rojection of the element, and whose altitude is equal to
the distance of the element from the upper surface of the
fluid.
The distance z has been estimated as positive from the
surface of the fluid downwards. If 9 < 90°, Ave have cos?
positive ; hence, p' will be positive, which shows that the
vertical pressure is exerted downwards. If 90°, we
have cosp negative ; hence, p' is negative, which shows that
the vertical pressure is exerted upwards (see Fig. 135).
Suppose the interior surface of a vessel containing a heavy
fluid to be divided into elementary portions, whose areas
are denoted by s, s\ s", &c. ; denote the distances of these
24:0 MECHANICS.
elements below the upper Surface, by z, z', z", &c. From
the principle just demonstrated, the pressures upon these
surfaces will be denoted by icsz, ws'z, ics"z", &c, and the
entire pressure upon the interior of the vessel will be
equal to,
w(sz + s'z + s"z" + &c.) ; or, to x 2(sz).
Let Z denote the depth of a column of the fluid, whose
base is equal to the entire surface pressed, and whose weight
is equal to the entire pressure, then will this pressure be
equal to w(s + s' -+- *" + &c)Z; or, wZ . Is. Equating
these values, we have,
w.2(sz) =wZ.l(s), .'. Z=^j • (1^3.)
The second member of (143), (Art. 51), expresses the
distance of the centre of gravity of the surface pressed,
below the free surface of the fluid. Hence,
The entire pressure of a heavy fluid upon the interior of
the containing vessel, is equal to the weight of a volume of
the fluid, u'/wse base is equal to the area of the surface
■pressed, and ichose altitude is equal to the distance of the
centre of gravity of the surface from the free surface of the
fluid.
EXAMPLES.
1. A hollow sphere is filled with a liquid. How does the
entire pressure, on the interior surface, compare with the
weight of the liquid ?
SOLUTION.
Denote the radius of the interior surface of the sphere,
by r, and the weight of a unit of volume of the liquid, by
w. The entire surface pressed is measured by 4c?- 2 ; and,
since the centre of gravity of the surface pressed is at a
distance r below the surface of the liquid, the entire pre*
MKCHAXK.S OF LIQUIDS. 241
sure on tne interior surface will be measured by the
expression,
w X 4^r 5 x r — 4«tor*.
But the weight of the liquid is equal to
Hence, the entire pressure is equal to three times the
weight of the liquid.
2. A hollow cylinder, with a circular base, is filled with a
liquid. How does the pressure on the interior surface com-
pare with the weight of the liquid?
SOLUTION.
Denote the radius of the base of the cylinder, by r, and
the altitude, by h. The centre of gravity of the lateral
surface is at a distance below the upper surface of the fluid
equal to \h. If we denote the weight of the unit of volume
of the liquid, by w, we shall have, for the entire pressure on
the interior surface,
whxr* + 2wxr . \h* = wirrh{r -f- h).
But the weight of the liquid is equal to
wtr^h.
T + h
Hence, the total pressure is equal to times the
iceight of the liquid.
If we suppose h = r, the pressure will be twice the
weight.
If we suppose r = 2h, we shall have the pressure equal
to \ of the weight.
If we suppose h =; 2r, the pressure will be equal to three
times the weight, and so on.
11
242 MECHANICS.
In all cases, the total pressure will exceed tbe weight of
the liquid.
3. A right coue, with a circular base, stands on its base,
and is tilled with a liquid. How does the pressure on the
internal surface compare with the weight of the liquid ?
solution.
Denote the radius of the base, by r, and the altitude, by
A, then will the slant height be equal to
^/hF^?.
The centre of gravity of the lateral surface, below the
upper surface of the liquid is equal to §A. If we denote
the weight of a unit of volume of the liquid, by w, we shall
have, for the total pressure on the interior surface,
wvr'h + %w*rhi/h* + r* = w«rh{r + f -/A 2 + r 3 ).
But the weight of the liquid is equal to
%ioirr*h = w*rh x £r.
3r + 2v / A r T~r a
Hence, the total pressure is equal to
times the iceight.
4. Required the relation between the pressure and the
weight in the preceding case, when the cone stands on its
vertex.
SOLUTION.
The total pressure is equal to
^wrrhyh 2 4- r° ;
■\Zh a -f r*
and, consequently, the pressure is equal to — times
the weight of the liquid
MECHANICS OF LIQUIDS. 243
5. What is the pressure on the lateral faces of a cubical
vessel filled with water, the edges of the cube being 4 feet,
and the weight of the water 62^ lbs. per cubic foot ?
A?is. 8000 lbs.
6. A cylindrical vessel is filled with water. The height
of the vessel is 4 feet, and the radius of the base 6 feet.
What is the pressure on the lateral surface ?
Ans. 18850 lbs., nearly.
Centre of Pressure on a Plane Surface.
156. Let ABCD represent a plane, pressed by a fluid
on its upper surface, AB its intersec-
tion with the free surface of the fluid, , j±
G its centre of gravity, the centre y- ^^^^^
of pressure, and s the area of any
cept the scale-pans have hooks at-
tached to their lower surfaces for
the purpose of suspending bodies.
The suspension is effected by a
fine platinum wire, or by some —
other material not acted upon by
the liquids employed.
n
Fig. 144
To determine the Specific Gravity of an Insoluble Body.
161. Attach the suspending wire to the first scale-pan,
and after allowing it to sink in a vessel of water to a certain
depth, counterpoise it by an equal weight, attached to the
hook of the second scale-pan. Place the body in the first
scale-pan, and counterpoise it by weights in the second pan.
These weights will give the weight of the body in air.
Xext, attach the body to the suspending wire, and immerse
it in the water. The buoyant effort of the Mater will be
equal to the weight of a volume of water equivalent to that
of the body (Art. 157) ; hence, the second pan will descend.
Restore the equilibrium by- weights placed iti the first pan.
These weights will give the weight of the displaced water.
254 MECHANICS.
Divide the weight of the body in air by the weight just
found, and the quotient will be the specific gravity sought.
If the body will not sink in water, determine its weight in
air as before ; then attach to it a body so heavy, that the
combination will sink ; find, as before, the loss of weight of
the combination, and also the loss of weight of the heavier
body ; take the latter from the former, and the difference
will be the loss of weight of the lighter body ; divide its
weight in air by this weight, and the quotient will be the
specific gravity sought.
If great accuracy is required, account must be taken of
the buoyant effort of the air, which, when the body is very
light, and of considerable dimensions, will render the appa-
rent weight less than the true weight, or the weight in
vacuum. Since the weights used in counterpoising are
always very dense, and of small dimensions, the buoyant
effort of the air upon them may always be neglected.
To^determine the true weight of a body in vacuum : let
to denote its weight in air, w' its weight in water, and IF its
weight in vacuum ; then will W — ?c, and IV — w', denote
its loss of weight in air and water ; denote the specific
gravity of air referred to water, by s. Since the losses of
weight in air and water arc proportional to their specific
gravities, we have,
W — w : W — w' : : s : 1 ; or, W — to = * W — sio\
1 — s
This weight should be used, instead of the weight in air.
To determine the Specific Gravity of Liquids.
16*2. First Method. — Take a vial with a narrow neck,
and weigh it ; fill it with the liquid, and weigh again ;
empty out the liquid, and fill with water, and weigh again ;
deduct from the last two weights, respectively, the weight
of the vial; these results will give the weights of- equal
MECHANICS OF LIQUID?. 255
volumes of the liquid and of water. Divide the former by
the latter, and the quotient will be the specific gravity
sought.
Second Method. — Take a heavy body, that will sink both
in the liquid and in water, and which will not be acted upon
by either ; determine its loss of weight, as already explained,
first in the liquid, then in water; divide the former by the
latter, and the quotient will be the specific gravity sought.
The reason is evident.
Third Method. — Let AB and CD represent two
graduated glass tubes of half an inch in
diameter, open at both ends. Let their
upper ends communicate with the receiver
of an air-pump, and their lower ends dip
into two cisterns, one containing distilled
water, and the other the liquid whose
specific gravity is to be determined. Let
the air be partially exhausted from the
receiver by means of an air-pump ; the liquids will rise in
the tubes, but to different heights, these being inversely as
the specific gravities of the liquids. If we divide theheigl t
of the column of water by that of the other liquid, the
quotient will be the specific gravity sought. By creating
different degrees of rarefaction, the columns will rise to
different heights, but their ratios ought to be the same. We
are thus enabled to make a series of observations, each cor-
responding to a different degree of rarefaction, from which
a more accurate result can be had than from a single obser-
vation.
To determine the Specific Gravity of a Soluble Body.
163. Find its specific gravity by the method already
given, with respect to some liquid in which it is not soluble,
and find also the specific gravity of this liquid referred to
water; take the product of these specific gravities, and it
will be the specific gravity sought. For, if the body is m
times heavier than an equivalent volume of the liquid used,
256
MECHANICS.
and this is n times heavier than an equivalent volume of
water, it follows that the body is mn times heavier than its
volume of water, whence the rule.
The auxiliary liquid, in some cases, might be a saturated solu-
tion of the given body in water ;. the rule remains unchanged.
To determine the Specific Gravity of the Air.
164. Take a hollow globe, fitted with a stop-cock, to
shut off communication with the external air, and, by means
of the air-pump or condensing syringe, pump in as much air
as is convenient, close the stop-cock, and weigh the globe
thus filled. Provide a glass tube, graduated so as to show
cubic inches and decimals of a cubic
inch, and, having tilled it with mer-
cury, invert it over a mercury bath.
Open the stopcock, and allow the com-
pressed air to escape into the inverted
tube, taking care to bring the tube
into such a position that the mercury
without the tube is at the same level
as within. The reading on the tube
will give the volume of the escaped air. Weigh the globe
again, and subtract the weight thus found from the first
weight ; this difference will indicate the weight of the
escaped air. Having reduced the measured volume of air
to what it would have occupied at a standard temperature
and barometric pressure, by means of rules yet to be
deduced, compute the weight of an equivalent volume of
water; divide the weight of the corrected volume of air by
that of an equivalent volume of distilled water, and the
quotient will be the specific gravity sought.
To determine the Specific Gravity of a Gas.
IG5. Take a glass globe of suitable dimensions, fitted
with a stop-cook for shutting off communication with the
atmosphere. Fill the globe with air, and determine the
weight of the globe thus filled referred to a vacuum, as
already explained. From the known volume of the globe
Fig. 146.
MECHANICS OF LIQUIDS. 257
and the specific gravity of air, the weight of the contained
air can be computed ; subtract this from the previous
weight, and we shall have the true weight of the globe
alone; determine in succession the weights of the globe
filled with water and with the gas in vacuum, and from each
subtract the weight of the globe ; divide the latter result by
the former; the quotient will be the specific gravity required.
Hydrometers.
166. A hydrometer is a floating body, used for the pur-
pose of determining specific gravities. Its construction de-
pends upon the principle of floatation. Hydrometers are
of two kinds. 1. Those in which the submerged volume is
constant. 2. Those in which the weight of the instrument
remains constant.
Nicholson's Hydrometer.
167. This instrument consists of a hollow brass cylinder
A, at the lower extremity of which is fastened
a basket B, and at the upper extremity a wire,
bearing a scale-pan C. At the bottom of the
basket is a ball of glass E, containing mer-
cury, the object of which is, to cause the in-
strument to float in an upright position. By
means of this ballast, the instrument is ad-
justed so that a weight of 500 grains, placed
in the pan C\ will sink it in distilled water to
a notch 2>, filed in the neck.
To determine the specific gravity of a solid Fig. 147.
which weighs less than 500 grains. Place the
body in the pan C, and add weights till the instrument
sinks, in distilled water, to the notch J). The added
weights, substracted from 500 grains, will give the weight
of the body in air. Place the body in the basket JB, which
generally has a reticulated cover, to prevent the body from
floating away, and add other weights to the pan, until the
instrument again sinks to the notch D. The weights last
added give the weight of the water displaced by the body.
258 MECHANICS.
Divide the first of these -weights by the second, and tne
quotient will be the specific gravity required.
To find the specific gravity of a liquid. Having carefully
weighed the instrument, place it in the liquid, and add
weights to the scale-pan till it sinks to D. The weight of
the instrument, plus the sum of the weights added, will be
the weight of the liquid displaced by the instrument. Next,
place the instrument in distilled water, and add weights till
it sinks to I). The weight of the instrument, plus the added
weights, gives the weight of the displaced water. Divide
the first result by the second, and the quotient will be the
specific gravity required. The reason for this rule is evident.
A modification of this instrument, in which the basket B,
is omitted, is sometimes constructed for determining specific
gravities of liquids only. This kind of hydrometer is
generally made of glass, that it may not be acted upon
chemically, by the liquids into which it is plunged. The
hydrometer just described, is generally known as Fahren
heiVs hydrometer, or Fahrenheit's areometer.
Scale Areometer.
168. The scale areometer is a hydrometer whose weight
remains constant ; the specific gravity of a liquid is made
known by the depth to which it sinks in it. The
instrument consists of a hollow glass cylinder A, n
with a stem (7, of uniform diameter. At the
bottom of the cylinder is a bulb B, containing
mercury, to make the instrument float upright.
