L
- L
- the Roman numeral 50.
- L, l
- symbols for the liter. The lowercase l is the official symbol, but because
it is so easily confused with the numeral 1 the SI
permits the capital letter L to be used instead. Sometimes a script version
of the lowercase letter is used, but this is not approved by the SI.
- láb
- the traditional Hungarian foot, equal to about 31.6 centimeters (12.44 English
inches). This was the Hungarian version of the Viennese fuß.
- labor
- a traditional unit of area in Latin American countries. The labor is equal
to the area of a square 1000 varas on a side,
or 0.04 legua. Using the Texas standard for the vara (33 1/3 inches), this
is equivalent to 177.136 acres or 71.685 hectares.
The word labor means work in Spanish, as it does in English. As a unit
it represents the area that could be cultivated by a single farmer, somewhat
like the old English hide.
- lachter
- a traditional unit of length used in mining in Germany and other German-speaking regions. A variation on the fathom, the unit varied regionally (and even from mine to mine) but it was usually close to 2 meters (about 78.74 inches). The lachter was divided into 8 spann or into 80 lachterzoll.
- lachterzoll
- the "lachter inch," a traditional unit of length equal to 1/80 lachter (previous entry). This was roughly 2.5 cm, about the same as the English inch but a bit shorter than the standard German zoll. In the 1900s, a decimal lachterzoll equal to 1/100 lachter was also used; this was about 2 cm (0.78 inch or 0.76 zoll).
- lai
- see rai.
- lakh or lac
- a traditional unit of quantity in India, equal to 10^{5} or 100
000. In India the lakh is used commonly instead of the million and commas
are used to isolate the number of lakh; for example, the number 5 300 000
is called 53 lakh and written "53,00,000". See also crore.
- lambda [1]
- a metric unit of volume equal to the cubic millimeter (mm^{3}) or
microliter (µL). The lambda has been used in chemistry in measuring very
small samples. The symbol is the lower case Greek letter lambda.
- lambda [2]
- a unit of relative distance used in the design of integrated circuits in
microelectronics. These circuits are usually designed to be "scalable," so
that the same design can be repeated on ever-smaller chips as technology improves.
The "feature size" of a design is the width of its smallest element, and one
lambda equals one half the feature size. Lambdas of a few tenths of a micrometer
(micron) are common.
- lambert (La or Lb or L)
- a CGS unit of luminance. Luminance is the luminous
intensity of a surface, measuring the intensity of the light emitted (or reflected)
in all directions per unit of area of the surface. One lambert is the luminance
of a surface that emits or reflects one lumen per square centimeter. The lambert
is a large unit, and practical measurements tend to be in millilamberts (mLa).
A geometric calculation shows that a surface area having an intensity of one
candela per square meter emits a total light flux of pi lumens per square
meter; as a result, one lambert equals 10^{4}/pi
= 3183.099 candelas per square meter and
1 millilambert equals 3.183 099 candelas
per square meter. The lambert honors the German physicist Johann Lambert (1728-1777),
who showed that the illuminance of a surface is inversely proportional to
the square of the distance from the light source. There has never been agreement
on the symbol for the lambert; La is a good choice because it is unlikely
to be confused with any other unit symbol.
- lanac
- a traditional unit of land area in countries of the former Yugoslavia. Usage
varies. In Serbia, the lanac is equal to about 0.56 hectare
(1.38 acres), making it the Serbian equivalent
of the Austrian joch and Croatian jutro.
In Croatia, however, the lanac is a larger unit equal to 0.7193 hectare (1.777
acres). The word "lanac" means "chain", presumably referring to surveyors'
use of chains in measuring land.
- land mile
- the ordinary statute mile, equal to 5280 feet
or 1609.344 meters, is sometimes called a "land mile" to distinguish it from
the nautical mile. Similarly, a land
league is equal to 3 statute miles (5280 yards or 4828.03 meters) as distinguished
from 3 nautical miles.
- lane meter
- a unit of deck area for "roll on/roll off" ships: cargo vessels designed
so that containers or other cargo can be rolled on and off the decks of the
ship. A lane is a strip of deck 2 meters wide. A lane meter is an area of
deck one lane wide and one meter long, that is, 2 square meters (21.528 square
feet).
