points. Let A (see fig.
3) be the principal point distance of the object and A´ that of the
image, we then have

A = HO = HF + FO = FO - FH = x - f,
A´ = H´O´ = H´F´ + F´O´ = F´O´ - F´H´ = x´ - f´,

whence

x = A + f and x´ = A´ + f´.

Using xx´ = ff´, we have (A + f)(A´ + f´) = ff´, which leads to AA´ +
Af´ + A´f = O, or

f´ f
1 + -- + - = O;
A´ A

this becomes in the special case when f = -f´,

1 1 1
-- - -- = --.
A´ A f

To express the linear magnification in terms of the principal point
distances, we start with equation (4) (f - 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 [beta] = -fA´/f´A.

In the special case of f = -f´, this becomes [beta] = 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 [gamma] = 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, F1 is the first
principal focus of the first, and F´2 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´1 of the first system,
intersecting the axis at an angle w´1. The point F´1 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 F´1 in F´ by the second system leads to the relations F2F´1 = x2,
and F´2F´ = x´2, whence x2x´2 = f2f´2. Denoting the distance between
the adjacent focal planes F´1, F2 by [Delta], we have [Delta] = F´1F2
= -F2F´1, so that x´2 = -f2f´2/[Delta]. A similar ray parallel to the
axis at a distance y proceeding from the image-side will intersect the
axis at the focal point F2; 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 x1x´1 = f1f´1, and since x´1 =
F´1F2 = [Delta], we have x1 = f1f´1/[Delta] 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´1 = y/f´1. The variation of the angle w´1 by the second
system is deduced from the equation to the convergence, viz. [gamma] =
tan w´2/tan w2 = -x2/f´2 = [Delta]/f´2, and since w2 = w´1, we have
tan w´2 = ([Delta]/f´2) tan w´1. Since w´ = w´2 in our system of
notation, we have

y yf´2 f´1.
f´ = ------ = --------------- = -----------. (5)
tan w´ [Delta] tan w´1 f´2/[Delta]

By taking a ray proceeding from the image-side we obtain for the first
principal focal distance of the combination

f = -f1f2/[Delta].

In the particular case in which [Delta] = 0, the two focal planes F´1,
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.

So far we have assumed that all the rays proceeding from an
object-point are exactly united in an image-point after transmission
through the ideal system. The question now arises as 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
refracting surface. Let such a surface divide media of refractive
indices n and n´, the former being to the left. The point where the
axis intersects the surface is the vertex S (fig. 7). Denote the
distance of the axial object-point O from S by s; the distance from 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. From the
triangle O´PC we have sin u = (r/c) sin i, and from the triangle O´PC
we have sin u´ = (r/c´) sin i´. By Snell's law we have n´/n = sin
i/sin i´, and also [phi] = u´ + i´. Consequently c´ 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 i/c sin i´ = n´c´/nc. Also c = OC = CS + SO = -SC
+ SO = s - r, and similarly c´ = s´ - r. Substituting these values we
obtain

p´ n´(s´ - r) n(s - r) n´(s´ - r)
-- = ----------, or -------- = ----------. (6)
p n(s - r) p p´

To obtain p and p´ we use the triangles OPC and O´PC; we have p² = (s
- r)² + r² + 2r(s - r) cos [phi], p´² = (s´ - r)² + r² + 2r(s´ - r)
cos [phi]. Hence if s, r, n and n´ be constant, s´ must vary as [phi]
varies. The refracted rays therefore do not reunite in a point, and
the deflection is termed the spherical aberration (see ABERRATION).

Developing cos [phi] in powers of [phi], we obtain

/ [phi]² [phi]^4 [phi]^6 \
p² = (s - r)² + r² + 2r(s - r) ( 1 - ------ + ------- - ------- + ...),
\ 2! 4! 6! /

and therefore for such values of [phi] for which the second and higher
powers may be neglected, we have p² = (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

n´ n n´- n
-- = -- + -----. (7)
s´ s r

This relation shows that in a very small central aperture in which the
equation p = s 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. Gauss 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´
is measured from F´. The focal lengths are directly deducible from
equation (7):--

f´ = -n´r/(n´ - n) (8)

f = nr/(n´ - n). (9)

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.

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 r1, r2 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 F1 be the first principal focus of the first system of
radius r1, and F1´ the second principal focus; and let S1 be its
vertex. Denote the distance F1 S1 (the first principal focal length)
by f1, and the corresponding distance F´1 S1 by f´1. Let the
corresponding quantities in the second system be denoted by the same
letters with the suffix 2.