By introducing a suitable quantity of mercury,
the instrument may be adjusted so as to float at
any desired point of the stem. When it is de-
signed to determine the specific gravities of liquids,
both heavier and lighter than water, it is bal- B o
lasted so that in distilled water, it will sink to the n.. 14-.
middle of the stem. This point is marked on the
stem with a file, and since the specific gravity of water is 1,
it is numbered 1 on the scale. A liquid is then formed by
dissolving common salt in water whose specific gravity is
MECHANICS OF LIQUIDS. 259
1.1, and the instrument is allowed to float freely in it; the
point E, to which it then sinks, is marked on the stem, and
the intermediate part of the scale, HE, is divided into 10
equal parts, and the graduation continued above and below
throughout the stem. The scale thus constructed is marked
on a piece of paper placed within the hollow stem. To use this
hydrometer, we have simply to put it into the liquid and
allow it to come to rest; the division of the scale which cor-
responds to the surface of floatation, makes known the spe-
cific gravity of the liquid. The hypothesis on which this
instrument is graduated, is, that the increments of specific
gravity are proportional to the increments of the submerged
portion of the stem. This hypothesis is only approximately
true, but it approaches more nearly to the truth as the dia-
meter of the stem diminishes.
When it is only desired to use the instrument for liquids
heavier than water, the instrument is ballasted so that the
division 1 shall come near the top of the stem. If it is to
be used for liquids lighter than water, it is ballasted so that
the division 1 shall fall near the bottom of the stem. In
this case we determine the point 0.9 by using a mixture of
alcohol and water, the principle of graduation being the same
as in the first instance.
Volumeter.
169. The volumeter is a modification of the scale areo-
meter, differing from it only in the method of graduation.
The graduation is effected as follows : The instru-
ment is placed in distilled water, and allowed to
come to a state of rest, and the point on the stem
where the surface cuts it, is marked with a file.
The submerged volume is then accurately deter-
mined, and the stem is graduated in such a man-
ner that each division indicates a volume equal to
a hundredth part of the volume originally sub-
merged. The divisions are then numbered from
the first mark in both directions, as indicated in
the figure. To use the instrument, place it in the * lg ' 149 '
iquid, and note the division to which it sinks ;
C
A
B V
260 MECHANICS.
divide 100 by the number indicated, and the quotient will
be the specific gravity sought. The principle employed is,
that the specific gravities of liquids are inversely as the vol-
umes of equal weights. Suppose that the instrument indi-
cates x parts ; then the weight of the instrument displaces
x parts of the liquid, whilst it displaces 100 parts of
water. Denoting the specific gravity of the liquid by S, and
that of water by 1, we have,
S: 1 ::100 : as, .-. S = — •
x
A table may be computed to save the necessity of per
forming the division.
Densimeter.
170. The densimeter is a modification of the volum-
eter, and admits of use when only a small portion of the
liquid can be had, as is often the case in examining
animal secretions, such as bile, chyle, &c. The S
construction of the densimeter differs from that of
the volumeter, last described, in having a small
cup at the upper extremity of the stem, destined
to receive the fluid whose specific gravity is to be
determined.
The instrument is ballasted so that when the cup
is empty, the densimeter will sink in distilled water
to a point J5, near the bottom of the stem. This
point is the of the instrument. The cup is then
filled with distilled water, and the point C, to Fig. 150
which it sinks, is marked; the space BC, is divi-
ded into any number of equal parts, say 10, and the grad-
uation is continued to the top of the tube
To use the instrument, place if in distilled water, and fill
the cup with the liquid in question, and note the division to
which it sinks. Divide 10 by the number of this division,
and the quotient will be the specific gravity requ'red. The
principle of the densimeter is the same as that of the volu-
meter.
MECHANICS OF LIQUIDS. 261
Centesimal Alcoholometer of Gay Lussac.
171. This instrument is the same in construction as the
scale areometer ; the graduation is, however, made on a diff-
erent principle. Its object is, to determine the percentage <>t
alcohol in a mixture of alcohol and water. The graduation is
made as follows : the instrument is first placed in absolute
alcohol, and ballasted so that it will sink nearly to the top
of the stem. This point is marked 1 00. Next, a mixture
of 95 parts of alcohol and 5 of water, is made, and the point
to which the instrument sinks, is marked 95. The inter-
mediate space is divided into 5 equal parts. Next, a mix-
ture of 90 parts of alcohol and 10 of water is made; the
point to which the instrument sinks, is marked 90, and the
space between this and 95, is divided into 5 equal parts. In
this manner, the entire stem is graduated by successive
operations. The spaces on the scale are not equal at differ-
ent points, but, for a space of five parts, they may be re-
garded as equal, without sensible error.
To use the instrument, place it in the mixture of alcohol
and water, and read the division to which it sinks ; this will
indicate the percentage of alcohol in the mixture.
In all of the instruments, the temperature has to be taken
into account ; this is usually effected by means of correc-
tions, which are tabulated to accompany the different
instruments.
On the principle of the alcoholometer, are constructed a
great variety of areometers, for the purpose of determining
the degrees of saturation of wines, syrups, and other liquids
employed in the arts.
In some nicely constructed hydrometers, the mercury
used as ballast serves also to fill the bulb of a delicate ther-
mometer, whose stem rises into the cylinder of the instru-
ment, and thus enables us to note the temperature of the
fluid in which it is immersed.
EXAMPLES.
1. A cubic foot of water weighs 1000 ounces. Required
262 MECHANICS.
the weight of a cubical block of stone, one of whose edges
is 4 feet, its specific gravity being 2.5. Ans. 10000 lbs.
2. Required the number of cubic feet in a body whose
weight is 1000 lbs., its specific gravity being 1.25.
Ans. 12.8.
3. Two lumps of metal weigh respectively 3 lbs., and 1 lb.,
and their specific gravities are 5 and 9. What will be the
specific gravity of an alloy formed by melting them together,
supposing no contraction of volume to take place.
Ans. 5.625.
4. A body weighing 20 grains has a specific gravity of 2.5.
Required its loss of weight in water. Ans. 8 grains.
5. A body weighs 25 grains in water, and 40 grains in a
liquid whose specific gravity is .7. What is the weight of
the body in vacuum ? Ans. 75 grains.
6. A Nicholson's hydrometer weighs 250 grains, and it
requires an additional weight of 336 grains to sink it to the
notch in the stem, in a mixture of alcohol and water. What
is the specific gravity of the mixture? Ans. .781.
7. A block of wood is found to sink in distilled water till
•£ of its volume is submerged. What is its specific gravity ?
Ans. .875.
8. The weight of a piece of cork in air, is f oz. ; the
weight of a piece of lead in water, is 6| oz. ; the weight of
the cork and lead together in Mater, is 4 t J-q oz. What is
the specific gravity of the cork ? Ans, 0.24.
9. A solid, whose weight is 250 grains, weighs in water,
147 grains, and, in another fluid, 120 grains. What is the
specific gravity of the latter fluid ? Ans. 1.26°.
10. A solid weighs 60 grains in air, 40 in water, and 30 in
an acid. What is the specific gravity of the acid ?
Ans. 1.5.
MECHANICS OF LIQUIDS.
:63
The following table of the specific gravity of some of the
most important solid and fluid bodies, is compiled from a
table given in the Ordnance Manual.
TABLE OF SPECIFIC GRAVITIES OF SOLIDS AND LIQUIDS.
SOLIDS.
SPEC. GKAV.
80LIDS.
SPEC. 6EAV.
Antimony, cast
6.712
8.396
8 788
19.361
7.788
7.207
11.352
13.598
13.580
22.069
20.337
10.511
7.291
6.861
1.900
2.784
1.270
3.521
1 500
2.168
1.822
Limestone
Marble, common ....
Salt, common
Sand
Slate
Stone, common
Tallow
Boxwood
3.180
2.686
Copper, cast
Gold, hammered
Iron, bar
Iron, cast
2.130
1.800
2.672
2.520
. 945
Mercury at 32° F
" at 60°
Platina, rolled
" hammered. . .
Silver, hammered. . . .
0.912
Cedar
Cherry
Lignum vitae
Mahoganv
. 596
0.715
1.333
<>.8o4
Tin, cast
1.170
Zinc, cast
Bricks
Chalk
Coal, bitumiuous
Diamond
Earth, common
Gvpsum
Pine, yellow
Nitric acid
Sulphuric acid
Alcohol, absolute... .
Ether, sulphuric ....
Sea water
Olive oil
0.660
1.217
1.841
0.792
0.715
1.026
0.915
Ivory
Oil of Turpentine . . .
0.870
Thermometer.
172. A thermometer is an instrument used for measur-
ing the temperatures of bodies. It is found, by observation,
that almost all bodies expand when heated, and contract
when cooled, so that, other things being equal, they always
occupy the same volumes at the same temperatures. It is
also found that different bodies expand and contract in a
different ratio for the same increments of temperature. As
a general rule, liquids expand much more rapidly than solids,
and gases much more rapidly than liquids. The construc-
tion of the thermometer depends upon this principle of
unequal expansibility of different bodies. A great variety
of combinations have been used in the construction of ther-
: as
J
Fig. 151.
264: MECHANICS.
mometers, only one of which, the common mercurial ther
mometer, will be described.
The mercurial thermometer consists of a cylindrical or
spherical bulb A, at the upper extremity of which,
is a narrow tube of uniform bore, hermetically
sealed at its upper end. The bulb and tube are
nearly filled with mercury, and the whole is
attached to a frame, on which is a scale for deter-
mining the temperature, which is indicated by the
rise and fall of the mercury in the tube.
The tube should be of uniform bore through-
out, and, when this is the case, it is found that
the relative expansion of the mercury and glass
is very nearly uniform for constant increments of
temperature. A thermometer maybe constructed
and graduated as follows : A tube of uniform
Lore is selected, and upon one extremity a bulb is
blown, which may be cylindrical or spherical ; the former
shape is, on many accounts, the preferable one. At the
other extremity, a conical-shaped funnel is blown open at
the top. The funnel is filled with mercury, which should be
of the purest quality, and the whole being held vertical, the
heat of a spirit-lamp is applied to the bulb, which expand-
ing the air contained in it, forces a portion in bubbles up
through the mercury in the funnel. The instrument is next
allowed to cool, when a portion of mercury is forced down
the capillary tube into the bulb. By a repetition of this
process, the entire bulb may be filled with mercury, as well
as the tube itself. Heat is then applied to the bulb, until
the mercury is made to boil ; and, on being cooled down to
a little above the highest temperature which it is desired to
measure, the top of the tube is melted off by means of a
jet of flame, urged by a blow-pipe, and the whole is her-
metically sealed. The instrument, thus prepared, is attached
to a frame, and graduated as follows:
The instrument is plunged into a bath of melting ice,
and, after being allowed to remain a sufficient time for the
MECHANICS OF LIQUIDS. 2G5
parts of the instrument to take the uniform temperature of
the melting ice, the height of the mercury in the tube is
marked on the scale. This gives the freezing point of the
scale. The instrument is next plunged into a bath of boiling
water, and allowed to remain long enough for all of the parts
to acquire the temperature of the water and steam. The
height of the mercury is then marked on the scale. This
gives the boiling point of the scale. The freezing and
boiling points having been determined, the intermediate
space is divided into a certain number of equal parts,
according to the scale adopted, and the graduation is then
continued, both upwards and downwards, to any desired
extent. Three principal scales are used. Fahrenheit's
scale, in which the space between the freezing and boiling
point is divided into ISO equal parts, called degrees, the
freezing point being marked 32°, and the boiling point 212°.
In this scale, the point is 32 degrees below the freezing
point. The Centigrade scale, in which the space between
the fixed points is divided into 100 equal parts, called
degrees. The of this scale is at the freezing point.
Reaumur's scale, in which the same space is divided into
80 equal parts, called degrees. The of this scale also is
at the freezing point.
If we denote the number of degrees on the Fahrenheit,
Centigrade, and Reaumur scales, by F, C, and R respec-
tively, the following formula will enable us to j:>ass from
any one of these scales to any other :
±{F° -32) = \C° = Ji2°.
The scale most in use in this country is Fahrenheit's
The other two are much used in Europe, particularly the
Centigrade scale.
Velocity of a liquid flowing through a small orifice.
173. Let ABD represent a vessel, having a very small
orifice at its bottom, and filled with any liquid.
12
266
MECHANICS.
Denote the area of the orifice, by a, and its
depth below the upper surface, by h. Let D
represent an infinitely small layer of the liquid
situated nt the orifice, and denote its height,
by h '. This layer is (Art. 155) urged down-
wards by a force equal to the weight of a
column of the liquid whose base is equal to the orifice, and
whose height is h ; denoting this pressure, by p, and the
weight of a unit of volume of the liquid, by 10, we shall
have,
p = wah.
If the element is pressed downwards by its own weight
alone, this pressure being denoted by^', we have,
p — wah!.
Dividing the former equation by the latter, member by
member, we have,
p h
p' ~ A ,;
that is, the pressures are to each other as the heights h
and h'.