- langdo
- a traditional unit of land area in Bhutan. Like many European land units,
the langdo is the area that a team of oxen can plow in a day. If the land
is dry, a langdo is about 1/7 hectare or
1/3 acre; for a wet paddy field the langdo is
about 1/10 hectare or 1/4 acre.
- langley (Ly)
- a CGS unit of heat transmission equal to one thermochemical
calorie per square centimeter, or exactly 41.84 kilojoules per square meter
(kJ/m^{2}). Named for the American astronomer Samuel P. Langley (1834-1906),
the langley is used to express the rate of solar radiation received by the
earth.
- lap
- an informal unit of distance used in athletic competitions. In athletics
("track"), a lap is the length of one trip around a running track. This
may vary from track to track, but at the level of serious competition most
tracks have a standard length. In English speaking countries this was formerly
1/4
mile (1320 feet
or 402.336 meters). Tracks used in most competitions today have a length
of exactly 400 meters (1312.34 feet). In swimming, a lap is one tour of
the pool, that is, twice the length of the pool, a distance of exactly 100
meters (328.08 feet) in Olympic-size pools but only 50 meters (164.04 feet)
in many recreational and "short-course"
pools.
- last
- an ancient northern European unit used in measuring large quantities, either
by mass or volume or both. "Last" is the German word for "load," a meaning
which also survives in the English word "ballast." Generally the last is approximately
4000 pounds (about 1800 kilograms) as a mass
unit or 80 bushels (about 3 cubic meters)
as a volume unit. In the U.S., a last of wool was formerly 12 sacks at 364
pounds per sack, or 4368 pounds (1981.3 kilograms). In the Netherlands, the
last is a metric unit of volume equal to exactly 3 cubic meters. In Germany,
where the last was previously a volume unit equal to 2.819 cubic meters, it's
also a metric unit of mass equal to 2 metric tons (about 4409 pounds). In
Britain, a Riga last of timber is 80 cubic feet (2.265 cubic meters)
of square-sawn timber.
- lb, lbf, lbm
- lb is the traditional symbol in English, Spanish, and Italian for the pound,
derived from the Latin word libra for the Roman version of the same
unit. The symbols lbf and lbm are used to distinguish between pounds of force
and pounds of mass, respectively.
- LD
- see lunar distance (below).
- lea
- a traditional unit of length for yarn, varying with the weight of the fibers
in the yarn. Typically a lea of wool is 80 yards (73 meters); a lea of cotton
or silk 120 yards (110 meters); and a lea of linen 300 yards (274 meters).
For cotton and wool, a lea is equal to 1/7 hank.
A lea is sometimes called a skein.
- league
- a traditional unit of distance. Derived from an ancient Celtic unit and
adopted by the Romans as the leuga, the league became a common unit
of measurement throughout western Europe. It was intended to represent,
roughly, the distance a person could walk in an hour. The Celtic unit seems
to have been rather short (about 1.5 Roman miles, which is roughly 1.4
statute miles or 2275 meters), but the unit grew longer over time. In many
cases it was equal to 3 miles, using whatever
version of the mile was current. At sea, the league was most often equal
to 3 nautical
miles, which is 1/20 degree [2],
3.45 statute miles, or exactly 5556 meters. In the U.S. and Britain, standard
practice is to define the league to be 3 statute miles (about 4828.03 meters)
on land or 3 nautical miles at sea. However, many occurrences of the "league"
in English-language works are actually references to the Spanish league (the
legua), the Portuguese league (legoa) or the French league (lieue). For
these units, see below on this page. In the classic Jules Verne novel Twenty
Thousand Leagues under the Sea (Vingt Mille Lieues sous les Mers)
the unit in the title is the French metric lieue, equal to exactly
4000 meters.
- leap
- a traditional Welsh unit of distance equal to 6 feet 9 inches or 2.0574
meters.
- leap second
- an extra second added at the end of a day (June 30 or December 31) to realign
timekeeping with the earth's rate of rotation. See day
for details.
- leap year
- a unit of civil time equal to 366 days. See year
[2]. Normally, the day of the week on which a specific date falls
advances by one day from year to year. For example, August 1 falls on Tuesday
in 2006 and on Wednesday in 2007. But following the addition of a extra
day on February 29, a date "leaps" over a day of the week: in 2008, a leap
year, August 1 leaps over Thursday to fall on Friday. A leap year is sometimes
called a bissextile year.