By equations (8) and (9) we have

r1 nr1 nr2 r2
f1 = -----, f´1 = - -----, f2 = - -----, f´2 = -----,
n - 1 n - 1 n - 1 n - 1

f2 having the opposite sign to f1. Denoting the distance F´1F2 by
[Delta], we have [Delta] = F´1F2 = F´1S1 + S1S2 + S2F2 = F´1S1 + S1S2
- F2S2 = f´1 + d - f2.

Substituting for f´1 and f2 we obtain

nr1 nr2
[Delta] = ----- + d + -----.
n - 1 n - 1

Writing R = [Delta](n - 1), this relation becomes

R = n(r2 - r1) + d(n - 1).

We have already shown that f (the first principal focal length of a
compound system) = -f1f2/[Delta]. Substituting for f1, f2 and [Delta]
the values found above, we obtain

r1r2n r1r2n
f = --------- = ------------------------------, (10)
(n - 1)R} (n - 1){n(r2 - r1) + d(n - 1)}

which is equivalent to

1 /1 1 \ (n-1)²d
-- = (n - 1)( -- - -- ) + -------.
f \r1 r2/ r1r2n

If the lens be infinitely thin, i.e. if d be zero, we have for the
first principal focal length.

1 /1 1 \
-- = (n - 1)( -- - -- ).
f \r1 r2/

By the same method we obtain for the second principal focal length

f´1f´2 nr1r2
f´ = ------- = - --------- = -f.
[Delta] (n - 1)R

The reciprocal of the focal length is termed the _power_ of the lens
and is denoted by [phi]. In formulae involving [phi] it is customary
to denote the reciprocal of the radii by the symbol [rho]; we thus
have [phi] = 1/f, [rho] = 1/r. Equation (10) thus becomes

(n - 1)²d[rho]1[rho]2
[phi] = (n - 1)([rho]1 - [rho]2) + ---------------------.
n

The unit of power employed by spectacle-makers is termed the _diopter_
or _dioptric_ (see SPECTACLES).

We proceed to determine the distances of the focal points from the
vertices of the lens, i.e. the distances FS1 and F´S2. Since F is
represented by the first system in F2, we have by equation (2)

f1f´1 f1f´1 nr1²
x1 = ----- = ------- = --------,
x´1 [Delta] (n - 1)R

where x1 = F1F, and x´1 = F´1F2 = [Delta]. The distance of the first
principal focus from the vertex S, i.e. S1F, which we denote by s_F
is given by s_F = S1F = S1F1 + F1F = -F1S1 + F1F. Now F1S1 is the
distance from the vertex of the first principal focus of the first
system, i.e. f1 and F1F = x1. Substituting these values, we obtain

r1 nr1² r1(nr1 + R)
s_F = - ----- - -------- = -----------.
n - 1 (n - 1)R (n - 1)R

The distance F´2F´ or x´2 is similarly determined by considering F´1
to be represented by the second system in F´.

We have

f2f´2 f2f´2 nr2²
x´2 = ----- = ------- = --------,
x2 [Delta] (n - 1)R

so that

r2(nr2 - R)
s_F´ = x´2 - f´2 = -----------,
(n - 1)R

where s_F´ denotes the distance of the second principal focus from
the vertex S2.

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 s_H = S1H, i.e.
the distance of the object-side principal point from the vertex of the
first surface, and s_H´ = S2H´, i.e. the distance of the image-side
principal point from the vertex of the second surface, then f = FH =
FS1 + S1H = -S1F + S1H = -s_F + s_H; hence s_H = s_F + f = -dr1/R.
Similarly s_H´ = s_F´ + f´ = -dr2/R. It is readily seen that the
distances s_H and s_H´ are in the ratio of the radii r1 and r2.

The distance between the two principal planes (the interstitium) is
deduced very simply. We have S1S2 = S1H + HH´ + H´S2, or HH´ = S1S2 -
S1H + S2H´. Substituting, we have

HH´ = d - s_H + s_H´ = d(n - 1)(r2 - r1 + d)/R.

The interstitium becomes zero, or the two principal planes coincide,
if d = r1 - r2.

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
and condensing, or divergent and dispersing; the term positive is
sometimes applied to the former, and the term negative to the latter.
Convergent lenses transform a parallel pencil into a converging one,
and increase the convergence, and diminish the divergence of any
pencil. Divergent lenses, on the other hand, transform a parallel
pencil into a diverging one, and diminish the convergence, and
increase the divergence of any pencil. In convergent lenses the first
principal focal distance is positive and the second principal focal
distance negative; in divergent lenses the converse holds.