AVere the element to fall through the small height h\
under the action of the pressure^', or its own weight, the
velocity generated would (Art. 115) be given by the
equation,
v' -.= y2gh'.
Denoting the velocity actually generated whilst the ele-
ment is falling throught the height h\ by r, and recol-
lecting that the velocities generated in falling through a
MECHANICS OF LIQUIDS. 267
given height, are to each other as the square roots of the
pressures, we shall have,
v : v' : : y ' p : V^/, .'. v = v' \J ~, •
Substituting for v' its value, just deduced, and for — , its
h p
value, — , we have
(150.)
Hence, we conclude that a liquid loill issue from a very
small orifice at the bottom of the containing vessel, with a
velocity equal to that acquired by a heavy body in falling
freely through a height equal to the depth of the orifice
below the surface of the fluid.
We have seen that the pressure due to the weight of a
fluid upon any point of the surface of the containing vessel,
is normal to the surface, and is always proportional to the
depth of the point below the level of the free surface.
Hence, if the side of a vessel be thin, so as not to affect the
flow of the liquid, and an orifice be made at any point, the
liquid will flow out in a jet, normal to the surface at the
opening, and with a velocity due to a height equal to that
of the orifice from the free surface of the fluid.
If the orifice is on the vertical side of a vessel, the initial
direction of the jet will be horizontal ; if it be made at a
point where the tangent plane is oblique to the horizon, the
initial direction of the jet will be oblique ; if the opening is
made on the upper side of a por-
tion of a vessel where the tangent
is horizontal, the jet will be
directed upwards, and will rise
to a height due to the velocity ;
that is, to the height of the Fig. 158.
upper surface of the fluid. This
T
"""%B
D f-
<' Y"
-^0
268 MECHANIC?.
can be illustrated experimentally, by introducing a tube near
the bottom of a vessel of water, and bending its outer
extremity upwards, when the fluid will be observed to rise
to the level of the upper surface of the water in the vessel.
Spouting of Liquids on a Horizontal Plane.
174. Let KL represent a vessel lilled with water. Let
D represent an orifice in its ver-
tical side, and BE the path
described by the spouting fluid.
We may regard each drop of
water as it issues from the orifice,
as a projectile shot forth hori- fcSl-— —^'jl 7
zontally, and then acted upon by ^^ ■ '''' ~^~iy
the force of gravity. Its path « ^
will, therefore, be a parabola,
and the circumstances of its motion will be made known by
a discussion of Equations (115) and (120).
Denote the distance BK, by A', and the distance BL, by
h. TVe have, from Equation (120), by making y equal to
h\ and x = KE,
■*„
KE =
9
But we have found that v = V 2gh ; hence, by substitu-
tion, we have,
KE = 2V M 7 -
If we describe a semicircle on EL, as a diameter, and
through D draw an ordinate BIT, we shall have, from a
well-known property of the circle,
BII = y/BK.BL =
Hence we have, by substitution,
KE = 2BH.
MECHANICS OF LIQUIDS. 269
Since there are two points on KL at which the ordinates
are equal, it follows that there are two oririces through
which the fluid will spout to the same distance on the
horizontal plane ; one of these will be as far above the
centre 0, as the other is below it.
If the orifice be at 0, midway between K and Z, the
ordinate OS will be the greatest possible, and the range
KE' will be a maximum. The range in this case will be
equal to the diameter of the circle LHK, or to the
distance from the level of the water in the vessel to the
horizontal plane.
If a semi-parabola EE'be described, having its axis ver-
tical, its vertex at X, and focus at iT, then may every point
P, within the curve, be reached by two separate jets issuing
from the side of the vessel ; every point on the curve can be
reached by one, and only one ; whilst points lying without
the curve cannot be reached by any jet whatever.
If the jet is directed obliquely upwards by a short pipe
A (Fig. 153), the path described by each particle will still be
the arc of a parabola ABC. Since each particle of the
liquid may be regarded as a body projected obliquely up-
ward, the nature of the path and the circumstances of the
motion will be given by Equation ( 115 ).
In like manner, a discussion of the same equation will
make known the nature of the path and the circumstances
of motion, when the jet is directed obliquely downwards by
means of a short tube.
Modifications due to extraneous pressure.
175. If we suppose the upper surface of the liquid, in
any of the preceding cases, to be pressed by any force, as
when it is urged downwards by a piston, we may denote the
height of a column of fluid whose weight is equal to the ex-
traneous pressure, by h '. The velocity of efflux will then be
given by the equation,
v = V2ff(A + h')<
270 MECHANICS.
The pressure of the atmosphere acts equally on the upper
surface and the surface of the opening ; hence, in ordinary
cases, it may be neglected ; but were the water to flow into
a vacuum, or into rarefied air, the pressure must be taken
into account, and this may be done by means of the formula
just given.
Should the flow take place into condensed air, or into any
medium which opposes a greater resistance than the atmos-
pheric pressure, the extraneous pressure would act upwards,
ti would be negative, and the preceding formula would
become,
v = \^2g(h — A'),
Coefficients of Efflux and Velocity.
176. When a vessel empties itself through a small orifice
at its bottom, it is observed that the particles of fluid near
the top descend in vertical lines; when they approach the
bottom they incline towards the orifice, the converging lines
of fluid particles tending to cross each other as they emerge
from the vessel. The result is, that the stream grows nar-
rower, after leaving the vessel, until it reaches a point at a
distance from the vessel equal to about the radius of the
orifice, when the contraction becomes a minimum, and below
that point the vein again spreads out. This phenomenon is
called the contraction of the vein. The cross section at the
most contracted part of the vein, is not far from T \\ of the
area of the orifice, when the vessel is very thin. If we de-
note the area of the orifice, by a, and the area of the least
cross section of the vein, by a', we shall have,
a' = ka,
in which k is a number 10 be determined by experiment.
This number is called the coefficient of contraction.
To find the quantity of water discharged through an ori-
fice at the bottom of the containing vessel, in a second, we
have only to multiply the area of the smallest cross section
MECHANICS OF LIQUIDS. 271
of the vein, by the velocity. Denoting the quantity dis-
charged in one second, by Q\ we shall have,
Q' = ha \/2(/h.
This formula is only true on the supposition that the
actual velocity is equal to the theoretical velocity, which is
not the case, as has been shown by experiment. The theo-
retical velocity has been shown to be equal to y^A, and
if we denote the actual velocity, by v\ we shall have,
in which I is to be determined by experiment ; this value of
I is slightly less than 1, and is called the coefficient of veloc-
ity. In order to get the actual discharge, we must replace
■\/2gh by l\^2(/h, in the preceding equation. Doing so,
and denoting the actual discharge per second, by Q, Ave have.
The product hi, is called the coefficient of efflux. It has
been shown by experiment, that this coefficient for orifices
in thin plates, is not quite constant. It decreases slightly, as
the area of the orifice and the velocity are increased ; and
it is further found to be greater for circular orifices than for
those of any other shape.
If we denote the coefficient of efflux, by ?n, we have,
In this equation, h is called the head of water. Hence,
we may define the head of water to be the distance from
the orifice to the piano, of the upper surface of the fluid.
The mean value of m corresponding to orifices of from
| to 6 inches in diameter, with from 4 to 20 feet head of
272
MKCIIANICS.
water, has been found to be about .615. If we take the
value of k = .64, we shall have,
m .615
1 = k = MO = ^
That is, the actual velocity is only T 9 y 6 y of the theoretical
velocity. This diminution is due to friction, viscosity,
the atmosphere is called the barometric column,
because it is generally measured by an instrument called a
barometer. In fact, the instrument just described, when
MECHANICS OF GASES AND VAPORS. 287
provided with a suit able scale for measuring the altitude of
the column, is 1 complete barometer. The height of the
barometric column fluctuates somewhat, even at the same
place, on account of changes of temperature, and other
causes yet to be considered.
Observation has shown, that the average height of the
barometric column at the level of the sea, is a trifle less than
30 inches.
The weight of a column of mercury 30 inches in height,
having a cross section of one square inch, is nearly 15
pounds. Hence, the unit of atmospheric pressure at the
level of the sea, is 15 pounds.
This unit is called an atmosphere, and is often employed
in estimating the pressure of elastic fluids, particularly in
the case of steam. Hence, to say that the pressure of steam
in a boiler is two atmospheres, is equivalent to saying, that
there is a pressure of 30 pounds upon each square inch of
the interior of the boiler. In general, when we say that the
tension of a gas or vapor is n atmospheres, we mean that
each square inch is pressed by a force of n times 15 pounds.
Mariotte's Law.
189. When a given mass of any gas or vapor is com-
pressed so as to occupy a smaller space, other things being-
equal, its elastic force is increased ; on the contrary, if its
volume is increased, its elastic force is diminished.
The law of increase and diminution of elastic force, first
discovered by Mariotte, and bearing his name, may be
enunciated as follows :
The elastic force of a given mass of any gas, ichose tem-
perature 'remains the same, varies inversely as the volume
which it occupies.
, As long as the mass remains the same, the density must
vary inversely as the volume occupied. Hence, from Mari-
otte's Law, it follows, that,
The elastic force of any gas, whose temperature remains
the same, varies as its density, and conversely, the density
varies as the elastic force.
ITTd
p P
K li
■ c
2S8 MECHANICS.
Mariotte's law may be verified in the case of atmosplierio
air, by the aid of an instrument called Mamotte's Tube.
This instrument consists of a tube AH CD, of uniform bore*
bent so that its two branches are parallel to each
other. The shorter branch AP, is closed at its
upper extremity, whilst the longer one remains
open for the reception of mercury. Between the
two branches of the tube, and attached to the
same frame with it, is a scale of equal parts for
measuring distances.
To use the instrument, place it in a vertical
position, and pour mercury into the tube, until it
just cuts off the communication between the two
branches The mercury will then stand at the
same level PC, in both branches, and the tension
of the confined air in AB, will be exactly equal to that of
the external atmosphere. If an additional quantity of mer-
cury be poured into the longer branch, the confined air in
the shorter branch will be compressed, and the mercury
will rise in both branches, but higher in the longer, than in
the shorter one. Suppose the mercury to have risen in the
shorter branch, to K, and in the longer one, to P. There
will be an equilibrium in the mercury lying below the hori-
zontal plane KK; there will also be an equilibrium between
the tension of the air in AK, and the forces which give rise
to that tension. These forces are the pressure of the exter-
nal atmosphere transmitted through the mercury, and the
weight of a column of mercury whose base is the cross-sec-
tion of the tube, and whose altitude is PK. If we denote
the height of the column of mercury which will be sustained
by the pressure of the external atmosphere, by h, the ten-
sion of the air in AK, will be measured by the weight of a
column of mercury, whose base is the cross-section of the*
tube, an3 whose height is li + PK. Since the weight is
proportional to the height, the tension of the confined air
will be proportional to h -f- PK.
Now, whatever may be the value of PK, it is found that,
MECHANICS OF GASES AND VAPORS. 289
AB . h
AK =
h + PK
If PK = h, we shall have, AK '= 1.4J5; if P/iT= 2A,
we shall have, ^4A" = %AB ; in general, if PK = nA, w
being any positive number, either entire or fractional, we
AB
shall have, AK — • Mariotte's Law was verified
in this manner by Dulong and Arago for all values of w, up
to n — 27. The law may also be verified when the pres-
sure is less than an atmosphere, by means of the following
apparatus. -
AK represents a straight tube of uniform bore, closed at
its upper and open at its lower extremity : CD
is a long cistern of mercury. The tube AK is
either graduated into equal parts, commencing
at A, or it has attached to it a scale of brass or
ivory.
To use the instrument, pour mercury into the
tube till it is nearly full ; place the finger over
the open end, and invert it in the cistern of mer-
cury, and depress it till the mercury stands at
the same level without, as within the tube, and
suppose the surface of the mercury in this case Fig; 164
to cut the tube at B. Then will the tension
of the confined air in AB, be equal to that of the external
atmosphere. If now the tube be raised vertically, the air in
AB will expand, its tension will diminish and the mercury
will fall in the tube, to maintain the equlibrium. Suppose
the level of the mercury in the tube to have reached
the point K In this position of the instrument the tension
of the air in AK, added to the weight of the column of mer-
cury, KE will be equal to the tension of the external air.
Now, it is found, whatever may be the value of KE, that
r,A
L B
h-EK
13
290 MECHANICS.
If EK = iA, we have, AK = 2AB; if EK :, §A, we
have, w4/r = 3^4^; in general, if EK = A, we have,
ATI ft + 1 '
n + 1
aIakiotte's law has been verified in this manner, for all
values of n, up to n — 111.
It is a law of Physics that, when a gas is suddenly com-
pressed, heat is evolved, and when a gas is suddenly ex-
panded, heat is absorbed ; hence, in making the experiment,
care must be taken to have the temperature kept uniform.
Gay Lussac's Law.