- legua [1]
- the Spanish league. The traditional legua is equal to 5000 varas,
which is close to 2.6 miles or 4.2 kilometers. Using the Texas definition
of the vara, the legua is 2.6305 miles, 13889
feet, or 4233.4 meters. Using the traditional
Spanish definition, it would be 2.597 miles, 13712 feet, or 4179.4 meters.
Technically, this unit was abolished by Philip II in 1568, but it remained
in rather wide use, especially in the Americas. During the late 18th and early
19th centuries, a league of 8000 varas (4.15 miles or 6680 meters) was legal
in Spain. At sea, Spanish sailors used the usual marine league (3 nautical
miles or 5556 meters) or Philip V's "geographical" league of 1/17.5 degree
(3.429 nautical miles or 6350.5 meters). At present, the legua is used informally
in Argentina and in other Spanish-speaking countries as a metric unit equal
to exactly 5 kilometers (3.107 miles).
- legua [2]
- a traditional Spanish unit of area equal to one square legua [1].
In Spanish-speaking Latin America and the southwestern states of the U.S.
land was customarily measured in leguas, with 1 legua equal to 25 labors (see
above) or 25 million square varas. Using the Texas definition of the vara
as the starting point, the legua is 4428.4 acres,
6.919 square miles, 1792 hectares, or 17.92
square kilometers. A slightly larger figure, 4439 acres (1796 hectares), is
used in California. Larger sizes, between 1800 and 1900 hectares, were formerly
used in some parts of South America. In Mexico and Texas, this unit is often
called a sitio.
- légua or legoa
- the Portuguese league, equal to 3 milhas (Portuguese miles). This is equal
to about 3.836 statute miles or 6174.1 meters.
- length (lg)
- an informal unit of distance. The distance between competitors in horse
races, boat races, and similar situations is naturally expressed in lengths,
with one length equal to the average length of a horse, boat, etc. In horse
racing, the length of a horse is often understood to be about 8 feet or 2.4
meters. However, since the horses are moving a different speeds the distance
between them as they near the finish line is changing. To avoid this uncertainty,
the reported distance in lengths is often computed as 5 times the difference
in their running times in seconds. This means the length is actually interpreted
as a unit of time equal to 1/5 second. (Since the speed of a thoroughbred
horse often exceeds 50 feet per second, this calculation understates the distance.)
- lethal dose (LD)
- a measure used in pharmacology to express the percentage of a population
killed by a dose of the substance being studied. The measurement is often
given as a subscript. For example, the potency of a drug or pesticide is commonly
expressed by stating the size of the LD_{50} dose: the amount of the
substance that kills 50% of the test population.
- lǐ or li
- a traditional unit of distance in China. A Confucian proverb widely misquoted
in the West as "a journey of a thousand miles begins with a single step" actually
says "a journey of a thousand li begins with a single step." Although the
traditional li was approximately 1/3 mile or 500 meters, the late Imperial
governments of China used a li of 1800 ch'ih,
which is 2115 feet, about 0.401 mile, or 644.65 meters. In modern China,
the li equals exactly 0.5 kilometer or 500 meters. In Chinese, the kilometer
itself is often called a gongli, or "metric
li" (see gong).
- liang
- a traditional Chinese weight unit. During the European colonial period the
liang was equal to 1/16 catty, 1/12 pound,
or about 37.8 grams; this made it the same as a tael.
In modern China, the liang equals 1/10 jin or
10 qian; this is exactly 50 grams (1.7637 ounces).
- libra or libbra (lb)
- a traditional unit of weight in Italian, Spanish, and Portuguese speaking
countries. The libra was the Roman unit from which the English pound
is descended; the symbol "lb" for the pound comes from this unit. The Roman
libra contained only 12 unciae (ounces) and was about 0.722 English pound.
The traditional Italian libbra was often of similar size, but a wide variety
of libbras were used in Italian markets over the centuries. The Spanish and
Portuguese units are larger, generally in the range from 1.011 to 1.016 English
pound (very close to 460 grams). The Spanish libra equals 16 onzas, and the
Portuguese libra equals 16 onças. The word "libra" is sometimes used
now for the kilogram, a much larger unit.