The four forms of lenses are interpretable by means of equation (10).

r1r2n
f = -------------------------------.
(n - 1) {n(r2 - r1) + d(n - 1)}

(1) If r1 be positive and r2 negative. This type is called biconvex
(fig. 9, 1). The first principal focus is in front of the lens, and
the second principal focus behind the lens, and the two principal
points are inside the lens. The order of the cardinal points is
therefore FS1HH´S2F´. The lens is convergent so long as the thickness
is less than n(r1 - r2)/(n - 1). The special case when one of the
radii is infinite, in other words, when one of the bounding surfaces
is plane is shown in fig. 9, 2. Such a collective lens is termed
_plano-convex_. As d increases, F and H move to the right and F´ and
H´ to the left. If d = n(r1 - r2)/(n - 1), the focal length is
infinite, i.e. the lens is telescopic. If the thickness be greater
than n(r1 - r2)/(n - 1), the lens is dispersive, and the order of the
cardinal points is HFS1S2F´H´.

(2) If r1 is negative and r2 positive. This type is called _biconcave_
(fig. 9, 4). Such lenses are dispersive for all thicknesses. If d
increases, the radii remaining constant, the focal lengths diminish.
It is seen from the equations giving the distances of the cardinal
points from the vertices that the first principal focus F is always
behind S1, and the second principal focus F´ always in front of S2,
and that the principal points are within the lens, H´ always following
H. If one of the radii becomes infinite, the lens is _plano-concave_
(fig. 9, 5).

(3) If the radii are both positive. These lenses are called
_convexo-concave_. Two cases occur according as r2 > r1, or < r1. (a)
If r2 > r1, we obtain the _mensicus_ (fig. 9, 3). Such lenses are
always collective; and the order of the cardinal points is FHH´F´.
Since s_F and s_H are always negative, the object-side cardinal
points are always in front of the lens. H´ can take up different
positions. Since s_H´ = -dr2/R = -dr2/{n(r2 - r1) + d(n - 1)}, s_H´
is greater or less than d, i.e. H´ is either in front of or inside the
lens, according as d < or > {r2 - n(r2 - r1)}/(n - 1). (b) If r2 < r1 the
lens is dispersive so long as d < n(r1 - r2)/(n-1). H is always behind
S1 and H´ behind S2, since s_H and s_H´ are always positive. The
focus F is always behind S1 and F´ in front of S2. If the thickness be
small, the order of the cardinal points is F´HH´F; a dispersive lens
of this type is shown in fig. 9, 6. As the thickness increases, H, H´
and F move to the right, F more rapidly than H, and H more rapidly
than H´; F´, on the other hand, moves to the left. As with biconvex
lenses, a telescopic lens, having all the cardinal points at infinity,
results when d = n(r1 - r2)/(n - 1). If d > n(r1 - r2)/(n - 1), f is
positive and the lens is collective. The cardinal points are in the
same order as in the mensicus, viz. FHH´F´; and the relation of the
principal points to the vertices is also the same as in the mensicus.

(4) If r1 and r2 are both negative. This case is reduced to (3) above,
by assuming a change in the direction of the light, or, in other
words, by interchanging the object- and image-spaces.

The six forms shown in fig. 9 are all used in optical constructions.
It may be stated fairly generally that lenses which are thicker at the
middle are collective, while those which are thinnest at the middle
are dispersive.

_Different Positions of Object and Image._--The principal points are
always near the surfaces limiting the lens, and consequently the lens
divides the direct pencil containing the axis into two parts. The
object can be either in front of or behind the lens as in fig. 10. If
the object point be in front of the lens, and if it be realized by
rays passing from it, it is called _real_. If, on the other hand, the
object be behind the lens, it is called _virtual_; it does not
actually exist, and can only be realized as an image.

When we speak of "object-points," it is always understood that the
rays from the object traverse the first surface of the lens before
meeting the second. In the same way, images may be either real or
virtual. If the image be behind the second surface, it is _real_, and
can be intercepted on a screen. If, however, it be in front of the
lens, it is visible to an eye placed behind the lens, although the
rays do not actually intersect, but only appear to do so, but the
image cannot be intercepted on a screen behind the lens. Such an image
is said to be _virtual_. These relations are shown in fig. 11.