190. If, whilst the volume of any gas or vapor remains
the same, its temperature be increased, its tension is in-
creased also. If the pressure remain the same, the volume
of the gas will increase as the temperature is raised. The
law of increase and diminution, as deduced by Gay Lussac,
whose name it bears, may be enunciated as follows :
In a given mass of any gas, or vapor, if the volume
remains the same, the tension varies as the temperature / if
the tension remains the same, the volume varies as the tem-
perature.
According to Regnault, if a given mass of atmospheric
air be heated from 32° Fahrenheit to 212°, the tension, or
pressure remaining constant, its volume will be increased by
the .3G65th part of the volume at 32°. Hence, the increase
of volume for each degree of temperature is the .00204th part
of the volume at 32°. If we denote the volume at 32° by v,
and the volume at the temperature t\ by v', we si all there-
fore have,
v' = v[l + .00204(2'- 32)] . . ( 152.)
Solving with reference to v, we have,
v'
" 1 + .00204(2'- 32) (lo3.)
Formula (153) enables us to compute the volume of any
MECHANICS OF GASES AND VAPOWS.
291
mass of air at 32°, knowing its volume at the temperature
t\ the pressure remaining constant.
To find the volume at the temperature t", we have simply
to substitute t" for t' in (152.) Denoting this volume by
v'\ we have,
V"= v[l + .00204(£" — 32)].
Substituting for v its value from (153), we get,
,1 4- .00204(r— 32)
1 + .00204(£' - 32)
(154.)
This formula enables us to compute the volume of any
mass of air, at a temperature t", when we know its volume
at the temperature t' ; and, since the density varies in-
versely as the volume, we may also, by means of the same
formula, find the density of any mass of air, at the temper-
ature t'\ when we have given its density at the tempera-
ture t\
Manometers.
191. A manometer is an instrument used for measuring
the tension of gases and vapors, and particularly of steam.
Two principle varieties of manometers are used for measur-
ing the tension of steam, the open manometer, and the
dosed manometer.
The open Manometer.
192. The open manometer consists, essentially, of an
open glass tube A J?, terminating below,
nearly at the bottom of a cistern EF.
The cistern is of wrought iron, steam
tight, and filled with mercury. Its dimen-
sions are such, that the upper surface of
the mercury will not be materially lowered,
when a portion of the mercury is forced
up the tube. ED is a tube, by means of
which, steam may be admitted from the
boiler to the surface of the mercury in the
cistern. This tube is sometimes filled with Fig. 165.
lb
292 MECHANICS.
water, through which the pressure of the steam is trans
mitted to the mercury.
To graduate the instrument. All communication with
the boiler is cut off", by closing the stop-cock E, and commu-
nication with the external air is made by opening the stop-
cock I). The point of the tube AB, to which the mercury
rises, is noted, and a distance is laid off", upwards, from this
point, equal to what the barometric column wants of 30
inches, and the point .//"thus determined, is marked 1. This
point will be very near the surface of the mercury in the
cistern. From the point II, distances of 30, 60, 90, &c,
inches are laid off upwards, and the corresponding points
numbered 2, 3, 4, &c. These divisions correspond to
atmospheres, and may be subdivided into tenths and
hundredths.
To use the instrument, the stop-cock D is closed, and a
communication made with the boiler, by opening the stop-
cock E. The height to which the mercury rises in the
tube, will indicate the tension of the steam in the boiler,
which may be read from the scale in terms of atmospheres
and decimals of an atmosphere. If the pressure in pounds
is wished, it may at once be found, by multiplying the
reading of the instrument by 15.
The principal objection to this kind of manometer, is its
want of portability, and the great length of tube required,
when high tensions are to be measured.
The closed Manometer.
193. The general construction of the closed manometer
is the same as that of the open manometer, with the excep-
tion that the tube AB is closed at the top. The air which
is confined in the tube, is then compressed in the same way
as in Makiotte's tube.
To graduate this instrument, We determine the division
II, as before. The remaining divisions are found by apply-
ing Mariotte's law.
Denote the distance in inches, from II to the top of the
MECHANICS OF GASES AND VAPORS. 293
tube, by I; the pressure on the mercury, expressed in
atmospheres, by ?i, and the distance in inches, from II to the
upper surface of the mercury in the tube, by x.
The tension of the air in the tube will be equal to that on
the mercury in the cistern, diminished by the weight of a
column of mercury, whose altitude is x. Hence, in atmos-
pheres, it is
x
The bore of the tube being uniform, the volume occupied
by the compressed air will be proportional to its height.
When the pressure is 1 atmosphere, the height is £; when
x
the pressure is n atmospheres, the height is I — x.
Hence, from Mariotte's law,
X 7 7
1 : n — — : : I — x : I .
30
Whence, by reduction,
x> _ (30rc H- l)x = — 30l(n — 1).
Solving, w T ith respect to cc, w r e have,
SOn + 1 , / ~T, ~ , /30^ + A 2
x —
The upper sign of the radical is not used, as it would give
a value for x, greater than /. Taking the lower sign, and, a?
a particular case, assuming I — 30 in., we have,
x - 15^ +-15 — y/ — 900(n — 1) + {Ion + 15) 2 .
Making n = 2, 3, 4, &c, in succession, we find for x, the
corresponding values, 11.46 in., 17.58 in., 20.92 in., &c.
These distances being set off from II, upwards, and marked
2, 3, 4, , the upper portion of the cistern,
made of glass, that the surface of the mercury
may be seen; Tv, a conical piece of ivory, pro-
jecting from the upper surface of the cistern :
when the surface of the mercury just touches
the point of the ivory, it is -it the of the
scale; CC represents the Lower part of the
cifftern, and is made of leather, or some other Fi 17(|
• p~T; r rk
Q 1[
J)
Fig. 169.
r
A
15
\
a
I
TT
MECHANICS OF GASES AND VAPORS. 297
flexible substance, and firmly attached to the glass part ;
J) is a screw, working through the bottom of the frame, and
against the bottom of the bag (7(7, through the medium of
a plate P. The screw D, serves to bring the surface of the
mercury to the point of the ivory piece E, and also to force
the mercury up to the top of the tube, when it is desired
to transport the barometer from place to place.
To use this barometer, it should be suspended vertically,
and the level of the mercury in the cistern brought to the
point of the ivory piece E, by means of the screw D ; \ a
smart rap with a key upon the frame will detach the mer-
cury from the glass to which it sometimes tends to adhere.
The sliding ring jV, is next run up or down by means of the
screw J/, till its lower edge appears tangent to the upper
surface of the mercury in the tube, and the altitude is read
from the scale. The height of the attached thermometer
should also be noted.
The requirements of a good barometer are, sufficient
width of tube, perfect purity of the mercury, and a scale
with a vernier accurately graduated and adjusted.
The bore of the tube should be as large as practicable, to
diminish the effect of capillary action. On account of the
mutual repulsion between the particles of the glass and mer-
cury, the mercury is depressed in the tube, and this depres-
sion increases as the diameter of the tube diminishes.
In all cases, this depression should be allowed for, and
corrected by means of a table computed for the purpose.
To secure purity of the mercury, it should be carefully
distilled, and after the tube is filled, it should be boiled over
a spirit-lamp, to drive off any bubbles of air that might ad-
here to the walls of the tube.
Uses of the Barometer.
199. The primary object of the barometer is, to meas-
ure the pressure of the atmosphere at any time or place. It
is used by mariners and others, as a weather-glass. It is
also extensively employed for determining the heights of
points on the earth's surface, above the level of the ocean.
13*
298 MECHANICS.
The principle on which it is employed for the latter pur
pose is, that the pressure of the atmosphere at any place
depends upon the weight of a column of air reaching from
the place to the upper limit of the atmosphere. As we as-
cend above the level of the ocean, the weight of the column
diminishes ; consequently, the pressure becomes less, a tact
which is shown by the mercury falling in the tube. We
shall investigate a formula for determining the difference of
level between any two points.
Difference of Level.
200. Let aB represent a portion of a vertical prism of
air, whose cross-section is one square inch. De-
note the pressure on the lower base .2?, by p, and
on the upper base aa\ by p' ; denote the density
of the air at JB y by <7, and at aa\ by d\ and sup-
pose the temperature throughout the column to
be 32° Fah.
Pass a horizontal plane bb\ infinitely near to
(/'/', and denote the weight of the elementary F1 m
volume of air ob, by to. Conceive the entire
column to be divided by horizontal planes into elementary
prisms, such that the weights of each shall be equal to w,
and denote their heights, beginning at «, by s, s\ s", &c.
From Makiotte's law, we shall have,
y _ f = z (V _ ) ;
or finally,
| = l(p> + w) - lp\
in which I denotes the Napierian logarithm.
In this equation, p' denotes the pressure on the prism ab ;
hence, p' -f uo denotes the pressure on the next prism
below, that is, on the prism be.
w
If we substitute this value of — in Equation (155), we
shall have, for the height of the prism ab,
Substituting in succession for p', the values p'-\- w,p' + 2w>,
p' + 3w, &c, we shall find the heights of the elementary
prisms be, ed, &v.. Wo shall therefore have,
300 MECHANICS.
8 = !k [l{pf+ ^-^
* n '= J^P(*'+ nw? ) - ? 0p' + ( ;i - WJ-
If w denote the number of elementary prisms in AB, the
sum of the first members will be equal to AB. Adding the
equations member to member, and denoting the sum of the
first members by 2, we have,
Because nw denotes the weight of the column of nil A 2?,
we shall have, p' + nw = p, hence,
«= %-l£ (150.)
dg p'
Denoting ihe modulus of the common system of loga-
rithms by 31, and designating common logarithms by the
symbol log, we shall have,
Mz = §- log ^ , or z = J?— log ^ •
dg G y Mdg & p'
Now, the pressures jo and// are measured by the heights of
the columns of mercury which they will support; denoting
these heights by JET and II, we have,
p _ II
p ' ~ II' '
MECHANICS OF GASES AND VAPORS. 301
whence, by substitution,
z = ik los ^ r • * ' (157,)
We have supposed the temperature, both of the air and
mercury, to be 32°. In order to make the preceding for-
mula general, let T represent the temperature of the mer-
cury at B, T\ its temperature at a T and denote the cor-
responding heights of the barometric column by h and h f ;
also, let t denote the temperature of the air at B, and t' its
temperature at a.
t)
The quantity is the ratio of the density of the air at B,
to the corresponding pressure, the temperature being 32°.
According to Mariotte's law, this ratio remains constant,
whatever may be the altitude of B aK*\ o the level of the
ocean.
If we denote the latitude of the ph^o by £, we have,
(Art. 124),
g - g'{\ — 0.002695 cos2 7, l
It has been shown, by experiment, that, vl^n a column
of mercury is heated, it increases in length at thp. iV-e of
t9 ' 90 ths of its length at 32°, for each degree th^t the + ^)m-
perature is elevated. Hence,
*(> + ^?)-*
K
T' -32\ „., 9990 + T'-S2
V T 9990 /
9990
Dividing the second equation by the first, member by
member,
Ji_ II 9990 -f T— 32
h' ~ II'' 9990 + T— 32 *
302 MECHANICS.
TT
Dividing both terms of the fractional coefficient of ■= by
the denominator, and neglecting the quantity T — 32, in
comparison with 9990, we have,
* = 5£ + ^) - a* +-™> <*•-**
Whence, by reduction,
H_ _ h_ 1
H' " A'*l + .0001 (T- T') '
The quantity z denotes, not only the height, but also the
volume of the column of air aB, at 32°. When the tem-
perature is changed from 32°, the pressures remaining the
same, this volume will vary, according to the law of Gay
Lussac.
If we suppose the temperature of the entire column to be
a mean between the temperatures at B and cz, which we
may do without sensible error, the height of the column
will become, Equation (153),
S [~1 + .00204 ^y~ - 32 Yl = z[l+.00102(* + «'- 64)]
Hence, to adapt Equation (157) to the conditions pro-
posed, we must multiply the value of 2 by the factor,
1 + .00102(0 + t'- 64).
Substituting in Equation (157), for — and , we shall have,
Q-Aphxl(P) = Aphxl(^\ . (160.)
If we denote by c the number of cubic feet of gas, when the
pressure is p, and suppose it to expand till the pressure is y,
we shall have, Ah = c ; or, if A be expressed in square
Ah
feet, we shall have, c = Hence, by substitution,
'144 ' J
Finally, if we suppose the pressure at the high< st point to
he p', we shall have,
0. = a«qpX/f(Jp),
306 MECHANICS.
an equation which gives the quantity of work ot c cubic
feet of gas, whilst expanding from a pressure p, to a pres-
sure })'.
Efflux of a Gas or Vapor.
202. Suppose the gas to escape from a small orifice, and
denote its velocity by v. Denote the weight of a cubic
foot of the gas, by w, and the number of cubic feet dis-
charged in one second, by c, then will the mass escaping in
cw
one second, be equal to — , and its living force will be
cw
equal to — v 2 . But, from Art. 148, the living force is
double the accumulated quantity of work. If, therefore, we
denote the accumulated work by §, we shall have,
r\ cw 2
Q = —v.