- lieue
- the French league. A variety of lieue units were used for land measurement
in France, but generally these units were around 2.4-2.5 statute miles
in length. In the 18th century, the legal unit was the lieue de poste,
defined to equal 2000 toises or 2 milles
(2.4221 miles or 3898 meters). In metric France the lieue is now considered
to equal exactly 4 kilometers (2.4855 miles). See league (above). At sea,
the lieue was often taken to equal 1/25 degree
[2] or 2.4 nautical miles (4445
meters or 2.7619 miles); this unit was gradually replaced by the internationally
recognized 3 nautical miles (5556 meters or 3.452 miles). In the classic
Jules Verne novel Vingt Mille Lieues sous les Mers (Twenty Thousand
Leagues under the Sea) the unit
in the title is the metric lieue of exactly
4000 meters.
- light second
- a unit of distance equal to the distance light travels in a vacuum in one
second. In accordance with the official definition of the meter,
this distance is exactly 299 792 458 meters (about 186 282.4 miles).
Similarly, a light minute is 60 light seconds (about 17 987
547 kilometers or 11 176 944 miles) and a light day is 1440
light minutes (about 25.902 billion kilometers or 16.095 billion miles).
- light watt
- a unit measuring the relative power output of a light source. Calculating
the power delivered in the form of visible light is rather complicated. For
a monochromatic (single frequency) light source such as a laser, the power
in light watts equals 683V(l), where l is the wavelength of
the light and V(l) is the relative power in watts
per lumen (W/lm) required to produce a constant brightness
sensation in the eye at wavelength l. Values of V(l)
are defined by the International Commission on Illumination (CIE). The maximum
value of V(l) is 1/683 W/lm = 1.464 mW/lm at the wavelength
l = 555 nanometers (nm), the wavelength to which the eye is most sensitive.
When the source delivers light over a range of frequencies (as a light bulb
does), it is necessary to compute 683 times the integral of V(l) multiplied
by the fraction of energy delivered at wavelength l. See also candela
and talbot.
- light year (ly)
- a unit of distance used in astronomy. One light year is the distance that
light travels in one year through a vacuum (and, of course, most of the Universe
is close to being a vacuum). In the official definition of the International
Astronomical Union, the light year is based on the Julian year of exactly
365.25 days or 31 557 600 seconds and the defined speed of light c
= 299,792,458 meters per second. Thus one light year equals exactly 9 460
730 472 580.8 kilometers, or 9 460 730 472 580 800 meters. One light year
is approximately 5.880 trillion miles.
- ligne
- a traditional unit of distance in French speaking countries, equal to 1/12
pouce (French inch) and corresponding closely to the English line [1].
The Swiss ligne is used throughout the world by watchmakers; it equals about
2.256 millimeters (0.0888 inch) and is divided into 12 douzièmes. In
English the ligne is often pronounced "lean."
- ligula
- a Roman unit of liquid volume equal to 1/48 sextarius or about 11.07 milliliters.
The word literally means "a lick."
- line (li or ''') [1]
- a traditional unit of distance equal to 1/12 inch (about 2.1167 millimeters).
For measuring the thickness of buttons, there is also a smaller line equal
to 1/40 inch (0.635 millimeter). The line is called the ligne (see
above) in French, the linea in Spanish, the linie in German,
and the liniya in Russian.
- line (li) [2]
- a former name for the maxwell, the CGS
unit of magnetic flux. The unit was called the line because magnetic fields
were traditionally represented by lines depicting the direction of the field;
the idea was to quantify the strength of these lines. This is a small unit,
so fields were often measured in megalines; one megaline is equal to 0.01
weber.
- line [3]
- a traditional unit of area used in printing and advertising, equal to 1/14
inch (1.814 millimeters) multiplied by the width of the printed line. This
usage is short for agate line.
- linear foot (or "lineal" foot) (ft or lf)
- terms used loosely to describe a one-foot
length of any long, narrow object. The correct term is linear foot;
the word "lineal" refers to a line of ancestry, not to length. Boards, pipes,
and fencing are typical objects measured and sold by the linear foot. In the
moving industry, a linear foot is a one-foot length of a moving van, usually
a volume of about 72 cubic feet (roughly 2 cubic meters). Occasionally the
term "linear foot" is used as an alternate name for the board
foot, but this is appropriate only if the board is 12 inches wide. Terms
such as linear meter and linear yard are used in a similar way
to indicate one-meter or one-yard lengths.