By referring to the equations given above, it is seen that a thin
convergent lens produces both real and virtual images of real objects,
but only a real image of a virtual object, whilst a divergent lens
produces a virtual image of a real object and both real and virtual
images of a virtual object. The construction of a real image of a real
object by a convergent lens is shown in fig. 3; and that of a virtual
image of a real object by a divergent lens in fig. 12.

_The optical centre of a lens_ is a point such that, for any ray which
passes through it, the incident and emergent rays are parallel. The
idea of the optical centre was originally due to J. Harris (_Treatise
on Optics_, 1775); it is not properly a cardinal point, although it
has several interesting properties. In fig. 13, let C1P1 and C2P2 be
two parallel radii of a biconvex lens. Join P1P2 and let O1P1 and O2P2
be incident and emergent rays which have P1P2 for the path through the
lens. Then if M be the intersection of P1P2 with the axis, we have
angle C1P1M = angle C2P2M; these two angles are--for a ray travelling
in the direction O1P1P2O2--the angles of emergence and of incidence
respectively. From the similar triangles C2P2M and C1P1M we have

C1M : C2M = C1P1 : C2P2 = r1 : r2. (11)

Such rays as P1P2 therefore divide the distance C1C2 in the ratio of
the radii, i.e. at the fixed point M, the optical centre. Calling S1M
= s1, S2M = s2, then C1S1 = C1M + MS1 = C1M - S1M, i.e. since C1S1 =
r1, C1M = r1 + s1, and similarly C2M = r2 + s2. Also S1S2 = S1M + MS2
= S1M - S2M, i.e. d = s1 - s2. Then by using equation (11) we have s1
= r1d/(r - r2) and s2 = r2d/(r1 - r2), and hence s1/s2 = r1/r2. The
vertex distances of the optical centre are therefore in the ratio of
the radii.

The values of s1 and s2 show that the optical centre of a biconvex or
biconcave lens is in the interior of the lens, that in a plano-convex
or plano-concave lens it is at the vertex of the curved surface, and
in a concavo-convex lens outside the lens.

_The Wave-theory Derivation of the Focal Length._--The formulae above
have been derived by means of geometrical rays. We here give an
account of Lord Rayleigh's wave-theory derivation of the focal length
of a convex lens in terms of the aperture, thickness and refractive
index (_Phil. Mag._ 1879 (5) 8, p. 480; 1885, 20, p. 354); the
argument is based on the principle that the optical distance from
object to image is constant.

"Taking the case of a convex lens of glass, let us suppose that
parallel rays DA, EC, GB (fig. 14) fall upon the lens ACB, and are
collected by it to a focus at F. The points D, E, G, equally distant
from ACB, lie upon a front of the wave before it impinges upon the
lens. The focus is a point at which the different parts of the wave
arrive at the same time, and that such a point can exist depends upon
the fact that the propagation is slower in glass than in air. The ray
ECF is retarded from having to pass through the thickness (d) of glass
by the amount (n - 1)d. The ray DAF, which traverses only the extreme
edge of the lens, is retarded merely on account of the crookedness of
its path, and the amount of the retardation is measured by AF - CF. If
F is a focus these retardations must be equal, or AF - CF = (n - 1)d.
Now if y be the semi-aperture AC of the lens, and f be the focal
length CF, AF - CF = [root](f² + y²) - f = ½y²/f approximately, whence

f = ½y²/(n - 1)d. (12)

In the case of plate-glass (n - 1) = ½ (nearly), and then the rule
(12) may be thus stated: _the semi-aperture is a mean proportional
between the focal length and the thickness_. The form (12) is in
general the more significant, as well as the more practically useful,
but we may, of course, express the thickness in terms of the
curvatures and semi-aperture by means of d = ½y²[r1^(-1) - r2^(-1)].
In the preceding statement it has been supposed for simplicity that
the lens comes to a sharp edge. If this be not the case we must take
as the thickness of the lens the difference of the thicknesses at the
centre and at the circumference. In this form the statement is
applicable to concave lenses, and we see that the focal length is
positive when the lens is thickest at the centre, but negative when
the lens is thickest at the edge."

_Regulation of the Rays._

The geometrical theory of optical instruments can be conveniently divided into four parts: (1) The relations of the positions and sizes of objects and their images (see above); (2) the different aberrations from an ideal image (see ABERRATION); (3) the intensity of radiation in the object- and image-spaces, in other words, the alteration of brightness caused by physical or geometrical influences; and (4) the regulation of the rays (_Strahlenbegrenzung_).