* 2g
But the accumulated work is due to the expansion of the
gas, and if we denote the pressure within the orifice, by p y
and without, by^', we shall have, from Art. 201,
q = lucp x i(^y
Equating the second members, we have,
™v' = lUcpxl(£)
Whence,
•-»>/¥*<£)
Substituting for g, its value, 32i ft., we have, aftei
reduction,
96
v^¥) • • • < i6i >
MECHANICS OF GASES AND VArURS. 307
When the difference between p and p' is small, the pre-
ceding formula can be simplified.
Since — = 1 + ^ ~ , we have, from the logarithmic
p p
series,
When p —p' is very small, the second, and all succeeding
terms of the development, may be neglected, in comparison
with the first term. Hence,
©
p-p
P
Substituting, in the formula above deduced, we have,
V to p
or, since — is, under the supposition just made, equal to 1,
we have, finally,
v = 96
v 7 ^ ("*>
Coefficient of Efflux.
203. When air issues from an orifice, the section of the
current undergoes a change of form, analagous to the con-
traction of the vein in liquids, and for similar reasons. If
we denote the coefficient of efflux, by ft, the area of the
orifice, by A, and the quantity of air delivered in n seconds,
by Q, we shall have, from Equation (161),
Q = w^vf^O •
MECHANICS.
According to Koch, the value of k is equal to .58, when
the orifice is in a thin plate ; equal to .74, when the air
issues through a tube 6 times as long as it is wide ; and
equal to .85, when it issues through a conical nozzle 5 times
as long as the diameter of the oritice, and whose sides have
a convergence of 6° to the axis.
The preceding principles are applicable to the distribution
of gas, to the construction of blowers, and, in general, to a
great variety of pneumatic machines.
Steam.
204. If water be exposed to the atmosphere, at ordinary
temperatures, a portion is converted into vapor, which mixes
with the atmosphere, constituting one of the permanent
elements of the aerial ocean. The tension of watery vapor
thus formed, is very slight, and the atmosphere soon ceases
to absorb any more. If the temperature of the water be
raised, an additional amount of vapor is evolved, and of
greater tension. When the temperature is raised to that
point at which the tension of the vapor is equal to that of
the atmosphere, ebullition commences, and the vaporization
goes on with great rapidity. If heat be added beyond the
point of ebullition, neither the water nor the vapor will
increase in temperature till all of the water is converted into
steam. When the barometer stands at 30 inches, the boil-
ing point of pure water is 212° Fab. We shall suppose, in
what follows, that the barometer stands at 30 inches. After
the temperature of the water is raised to 212°, the addi-
tional heat that is added becomes latent in the vapor
evolved.
If heat be applied uniformly, it is found by experiment
thai it takes 5i times as much to convert all of the water
into Steam as it requires to raise it from :",2° to 212°. Hence,
the entire amount of heat which becomes latent is
5^ X (212° — 32°) = 990°. That the heat applied becomes
latent, may be shown experimentally as follows :
Let a cubic inch of water be converted into steam at
MECHANICS OF GASES AND VAPORS. 309
212°, and kept in a close vessel. Now, if 5^ cubic inches
of water at 32° be injected into the vessel, the steam will all
be converted into water, and the 6^ cubic inches of water
will be found to have a temperature of 212". The heat
that was latent becomes sensible again.
When water is converted into steam under any other
pressure than that of the atmosphere, or 15 pounds to the
square inch, it is found that, although the boiling point will
be changed, the entire amount of heat required for convert-
ing the water into steam will remain unchanged.
If the evaporation takes place under such a pressure, that
the boiling point is but 150°, the amount of heat which
becomes latent is 1052°, so that the latent heat of the
steam, plus its sensible heat, is 1202°. If the pressure under
which vaporization takes place is such as to raise the boiling
point to 500°, the amount of heat which becomes latent is
702°, the sum 702° + 500° being equal to 1202°, as before.
Hence, we conclude that the same amount of fuel is
required to convert a given amount of water into steam, no
matter what may be the pressure under which the evapora-
tion takes place.
When water is converted into steam under a pressure of
one atmosphere, each cubic inch is expanded into about
1700 cubic inches of steam, of the temperature of 212° ; or,
since a cubic foot contains 1728 cubic inches, we may say,
in round numbers, that a cubic inch of xoater is converted
into a cubic foot of steam.
If water is converted into steam under a greater or less
pressure than one atmosphere, the density will be increased
or diminished, and, consequently, the volume will be dimin-
ished or increased. The temperature being also increased
or diminished, the increase of density or decrease of volume
will not be exactly proportional to the increase of pressure ;
but, for purposes of approximation, we may consider the
densities as directly, and the volumes as inversely propor-
tional to the pressures under which the steam is generated.
Under this hypothesis, if a cubic inch of water be evapo-
310 MECHANICS.
rated under a pressure of a half atmosphere, it will afford
two cubic feet of steam; if generated under a pressure of
two atmospheres, it will only afford a half cubic foot of steam.
Work of Steam.
205. When water is converted into steam, a certain
amount of work is generated, and, from what has been shown,
this amount of work is very nearly the same, whatever may
be the temperature at which the water is evaporated.
Suppose a cylinder, whose cross-section is one square
inch, to contain a cubic inch of water, above which is an air-
tight piston, that may be loaded with weights at pleasure.
In the first place, if the piston is pressed down by a weight
of 15 pounds, and the inch of water converted into steam,
tha weight will be raised to the height of 1728 inches, or
144 feet. Hence, the quantity of work is 144 x 15, or,
2160 units. Again, if the piston be loaded with a weight
of 30 pounds, the conversion of water into steam will give
but 864 cubic inches, and the weight will be raised through
72 feet. In this case, the quantity of work will be 72 x 30,
or 2160 units, as before. We conclude, therefore, that the
quantity of work is the same, or nearly so, whatever may be
the pressure under which the steam is generated. We also
conclude, that the quantity of work is nearly proportional to
the fuel consumed.
Besides the quantity of work developed by simply con-
verting an amount of water into steam, a further quantity
of work is developed by allowing the steam to expand after
entering the cylinder. This principle is made use of in
steam engines working expansively.
To find the quantity of work developed by steam acting ex-
pansively. Let AJ3 represent a cylinder, closed at
A, and having an air-tight piston D. Suppose the
steam to enter at the bottom of the cylinder, and to
push the piston upward to (7, and then suppose
the opening at which the steam enters, to be
closed. If the piston is not too heavily loaded,
the steam will continue to expand, and the piston Fig m
MECHANICS OF GASES AND VAPORS. 311
will be raised to some position, B. The expansive force
of the steam will obey Mariotte's law, and the quantity of
work due to expansion will be given by Equation ( 1G0).
Denote the area of the piston in square inches, by A ; the
pressure of the steam on each square inch, up to the moment
when the communication is cut oft*, by p ; the distance A C,
through which the piston moves before the steam is cut oft*,
by A ; and the distance AD, by nh.
If we denote the pressure on each square inch, when the
piston arrives at J5, by p\ we shall have, by Mariotte's
law,
P
p : p f : : nh : A, . • . p' — — ,
an expression which gives the limiting value of the load of
the piston.
The quantity of work due to expansion being denoted by
2, we shall have, from Equation (160),
q = Aph x I (-7- ) — Aphl (n)*
If we denote the quantity of work of the steam, whilst
the piston is rising to (7, by q", we shall have,
q" = Aph.
Denoting the total quantity of work during the entire stroke
of the piston, by Q, we shall have,
Q = Aph[l -f l(n)] . . . (163.)
Experimental Formulas.
206. Numerous experiments have been made for the
purpose of determining the relation existing between the
elasticity and temperature of steam in contact with the
water by which it is produced, and many formulas, based
312 MECHANICS.
upon these experiments, have been given, two of which arc
subjoined :
The formula of Duloxg and Arago is,
p = (1 + .007153*) 3 ,
in which p represents the tension in atmospheres, and t the
excess of the temperature above 100° Centigrade.
Tredgold's formula is,
t = 0.85 y^ — 75,
in which t is the temperature, in degrees of the Centigrade
thermometer, and p the pressure, expressed in centimeters
of the mercurial column.
HYDRAULIC AND PXIXMATIC MACHINES.
313
CHAPTEE IX.
HYDRAULIC AND PNEUMATIC MACHINE8.
Definitions.
207. Hydraulic machines are those used in raising and
distributing water, such as pumps, siphons, hydraulic rams,
&c. The name is also applied to those machines in which
water power is the motor, or in which water is employed to
transmit pressures, such as water-wheels, hydraulic presses, &c.
Pneumatic machines are those employed to rarefy and
condense air, or to impart motion to the air, such as air-
pumps, ventilating-blowers, , the distance from the
upper surface of the water in the reservoir to the highest
position of the piston, by «, and the height at which the
water ceases to rise in the pump, by x. The distance from
the surface of the water in the pump to the highest position
of the piston will then be equal to a — se, and the distance
to the lowest position of the piston, will be a — p — x.
Denote the height at which the atmospheric pressure will
sustain a column of water in vacuum, by A, and the weight of
a column of water, whose base is the cross-section of the
pump, and whose altitude is 1, by w ; then will wh denote
the pressure of the atmosphere exerted upwards through the
water in the reservoir and pump.
Xow, when the piston is at its lowest position, in order
that it may not thrust open the piston valve and escape, the
pressure of the confined air must be exactly equal to that
of the external atmosphere; that is, equal to ich. When the
316 MECHANICS.
piston is at its highest position, the confined air will be rare-
fied, the volume occupied being proportional to its height.
Denoting the pressure of the rarefied air by toh\ we shall
have from Mariotte's law,
wh : wh' :: a — x : a — p — x.
• wh' = wh
a — x
If the water does not rise when the piston is at its highest
position, the pressure of the rarefied air, plus the weight of
the column already raised, will be equal to the pressure of
the external atmosphere; or
a — p — x
wh + wx = wh.
a — x
Solving this equation with respect to #, we have,
If we have,
_ a ± yet? — 4ph
Aph > a 2 ; or, p > -^ ,
the value of x will be imaginary, and there will be no point
at which the water will cease to rise. Hence, the, above
inequality expresses the relation that must exist, in order
that the pump may be eifective. This condition expressed
in words, gives the following rule :
The pump will be effective, when the play of the piston is
greater tl(>fffft.; or, p>2^it.
To find the quantity of work required to make a double
stroke of the piston, after the water reaches the level of the
spout.
In depressing the piston, no force is required, except that
necessary to overcome the inertia of the parts and the fric-
tion. Neglecting these for the present, the quantity of
work in the downward stroke, may be regarded as 0. In
raising the piston, its upper surface will be pressed down-
wards, by the pressure of the atmosphere w7i, plus the weight
of the column of water from the piston to the spout ; and it
will be pressed upwards, by the pressure of the atmosphere,
transmitted through the pump, minus the weight of a
column of water, whose cross-section is equal to that of the
barrel, and whose altitude is the distance from the piston to
the surface of the water in the reservoir. If we subtract
the latter pressure from the former, the difference will be
the resultant downward pressure. This difference will be
equal to the weight of a column of water, whose base is the
cross-section of the barrel, and whose height is the distance
of the spout above the reservoir. Denoting the height by
If, the pressure will be equal to wH. The path through
which the pressure is exerted during the ascent of the
piston, is equal to the play of the piston, or p. Denoting the
quantity of work required, by Q, we shall have,
Q = wpll.
But wp is the weight of a volume of water, whose base is
the cross-section of the barrel, and whose altitude is the
play of the piston. Hence, the value of Q is equal to the
31S
MECHANICS.
quantity of work necessary to raise this volume of watei
from the level of the water in the reservoir to the spout.
This volume is evidently equal to the volume actually
delivered at each double stroke of the piston. Hence, the
quantity of work expended in pumping with the sucking
and lifting pump, all hurtful resistances being neglected, is
equal to the quantity of work necessary to lift the amount
of water, actually delivered, from the level of the water in
the reservoir to the height of the spout. In addition to this
work, a sufficient amount of power must be exerted, to
overcome the hurtful resistances. The disadvantage of this
pump, is the irregularity with which the force must act,
being in depressing the piston, and a maximum in raising
it. This is an important objection when machinery is em-
ployed in pumping ; but it may be either partially or entirely
overcome, by using two pumps, so arranged, that the piston
of one shall ascend as that of the other descends. Another
objection to the use of this kind of pump, is the irregularity
of flow, the inertia of the column of water having to be
overcome at each upward stroke. This, by creating shocks,
consumes a portion of the force applied.
Sucking and Forcing Pump.
210. This pump consists of a cylindrical barrel A, with
its attached sucking-pipe B, and
sleeping- valve #, as in the pump
just discussed. The piston C is
solid, and is worked up and down
in the barrel by means of a lever
F, attached to the piston-rod D.
At the bottom of the barrel, a
branch-pipe leads into an air-vessel
K, tli rough a second sleeping-valve
J\ which opens upwards, and closes
by its own weight. A delivery-
pipe 7/", enters the air-vessel at its
top, and terminates near its bottom.
To explain the action of this
Fig. 178.