- link
- a traditional unit of distance used by surveyors, equal to 0.01 chain.
In Britain, one link is exactly 0.66 feet, or
7.92 inches, or approximately 20.12 centimeters.
In the U.S., both 66-foot and 100-foot chains have been used; for a 100-foot
chain the link is the same as the foot.
- lippie
- a traditional unit of volume in Scotland equal to 1/4 Scots peck.
This was about 2.27 liters for wheat, peas, or beans and about 3.04 liters
for barley or oats.
- liter or litre (L or l)
- the common metric unit of volume. The liter was originally defined to be equal to exactly
one cubic decimeter,
that is, to the volume of a cube 0.1 meter (or 10 centimeters)
on a side. (This definition made it equal also to the volume occupied by a kilogram of water.) Unfortunately, the physical objects constructed to represent
the meter and kilogram disagreed slightly. As measured by the standard meter
and standard kilogram, the standard liter turned out to be about 1.000 028
cubic decimeters. This discrepancy plagued the metric system for a long
time. In 1901 an international congress accepted the discrepancy and formally
defined the liter to be exactly 1.000 028 dm^{3}. No one was particularly
happy with such an awkward definition, and in 1964 the CGPM repealed the
definition. In the SI, volumes are to be measured
in cubic meters, not in liters. However,
the SI
states that the liter "may be employed as a special name for the cubic decimeter."
Throughout this dictionary, the liter is used as a name for exactly 1 cubic
decimeter, 1000 cubic centimeters, or 0.001 cubic meter. In its renewed
guise as the cubic decimeter, the liter is approximately 61.023 744 cubic
inches. Compared to the customary volume units, the liter is a little more
than a U. S. liquid quart (1.056 688 qt or 33.814 fluid
ounces) but a little less than a U. S. dry quart (0.908 08 qt) or a
British Imperial quart (0.879 89 qt or 35.195 fluid ounces). Its name comes
from a French volume unit, the litron, which was in turn derived
from the Latin litra. The original symbol for the liter was the
lower case letter l, but since 1979 the upper case L has also been accepted.
The U.S. Department of Commerce specifies that L be used, at least by
businesses, to avoid confusion with the numeral 1. The unit is spelled liter in
the U.S. and litre in
Britain; there are many other spellings in various languages (see Spelling
of Metric Units).
- liter atmosphere (L·atm)
- a unit of work or energy used in the study of confined gases. The behavior
of gases is described, to a first approximation, by the ideal gas law PV =
nRT. The ideal gas law is really an energy equation in which the left hand
side, pressure P (in atmospheres) times volume V (in liters), measures the
potential energy in the confined gas. One liter atmosphere is equal to 101.325
joules, 0.09605 Btu
or 74.73 foot pounds.
- liter per 100 kilometers (L/100 km)
- a measure of fuel consumption rate for vehicles used widely in Europe and
elsewhere. Although this unit is specified in the regulations of several
countries, it violates SI rules for naming units. The equivalent SI
unit is centiliters per kilometer (cL/km). In the U.S., fuel consumption
is stated as the number of miles driven per U.S. gallon of fuel consumed;
a consumption rate of x liters
per 100 km equals exactly 100/x liters per kilometer or about 235.215/x miles
per gallon. In the British Commonwealth, fuel consumption was (and sometimes
still is) measured in miles per Imperial gallon; x liters
per 100 kilometers is equal to 282.481/x miles per Imperial gallon.
- liter per mil (L/mil)
- a measure of fuel consumption rate for vehicles used in Sweden. The
Scandinavian mil [4] is equal to 10 kilometers,
so 1 liter per mil equals 10 liters per 100 kilometers.
- livre
- a traditional unit of weight in French speaking countries and in Greece.
The livre corresponds to the English pound
and to the Spanish libra (see above). The livre is divided into 2 marcs or
into 16 onces. The French livre varied from market to market, but the official
standard from about 1350 to the introduction of the metric system was the
livre poids de marc or livre de Paris of 489.5 grams (1.079
English pounds). In modern France, the livre is used as an informal metric
unit equal to exactly 500 grams or 0.5 kilogram (1.1023 pounds). The traditional
Greek livre is also about 500 grams.
- load
- a traditional, generally informal, unit of volume. In U.S. landscaping and
some construction trades a load often means a cubic yard (0.764555 m^{3}).