HYDRAULIC AND PNEUMATIC MACHINES. 319
pump, suppose the piston C to be depressed to its lowest
limit. Now, if the piston be raised to its highest position,
the air in the barrel will be rarefied, its tension will be
diminished, the air in the tube j5, will thrust open the valve,
and a portion of it will escape into the barrel. The pres-
sure of the external air will then force a column of water
up the pipe i?, until the tension of the rarefied air, plus the
weight of the column of water raised, is equal to the tension
of the external air. An equilibrium being produced, the
valve G closes by its own weight. If, now, the piston be
again depressed, the air in the barrel will be condensed, its
tension will increase till it becomes greater than that of the
external air, when the valve F will be thrust open, and a
portion of it will escape through the delivery-pipe H. After
a few double strokes of the piston, the water will rise
through the valve G, and then, as the piston descends, it
will be forced into the air-vessel, the air will be condensed
in the upper part of the vessel, and, acting by its elastic
force, will force a portion of the water up the delivery-pipe
and out at the spout P. The object of the air-vessel is, to
keep up a continued stream through the pipe H, otherwise
it would be necessary to overcome the inertia of the entire
column of water in the pipe at every double stroke. The
flow having commenced, at each double stroke, a volume of
water will be delivered from the spout, equal to that of a
cylinder whose base is the area of the piston, and whose
altitude is the play of the piston.
The same relative conditions between the parts should
exist as in the sucking and lifting pump.
To find the quantity of work consumed at each double
stroke, after the flow has become regular, hurtful resistances
being neglected :
When the piston is descending, it is pressed downwards
by the tension of the air on its upper surface, and upwards
by the tension of the atmosphere, transmitted through the
delivery-pipe, plus the weight of a column of water whose
base is the area of the piston, and whose altitude is the
320 MECHANICS.
distance of the spout above the piston. This distance is
variable during the stroke, but its mean vame is the distance
of the middle of the play below the spout. The difference
between these pressures is exerted upwards, and is equal to
the weight of a column of water whose base is the area of
the piston, and whose altitude is the distance from the
middle of the play to the spout. The distance through
which the force is exerted, is equal to the play of the piston.
Denoting the quantity of work during the descending
stroke, by Q' ; the weight of a column of water, having a
base equal to the area of the piston, and a unit in altitude,
by w; and the height of the spout above the middle of the
the play, by A', we shall have,
Q' = wh' x p.
When the piston is ascending, it is pressed downwards
by the tension of the atmosphere on its upper surface, and
upwards by the tension of the atmosphere, transmitted
through the water in the reservoir and pump, minus the
weight of a column of water whose base is the area of the
piston, and whose altitude is the height of the piston above
the reservoir. This height is variable, but its mean value
is the height of the middle of the play above the Mater in
the reservoir. The distance through which this force is
exerted, is equal to the play of the piston. Denoting the
quantity of work during the ascending stroke, by Q'\ and
the height of the middle of the play above the reservoir, by
A", we have,
Q'' = wh" x p.
Denoting the entire quantity of work during a double strok
by Q, we have,
Q = C+ Q" = wp{h> + h").
But irp is the weight of a volume of water, the area of
whose base is that of the piston, and whose altitude is the
HYDRAULIC AND PNEUMATIC MACHINES.
321
play of the piston ; that is, it is the weight of the volume
delivered at the spout at each double stroke.
The quantity A' + A", is the entire height of the spout
above the level of the cistern. Hence, the quantity of work
expended, is equal to that required to raise the entire volume
delivered, from the level of the water in the reservoir to the
height of the spout. To this must be added the work
necessary to overcome the hurtful resistances, such as fric-
tion, &c.
If h' = A", we shall have, Q' = Q" ; that is, the quan-
tity of work during the ascending stroke, will be equal to
that during the descending stroke. Hence, the work of the
motor will be more nearly uniform, when the middle of the
play of the piston is at equal distances from the reservoir
and spout.
Fire Engine.
211. The fire engine is essentially a double sucking and
forcing pump, the two piston rods being so connected, that
when one piston ascends the other descends. The sucking
and delivery pipes are made of some flexible material, gen-
erally of leather, and are attached to the machine by means
of metallic screw joints.
The figure exhibits a cross-section of the essential part of
a Fire Engine.
A A' are the two barrels, C C the two pistons, con-
nected by the rods, D D ,
with the lever, E E '. B
is the sucking pipe, termi-
nating in a box from
which the water may en-
ter either barrel through
the valves, G G'. K is
the air vessel, common to
both pumps, and com-
municating with them by
the valves F F '. II is
the delivery pipe.
14*
Fi«t. 17ft.
322
MECHANICS.
The instrument is mounted on wheels for convenience of
transportation. The lever E E' is worked by means of
rods at right angles to the lever, so arranged that several
men can apply their strength in working the pump. The
action of the pump differs in no respect from that of the
forcing pump; but when the instrument is worked vigor-
ously, there is more water forced into the air vessel, the
tension of the air is very much augmented, and its elastic
force, thus brought into play, propels the water to a consider-
able distance from the mouth of the delivery pipe. It is
this capacity of throwing a jet of water to a great distance,
that gives to the engine its value in extinguishing fires.
A pump entirely similar to the fire engine in its construc-
tion, is often used under the name of the double action forc-
ing pump for raising water for other purposes.
The Rotary Pump.
212. The rotary pump is a modification of the sucking
and forcing pump. Its construction will be best understood
from the drawing, which represents a vertical section through
the axis of the sucking-pipe, and at right angles to axis of
the rotary portion of the pump.
A represents an annular ring of metal, which may be
made to revolve about its axis
0. D D is a second ring of
metal, concentric with the first,
and forming with it an inter-
mediate annular space. This
space communicates with the
sucking-pipe 7f, and the de-
livery pipe Z. Four radial
paddles C\ are disposed so as
to slide backwards and for-
wards through suitable open- Fig. 177.
ings, which are made in the
ring A, and which are moved around with it. G is a solid
guide, firmly fastened to the end of the cylinder enclosing
HYDRAULIC AND PNEUMATIC MACHINES. 323
the rotary apparatus, and cut as represented in the figure.
E E are two springs, attached to the ring X>, and acting by
their elastic force, to press the paddles firmly against the
guide. These springs are of such dimensions as not to
impede the flow of the water from the pipe Ji, and into the
pipe X.
When the axis is made to revolve, each paddle, as it
reaches and passes the partition II, is pressed against the
guide, but, as it moves on, it is forced, by the form of the
guide, against the outer wall D. The paddle then drives
the air in front of it, around, in the direction of the arrow-
head, and finally expels it through the pipe L. The* air
behind the paddle is rarefied, and the pressure of the exter-
nal air forces a column of water up the pipe. As the paddle
approaches the opening to the pipe X, the paddle is pressed
back by the spring E, against the guide, and an outlet into
the ascending pipe X, is thus provided. After a few revo-
lutions, the air is entirely exhausted from the pipe K. The
water enters the channel C C, and is forced up the pipe X,
from which it escapes by a spout at the top. The quantity
of work expended in raising a volume of water to the
spout, by this pump, is equal to that required to* lift it
through the distance from the level of the water in the cis-
tern to the spout. This may be shown in the same manner
as was explained under the head of the sucking and forcing-
pump. To this quantity of work, must be added the work
necessary to overcome the hurtful resistances, as fric-
tion, &c.
This pump is well adapted to machine pumping, the work
being very nearly uniform.
A machine, entirely similar to the rotary pump, might be
constructed for exhausting foul air from mines ; or, by re-
versing the direction of rotation, it might be made to force
a supply of fresh air to the bottom of deep mines.
Besides the pumps already described, a great variety
of others have been invented and used. All, however,
324
MECHANICS.
depend upon some modification of the principles that have
just been discussed.
The Hydrostatic Press.
213. The hydrostatic press is a machine for exerting
great pressure through small spaces. It is much used in
compressing seeds to obtain oil, in packing hay and bales of
goods, also in raising great weights. Its construction, though
requiring the use of a sucking-pump, depends upon the prin-
ciple of equal pressures (Art. 154).
It consists essentially of two vertical cylinders, A and B,
each provided with a solid pis-
ton. The cylinders communi-
cate by means of a pipe (7,
whose entrance to the larger
cylinder is closed by a sleeping
valve E. The smaller cylinder
communicates with the reser-
voir of water 7T, by a sucking-
pipe H, whose upper extremity
is closed bythe sleeping-valve D.
The smaller piston 7>', is worked up and down by the lever
G. By working the lever G, up and down, the water is
raised from the reservoir and forced into the larger cylinder
A] and when the space below the piston F is tilled, a force
of compression is exerted upwards, which is as many times
greater than that applied to the piston B, as the area of
i^is greater than B (Art. L54). This force may be util-
ized in compressing a body L, placed between the piston
and the frame of the press.
Denote the area of the larger piston by P, of the smaller,
by p, the pressure applied to 7>, by/, and that exerted at
F, by F; we shall have,
FiC, of which the outer one
is the longer. To use the instrument, the tube
is filled with the liquid in any manner, the end of
the longer branch being stopped with the finger
or a stop-cock, in which case, the pressure of the
atmosphere will prevent the liquid from escaping Fig. no.
HYDRAULIC AND PNEUMATIC MACHINFS. 327
at the other end. The instrument is then inverted,
the end C being submerged in the liquid, and the stop
removed from A, The liquid will begin to flow through
the tube, and the flow will continue till the level of the
liquid in the reservoir reaches that of the mouth of the
tube C.
To find the velocity with which water will issue from the
siphon, let us consider an infinitely small layer at the orifice
A This layer will be pressed downwards, by the tension
of the atmosphere exerted on the surface of the reservoir,
diminished by the weight of the water in the branch BD,
and increased by the weight of the water in the branch
BA. It will be pressed upwards by the tension of the
atmosjmere acting directly upon the layer. The difference
of these forces, is the weight of the water in the portion of
the tube DA, and the velocity of the stratum will be due
to that weight. Denoting the vertical height of DA, by h,
we shall have, for the velocity (Art. 173),
This is the theoretical velocity, but it is never quite
realized in practice, on account of resistances, which have
been neglected in the preceding investigation.
The siphon may be filled by applying the mouth to the
end A, and exhausting the air by suction. The
tension of the atmosphere, on the upper surface
of the reservoir, will press the water up the tube,
and fill it, after which the flow will go on as
before. Sometimes, a sucking-tube AD, is in-
serted near the opening A, and rising nearly to
the bend of the siphon. In this case, the opening
A, is closed, and the air exhausted through the
sucking-tube AD, after which the flow goes on as before.
The Wurtemburg Siphon.
215. In the Wurtemburg siphon, the ends of the tube are
328
MECHANICS.
GZ\
m
<&)
bent twice, at right-angles, as shown in the figure.
The advantage of this arrangement is, that the
tube, once filled, remains so, as long as the plane
of its axis is kept vertical. The siphon may be
lifted out and replaced at pleasure, thereby
stopping the flow at will.
It is to be observed that the siphon is only effectual when
the distance from the highest point of the tube to the level
of the water in the reservoir is less than the height at which
the atmospheric pressure will sustain a column of water in
a vacuum. This will, in general, be less than 3-4 feet.
Fig. 181.
The Intermitting Siphon.
216. The intermitting siphon is represented in the
figure. AB is a curved tube issuing
from the bottom of a reservoir. The
reservoir is supplied with water by a
tube E, having a smaller bore than
that of the siphon. To explain its
action, suppose the reservoir at first
to be empty, and the tube E to be
opened; as soon as the reservoir is
filled to the level of CD, the water
will begin to flow from the opening
B, and the flow once commenced, will continue till the
level of the reservoir is again reduced to the level CD',
drawn through the opening A. The flow will then cease
till the cistern is again filled to CD, and so on as before.
Fie-. 1S2.
Intermitting Springs.
217. Let A represent a subterranean cavity, communi-
Cating with the surface of the earth by
a channel AI><\ bent like a siphon.
Suppose the reservoir to lie fed by
percolation through the crevices, or
by a small channel D. When the
HYDRAULIC AND PNEUMATIC MACHINES. 329
water in the reservoir rises to the height of the horizontal
plane BD, the flow will commence at C, and, if the chan-
nel is sufficiently large, the flow will continue till the water
is reduced to the level plane drawn through C. An inter-
mission of flow will occur till the reservoir is again tilled,
and so on, intermittingly. This phenomena has been observed
at various places.
Siphon of Constant Flow.
218. We have seen that the velocity of efflux depends
upon the height of the water in the reservoir above the
external opening of the siphon. When the water is drawn
off from the reservoir, the upper surface sinks, this height
diminishes, and, consequently, the velocity continually
diminishes.
If, however, the shorter branch <7Z>, of the tube, be
inserted through a piece of cork large enough to float the
.siphon, the instrument will sink as the upper surface is
depressed, the height of DA will remain the same, and,
consequently, the flow will be uniform till the bend of the
siphon comes in contact with the upper edge of the reservoir.
By suitably adjusting the siphon in the cork, the velocity
of efflux can be increased or decreased within certain limits.
In this manner, any desired quantity of the fluid can be
drawn off" in a given time.
The siphon is used in the arts, for decanting liquids, when
it is desirable not to stir the sediment at the bottom of a
vessel. It is also employed to draw a portion of a liquid
from the interior of a vessel when that liquid is overlaid by
one of less specific gravity.
The Hydraulic Ram.
219. The hydraulic ram is a machine for raising watei
by means of shocks caused by the sudden stoppages of a
stream of water.
The instrument consists of a reservoir .7?, which is sup-
plied with water by an inclined pipe A ; on the upper surface
330
MECHANICS.
Fig. 184
of the reservoir, is an orifice which may be closed by
a spherical valve D\ this valve,
when not pressed against the
opening, rests in a metallic
framework immediately below
the orifice ; G is an air-vessel
communicating with the reser-
voir by an orifice F, which is
fitted with a spherical valve E\
this valve closes the orifice F,
except when forced upwards,
in which case its motion is restrained by a metallic frame
work or cage ; // represents a delivery-pipe entering the
air-vessel at its upper part, and terminating near the bot-
tom. At P is a small valve, opening inwards, to supply
the loss of air in the air-vessel, arising from absorption by
the water in passing through the air vessel.
To explain the action of the instrument, suppose, at first,
that it is empty, and all the parts in equilibrium. If a cur-
rent of water be admitted to the reservoir, through the in-
clined pipe A, the reservoir will soon be filled, and com-
mence rushing out at the orifice C. The impulse of the
water will force the spherical valve Z>, upwards, closing the
opening ; the velocity of the water in the reservoir will be
suddenly checked ; the reaction will force open the valve
F, and a portion of the water will enter the air-chamber G.
The force of the shock having been expended, the spherical
valves will both fall by their own weight ; a second shock
will take place, as before ; an additional quantity of water
will be forced into the air-vessel, and so on, indefinite!}'.
As the water is forced up into the air-vessel, the air becomes
compressed; and acting by its elastic force, it urges a stream
of water up the pipe // The shocks occur in rapid succes-
sion, and, at each shock, a quantity of water is forced into
the air-chamber, and thus a constant stream is kept up.
To explain the use of the valvt ]\ it maybe remarked that
water absorbs more air under a great pressure, than under
HYDRAULIC AND PNEUMATIC MACHINES. 331
a smaller one. Henee, as it passes through the air-chamber,
a portion of the air contained is taken up by the water and
carried out through the pipe H. But each time that the
valve D falls, there is a tendency to produce a vacuum
in the upper part of the reservoir, in consequence of the
rush of the fluid to escape through the opening. The pres-
sure of the external air then forces the valve P open, a
small portion of air enters, and is afterwards forced up with
the water into the vessel (r, to keep up the supply.
The hydraulic ram is only used where it is required to
raise small quantities of water, such as for the supply of a
house, or garden. Only a small fraction of the amount of
fluid which enters the supply-pipe actually passes out
through the delivery-pipe; but, if the head of water is
pretty large, the column may be raised to a great height.
Water is often raised, in this manner, to the highest points
of lofty buildings.
Sometimes, an additional air-vessel is introduced over the
valve E, for the purpose of deadening the shock of the
valve in its play up and down.
Archimedes' Screw.
220. This machine is intended for raising water through
small heights, and consists, in its simplest form, of a tube
wound spirally around a cylinder. This cylinder is mounted
so that its axis is oblique to the horizon, the lower end dip-
ping into the reservoir. When the cylinder is turned on its
axis, by a crank attached to its upper extremity, the lower
end of the tube describes a circumference of a circle, whose
plane is perpendicular to the axis. When the mouth of the
tube comes to the level of the axis and begins to ascend,
there will be a certain quantity of water in the tube, which will
flow so as to occupy the lowest part of the spire; and, if the
cylinder is properly inclined to the horizon, this flow will be
towards the upper end of the tube. At each revolution, an
additional quantity of water will enter the tube, and that
already in the tube will be forced, or raised, higher and
332
MECHANICS.
Fig. 1S5.
higher, till, at last, it will flow from the orifice at the upper
end of the spiral tube.
The Chain Pump.
221. The chain pump is an instrument for raising wate»
through small elevations. It consists
of an endless chain passing over two
wheels, A and J3, having their axes
horizontal, the one being below the
surface of the water, and the other
above the spout of the pump. At-
tached to this chain, and at right
angles to it, are a system of circular
disks, just fitting the tube CD. If
the cylinder A be turned in the di-
rection of the arrow-head, the buckets
or disks will rise through the tube
CD, carrying the water in the tube before them, until it
reaches the spout C, and escapes. The buckets thus emptied
return through the air to the reservoir, and so on perpetually.
One great objection to this machine is, the difficulty of
making the buckets fit the tube of the pump. Hence* there
is a constant leakage, requiring a great additional expend-
iture of force.
Sometimes, instead of having the body of the pump ver-
tical, it is inclined ; in which case it does not differ much
in principle from the wheel with fiat buckets, that has been
used for raising water.
The Air Pump.
222. The air pump is a machine for rarefying the air ir.
a closed space.
It consists of a cylindrical
barrel A, in which a piston
B, fitting air-tight, is work-
ed up and down by a \e\ ;•:■
C\ attached to a piston-rod
D. The barrel communi-
cates with an air-tight ves-
Fig. 186.
HYDRAULIC AND PNEUMATIC MACHINES. 333
sel E, called a receiver, by means of a narrow pipe. The
receiver, which is usually of glass, is ground so as to fit air-
tight upon a smooth bed-plate KK. The joint between the
receiver and plate may be rendered more perfectly air-tight
by rubbing it with a little oil. A stop-cock //, of a peculiar
construction, permits communication to be made at pleasure
between the barrel and receiver, or between the barrel and
the external air. When the stop-cock is turned in a partic-
ular direction, the barrel and receiver are made to commu-
nicate ; but on turning it through 90 degrees, the communi-
cation with the receiver is cut off, and a communication is
opened between the barrel and the external air. Instead of
the stop-cock, valves are often used, which are either opened
and closed by the elastic force of the air, or by the force
that works the pump. The communicating pipe should be
exceedingly small, and the piston B should, when at its low-
est point, fit accurately to the bottom of the barrel.
To explain the action of the air pump, suppose the pLston
to be depressed to its lowest position. The stop-cock H, is
turned so as to open a communication between the barrel
and receiver, and the piston is raised to its highest point by
a force applied to the lever C. The air which before occu-
pied the receiver and pipe, will expand so as to fill the bar-
rel, receiver, and pipe. The stop-cock is then turned so as to
cut oft' communication between the barrel and receiver, and
open the barrel to the external air, and the piston again de-
pressed to its lowest position. The rarefied air in the barrel
is expelled into the external air by the depression of the
piston. The air in the receiver is now more rarefied than at
the beginning, and by a continued repetition of the process
just described, any degree of rarefaction may be attained.
To measure the degree of rarefaction of the air in the
receiver, a siphon-gauge may be used, or a glass tube, 30
inches long, may be made to communicate at its upper
extremity with the receiver, whilst its lower extremity dips
into a cistern of mercury. As the air is rarefied in the
receiver, the pressure on the mercury in the tube becomes
334 MECHANICS.
less than that on the surface of the mercury in the cistern,
and the mercury rises in the tube. The tension of the air
in the receiver will be given by the difference between the
height of the barometric column and that of the mercury
in the tube.
To investigate a formula for computing the tension of the
air in the receiver, after any number of double strokes, let
us denote the capacity of the receiver in cubic feet, by r,
that of the connecting-pipe, by p, and the space between
the bottom of the barrel and the highest position of the
piston, by b. Denote the original tension of the air, by t ;
its tension after the first upward stroke of the piston, by t ';
after the second, third, ...»'*, upward strokes, by
*, r, . . . f.
The air which originally occupied the receiver and pipe,
fills the receiver, pipe, and barrel, after the first upward
stroke ; according to Mariotte's law, its tension in the two
cases varies inversely as the volumes occupied ; hence,
t • t' : : p + r + b : p + r, .% t' = t p + f •
In like manner, w r e shall have, after the second upward
stroke,
f : t" : : p + r + b : p + r, .-. t" = t'
p -f b + r
Substituting for t' its value, deduced from the preceding
equation, we have,
t" - 1( p+r V
In like manner, we find,
pArt \ s .
" = *(-
\p
b + r
HYDRAULIC AND PNEUMATIC MACHINES. 335
and, in general,
*> = «( . ;t: ) '•
If the pipe is exceedingly small, its capacity may be
neglected in comparison with that of the receiver, and we
shall then have,
< = , and then con-
fined as before, by turning the stop-cock E.
The principle of IIeko's ball is the same as that of the air-
chamber in the forcing pump and fire-engine, already ex-
plained.
HYDRAULIC AND PNEUMATIC MACHINE8.
337
i
3d
Fig. 188
Hero's Fountain.
225. Hero's fountain is constructed on the same prin-
ciple as Hero's ball, except that the compression of the air
is effected by the weight of a column of water, instead of by
aid of a condenser
A represents a cistern, similar to Hero's ball, with a tube
J5, extending nearly to the bottom of the cis-
tern. C is a second cistern placed at some
distance below A. This cistern is connected
with a basin D, by a bent tube E, and also
with the upper part of the cistern A, by a
tube F. When the fountain is to be used,
the cistern A is nearly filled with water,
the cistern C being empty. A quantity of
water is then poured into the basin _Z), which,
acting by its weight, sinks into the cistern C,
compressing the air in the upper portion of it
into a smaller space, thus increasing its tension.
This increase of tension acting on the surface
of the water in A, forces a jet through the tube J5, which
rises to a greater or less height according to the greater or
less increase of the atmospheric tension. The flow will con-
tinue till the level of the water in A, reaches the bottom of
the tube B. The measure of the compressing force on a
unit of surface of the water in G 7 , is the weight of a column
of water, whose base is a square unit, and whose altitude is
the difference of level between the water in D and C.
If Hero's ball be partially filled with water and placed
under the receiver of an air pump, the water will be ob-
served to rise in the tube, forming a fountain, as the air in
the receiver is exhausted. The principle is the same as
before, an excess of pressure on the water within the globe
over that without. In both cases, the flow is resisted by the
tension of the air without, and is urged on by the tension
within.
Wine-Taster and Dropping-Bottle.
226. The wine-taster is used to bring up a small por-
338 MECHANICS.
tion of wine or other liquid, from a cask. It
consists of a tube, open at the top, and terminat-
ing below in a very narrow tube, also open. When
it is to be used, it is inserted to any depth in the
liquid, which will rise in the tube to the level of
the upper surface of that liquid. The finger is
then placed so as to close the upper orifice of
the tube, and the instrument is raised out of the !g '
cask. A portion of the fluid escapes from the lower orifice,
until the pressure of the rarefied air in the tube, plus the
weight of a column of liquid, whose cross-section is that of
the tube, and whose altitude is that of the column of fluid
retained, is just equal to the pressure of the external air.
If the tube be placed over a tumbler, and the finger re-
moved from the upper orifice, the fluid brought up will
escape into the tumbler.
If the lower orifice is very small, a few drops may be
allowed to escape, by taking off the finger and immediately
replacing it. The instrument then constitutes the dropping
tube.
The Atmospheric Inkstand.
227. The atmospheric inkstand consists of a cylinder
A, which communicates by a tube with a
second cylinder B. A piston C, is moved JP
up and down in A, by means of a screw D.
Suppose the spaces A and B, to be filled
with ink. If the piston G is raised, the
pressure of the external air forces the ink to
follow it, and the part B is emptied. If the F ig. lto
operation be reversed, and the piston C
depressed, the ink is again forced into the space B. This
operation may be repeated at pleasure.
r c
APPENDIX.
The following notes contain elementary demonstrations
of those principles, which in the body of the work are
proved by means of the Calculus.
Note on Articles 64—70 ; pp. 72—76.
These articles may be omitted without at all impairing
the unity of the subject, the preceding principles being suf-
ficient to find the centre of gravity of all bodies, approxima-
tively.
Note on Articles 112—114; pp. 143—148.
The principal formulas in these articles may be deduced
as follows :
112. By definition, a body moves uniformly when it
passes over equal spaces in equal times; now if it passes
over a space v in one second, it will pass over t times that
space in t seconds ; that is, it will pass over a space vt. If
we suppose it to have passed over a space s' before the com-
mencement of the time t, we shall have for the entire space
passed over, and which may be denoted by s,
8 = Vt + %* (58.)
This equation corresponds to Equation (58) of the text.
113. The formulas of Article 113 may be omitted with-
out impairing the unity of the course. They are only of
use in Higher Mechanics, where the employment of the Cal-
culus is a necessity.
340 MECHANICS.
114. Uniformly varied motion, is that in which the
velocity increases or diminishes uniformly. In the former
case the motion is accelerated, in the latter it is retarded.
In both cases the moving force is constant.
Denote the moving force by/*, the mass moved being the
unit of mass.
According to Art. 24, the measure of the force is the ve-
locity impressed in a unit of time, that is, in 1 second. Now
from the principal of inertia, Art. 18, it follows that a force
will produce the same general effect upon a body, whethei
it finds the body at rest or in motion. Hence, the velocity
impressed in any second of time is constant ; that is, if the
velocity impressed in one second of time is f in t seconds
it will be t times f or ft. Denoting the velocity by v,
we shall have,
v =ft (69.)