In ordinary language in the U.S., a load often means the volume of a pickup
truck, a varying unit. In Britain prior to modern times, a load was sometimes
a standardized unit, but it varied with the commodity being carried. A typical
size was 40 bushels (roughly 1.4 cubic meters).
- long hundredweight
- the British hundredweight, equal to 112 pounds.
- long ton
- the traditional British ton, equal to 2240
pounds.
- longword
- a unit of information generally equal to 2 words
[2].
- lot
- a traditional unit of weight in German speaking countries, equal to about
1/2 ounce or 15 grams. "Lot" is the German
word for a lead plumbbob, so the unit represents a small lead weight.
- Lovibond color units
- see degree Lovibond.
- Lpf
- symbol for liters per flush, a specification found on toilets. U.S. government
regulations now require plumbers to install only low-flush toilets
of 6.0 Lpf or less. 1 Lpf = 0.264 U.S. gallon per
flush (gpf).
- lpi
- abbreviation for lines per inch, a unit used to state the resolution of
display devices (such as television or computer monitor screens), or to state
the line spacing of printed pages.
- lug [1]
- an old English name for a rod [1] (5.5 yards
or 5.0292 meters). In some parts of England this unit represented a longer
rod of 7 yards (6.4008 meters), a unit also
called the great lug.
- lug [2]
- a shallow box or crate for produce such as cherries, grapes, or peaches.
The size of a lug varies with the item it contains. Typical lugs hold about
16-28 pounds (7-13 kilograms) of produce in
a volume of roughly 1/3 bushel (about 12 liters).
This unit seems to be particularly common in produce markets in the midwestern
U.S.
- lumberg
- an older name for the talbot, the unit of
luminous (light) energy equal to 1 lumen second.
- lumen (lm)
- the SI unit for measuring the flux of light being
produced by a light source or received by a surface. The intensity of a light
source is measured in candelas. One lumen
represents the total flux of light emitted, equal to the intensity in candelas
multiplied by the solid angle in steradians
(1/(4·pi) of a sphere) into which the light
is emitted. Thus the total flux of a one-candela light, if the light is emitted
uniformly in all directions, is 4·pi lumens.
"Lumen" is a Latin word for light.
- lumen hour (lm h)
- a unit of quantity of light, equal to one lumen of light flux continued
for one hour. The lumen second (lm s) is defined similarly.
- lunar day
- another name for the tidal day, a unit
of time equal to 24 hours 50 minutes used in tidal predictions.
- lunar distance (LD)
- the average distance between the Earth and the Moon (technically, the length
of the semimajor axis of the Moon's orbit). This unit, equal to about 384
401 kilometers or 238 855 miles, is used to
measure the "miss distances" of asteroids passing near the Earth.
- lunar month, lunation
- names for the average interval between two successive new moons, a unit
of time equal to 29.530 59 days. See month [1].
- lusec
- a unit of power used to express the performance or leakage of vacuum pumps.
One lusec represents a flow of one liter per second at a pressure of one micrometer
(or micron) of mercury, or 1 L·µmHg/s.
Since "u" is sometimes used as a symbol for the micron, the name of the unit
is an acronym for liter-micron/second. One lusec is equivalent to 0.001 315
6 atm·cm^{3}/s or 133.3 Pa·cm^{3}/s, which is the
same as 133.3 microwatts.
- luster, lustre, lustrum
- a traditional unit of time equal to 5 years. In ancient Rome the Lustrum
was a ceremony of expiation and purification for the whole population of the
city, carried out every 5 years after the completion of the census. The use
of luster or lustrum as a unit of time in English was fairly common in well-educated
circles as long as "well-educated" meant classically educated; the unit has
pretty much disappeared today.
- lux (lx)
- the SI unit for measuring the illumination (illuminance)
of a surface. One lux is defined as an illumination of one lumen per square
meter or 0.0001 phot. In considering the various
light units, it's useful to think about light originating at a point and shining
upon a surface. The intensity of the light source is measured in candelas;
the total light flux in transit is measured in lumens (1 lumen = 1 candela·steradian);
and the amount of light received per unit of surface area is measured in lux
(1 lux = 1 lumen/square meter). One lux is equal to approximately 0.09290
foot candle.
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Checked and revised December 15, 2008.