If the body has a velocity v' at the beginning of the time
t, this velocity is called the initial velocity. Adding this to
the velocity imparted during the time t, we have,
V = v' + ft (67.)
With respect to the space passed over, it may be re-
marked that the velocity increases uniformly ; hence the
space passed over in any time, is the same that it would
have passed over in the same time, had it moved uniformly
during that time with its mean or average velocity. Now,
if a body start from a state of rest, its velocity at starting is
0, and at the end of the time t it is ft, Equation (69) ; the
average or mean of these is \ft. But the space described
in the time t, when the body moves with the uniform ve-
locity \ft, is (Equation 55) equal to \ft x t ; denoting
the space by *, we have,
s = ift* (70.)
Kin Equation (70), we make t — \, we have,
s - \f\ or, f= 2s;
APPENDIX. 341
that is, if a body moves from a state of rest, the space de-
scribed in the first second of time, is equal to half the
measure of the accelerating force ; or, the acceleration is
measured by twice the space passed over in one second of
time.
If we suppose that a body starts from rest before the be-
ginning of the tinie t, so as to pass over a space s r before
the beginning of t, it will during that time have acquired
some velocity, which we may denote by v'. The space
reckoned from the origin of spaces up to the position of the
body at the end of the time £, is made up of three parts ;
first, the space s\ called the initial space ; second, a space
due to the velocity v' during the time t, which is measured
by v't; third, a space due to the action of the incessant force
during the time t, which will (Equation 70) be equal to
ift 2 . Adding these together, we have finally,
8 = *'+ V't+ \f& . . . (68.)
If, in Equations (67) and (68), we suppose /to be essen-
tially positive, the motion will be accelerated ; if we suppose
it to be essentially negative, the motion will be retarded,
and these equations become
v = v' -ft (VI.)
s = 8 >+ v 't- \f& . . . . (72.)
Note on Article 121, pp. 163—164.
The formula deduced in the first part of this article is
needed in the investigations of Acoustics and Optics, and
can only be found by the Calculus This part of the article
may be omitted without impairing the unity of the course.
Note on Article 123, pp. 166—168.
This article, up to the end of Equation (95), may be re-
placed by the following demonstration:
U2
MECHANICS.
The simple pendulum.
123. A pendulum is a heavy body suspended from a
horizontal axis about which it is free to vibrate.
In order to investigate the circumstances of vibration, let
us first consider the hypothetical case of a single material
point, vibrating about an axis to which it is attached by a
rod destitute of weight. Such a pendulum is called a
simple pendulum. The laws of vibration in this case will
be identical with those explained in Art. 120, the arc ABC
being an arc of a circle.
Let AB C be the arc through
which the vibration takes place,
and denote its radius DA, by I.
The angle ABC is called the
amplitude of vibration ; half of
this angle, ABB, is called the
angle of deviation.
If the point starts from rest at
A, it will, on reaching any point
If, have a velocity v, due to the
height EK, denoted by A, (Art.
120). Hence,
(92.)
Let us suppose that the angle of deviation is so small, that
the chords of the arcs AB and HB, may be considered
equal to the arcs themselves. We shall have (Davies' Le-
gendre, Bk. IV., Prop. XXIIL, Cor.),
AB 2 = 21 x EB, and 1W = 21 x KB,
whence, by subtraction,
AB 2 - IIB 2 - 2l(EB - KB) = 21 x A.
APPENDIX. 34-3
Denoting AB by a, and HB by jc, and solving the
last equation, we have,
21
Substituting this value of h in (92) it becomes,
v = yJJ{a? - a?) .... (a.)
Now let us develop the arc ABC into a straight line
.4'.Z?' 0", and suppose a material point to start from A' at
the same time that the pendulum starts from A, and to
vibrate back and forth upon A'B' C with the same veloci-
ties as the pendulum ; then, when the pendulum is at any
point JET, this material point will be at the corresponding
point H\ and the times of vibration of the two will be
exactly the same.
To find the time of vibration along the line A'B'C, de-
scribe upon it a semi-circle A'3fC, and suppose a third
material point to start from A' at the same time as the
second, and to move uniformly around the arc with a ve-
locity equal to a \J j • Then will the time required for
this particle to reach C be equal to the space divided by
the velocity (Art. 112). Denoting this time by t, and re-
membering that A'B' = a, we shall have,
g v g
Make ITB' = x, and draw WM perpendicular to A' C\
and at M decompose the velocity of the third particle
MT into two components 3IN~ and MQ, respectively par-
allel and perpendicular to A'C.
344 mecha:sics.
We shall have for the horizontal component JIN",
MN = JIT cos TMN.
But, JUT = «\/f, and because JIT and JAY are re-
spectively perpendicular to B ' JI and II' JI, we have,
cos TJIX = cos B'JIIF = ~^ L - But JB'Jf = a,
and II'JI — J a? — # 2 ; hence, cos TJIX =
v a
Substituting these values in Equation (5), we have for
the horizontal velocity,
Mir = y^ - »?),
which is the same value as that obtained for v in Equa-
tion (a). Hence, we infer that the velocity of the third
material point in the direction of A' C is always equal to
that of the second point, consequently the times required
to pass from A' to C must be equal ; that is, the time
of vibration of the second point, and consequently of the
pendulum, must be ttyj - • Denoting this time by f, we
have,
< = V? {95 - ]
Note on Article 131, pp. 182—186.
This article may be omitted without impairing the unity
of the course. The results may be assumed if needed.
They can only be deduced by the Calculus by demon-
strations too tedious for an Elementary Course.
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In claiming for this series the first place among American text-books, of whatever
class, the publishers appeal to the magnificent record which its volumes have earneq
during the thirty-fine years of Dr. Charles Davies's mathematical labors. The unremit-
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among competitors that each of its predecessors had successively enjoyed in a course of
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During the period alluded to, many authors and editors in this department Iipvb
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and prosper, and lix a still (inner hold on the affection of every educated American.
One cause of this unparalleled popularity is found in the fact that the enterprise of the
author did not cease with the original completion of his books. Always a practical
teacher, he has incorporated in his text-books from time to time the advantages or every
improvement in methods of teaching, and every advance in science. During all the
years in which he has been laboring lie constantly submitted his own theories ami those
of others to the practical test of the class-room, approving, rejecting, or modifying
them as the experience thus obtained might suggest. In this way he has been aide
to produce an almost perfect series of class-books, in which every department of
mathematics lias received minute and exhaustive attention.
Upon the death of Dr. Davies, which took place in 1876, his work was immediately
taken up by his former pupil and mathematical associate of many years, Prof. W. G.
Peck, L.L.D., of Columbia College. By him, with Prof. J. H. Van Amringe, of Columbia
College, the original series is kept carefully revised and up to the times.
Davies's System is the ACKNOWLEDGES National Stan-dard for the United
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4th. The officials of the Government use it as authority in all cases Involving mathe-
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5th. Our great soldiers and sailors commanding the national armies and navies were
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being extensively used in every State in the Union.
It
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The best thoughts of these two illustrious mathematicians are combined in the
following beautiful works, which are tlie natural successors of Davies's Arithmetics,
sumptuously printed, and bound in crimson, green, and gold: —
Davies and Peck's Brief Arithmetic.
Also called the " Elementary Arithmetic." It is the shortest presentation of the sub-
ject, and is adequate for all grades in common schools, being a thorough introduction to
practical life, except for the specialist.
At first the authors play with the little learner for a few lessons, by object-teaching
and kindred allurements ; but he soon begins to realize that study is earnest, as he
becomes familiar with the simpler operations, and is delighted lo hud himself master of
important results.
The second part reviews the Fundamental Operations on a scale proportioned to
the enlarged intelligence of the learner. It establishes the General Principles and
Properties of Numbers, ami then ] loceeds to Fractions. Currency and the Metric
System are fully treated in connet won with Decimals. Compound Numbers and Re-
duction follow, and finally Percentage with all its varied appli cations.
An Index of words and principles concludes the book, for which every scholar and
most teachers will be grateful. How much time has been spent in searching for a half-
forgotten definition or principle in a former lesson !
Davies and Peck's Complete Arithmetic.
This work certainly deserves its name in the best sense. Though complete, it is not,
like most others which bear the same title, cumbersome. These authors excel in clear,
lucid demonstrations, teaching the science pure and simple, yet not ignoring convenient
methods and practical applications.
For turning out a thorough business man no other work is so well adapted. He will
have a clear comprehension of the science as a whole, and a working acquaintance
with details which must serve him well in all emergencies. Distinguishing features of
the book are the logical progression of the subjects and the great variety of practical
problems, not puzzles, which are beneath the dignity of educational science. A clear-
minded critic has said of Dr. Peck's work that it is free from that juggling with
numbers which some authors falsely call " Analysis." A series of Tables for converting
ordinary weights and measures into the Metric System appear in the later editions.
PECK'S ARITHMETICS.
Peck's First Lessons in Numbers.
This book begins with pictorial illustrations, and unfolds gradually the science of
numbers. It noticeably simplifies the subject by developing the principles of addition
and subtraction simultaneously ; as it does, also, those of multiplication and division.
Peck's Manual of Arithmetic.
This book is designed especially 'or those who seek sufficient instruction to carry
them successfully through practical life, but have not time for extended study.
Peck's Complete Arithmetic.
This completes the series but is a much briefer book than most of the complete
arithmetics, and is recommended not only for what it contains, but also for what is
omitted.
It may be said of Dr. Peck's books more truly than of any other series published, that
they are clear and simple in definition and rule, and that" superfluous matter of every
kind has been faithfully eliminated, thus magnifying the working value of the book
and saving unnecessary expense of time and labor.
19
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
Algebra. The student's progress in Algebra depends very Largely npon the proper treat-
ment of the four Fundamental Operation*. The terms Addition, Subtraction, Multiplication,
and Jjirision in Algebra have a wider meaning than in Arithmetic, and these operations
have been so defined as to include their arithmetical meaning ; so that the beginner
is sinrily called upon to enlarge his views of those fundamental operations. Much
attention has been given to the explanation of the negative sign, in order to remove the
well-known difficulties in the. use and interpretation of that sign. Special attention is
here called to " A Short Method of Removing Symbols of Aggregation," Art 76. On
account of their importance, the BUOJects ol Factoring, Greatrst Common lHrist>r, and
Least Common Multiple have been treated al greater length than is usual in elementary
works. In the treatment of Fractions, a method is used which is quite simple, and,
at the same time, more general than that usually employed. In connection with Radical
s the roots are expressed by fractional exponents, for the principles and rules
applicable to integral exponents may then be used without modification. The Equation
is made the chief subject of thought in this work. It is defined near the beginning,
and used extensively in every chapter. In addition to this, four chapters are devoted
exclusively to the subject of Equations. All Proportions are equations, and in their
treatment as such all the difficulty commonly connected with the subject of Proportion
disappears. The chapter on Logarithms will doubtless be acceptable to many teachers
who do not require the student to master Higher Algebra before entering upon the
study of Trigonometry.
HIGHER MATHEMATICS.
Peck's Manual of Algebra.
Bringing the methods of Bourdon within the range of the Academic Course.
Peck's Manual of Geometry.
By a method purely practical, and unembarrassed by the details which rather confuse
than simplify science.
Peck's Practical Calculus.
Peck's Analytical Geometry.
Peck's Elementary Mechanics.
Peck's Mechanics, with Calculus.
The briefest treatises on these subjects now published. Adopted by the great Univer-
sities : rale, Harvard, Columbia, Princeton, Cornell, &e
Macnie's Algebraical Equations.
Serving as a e plemenl to the more advanced treatises on Algebra, giving special
attention to the analysis and solution of equations with numerical coefficients.
Church's Elements of Calculus.
Church's Analytical Geometry.
Church's Descriptive Geometry. "With plates. £ vois.
These volumes constitute the "West Point Course" in their several departments.
Pr< it. Church was long the eminent professor of mathematics at West Point Military
Academy, and his works are standard in all the leading colleges.
Courtenay's Elements of Calculus.
A standard work of the very highest grade, presenting the most elaborate attainable
butv( y ol
Hackley's Trigonometry.
With applications to Navigation and Surveying, Nautical and Practical Geometry,
and Geodesy.
21
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
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GENERAL HISTORY.
Monteith's Youth's History of the United States.
A History of the United States for beginners. It is arranged upon the catechetical plan,
with illustrative maps and engravings, review questions, dates in parentheses (that their
study may be optional with the younger class of learners), and interesting biographical
sketches of all persons who have been prominently identified with the history of our
country.
Willard's United States. School and University Editions.
The plan of this standard work is chronologically exhibited in front of the titlepage.
The maps and sketches are found useful assistants "to the memory ; and dates, usually
so difficult to remember, are so systematically arranged as in a great degree to obviate
the difficulty. Candor, impartiality, and accuracy are the distinguishing features of
the narrative portion.
Willard's Universal History. New Edition.
The most valuable features of the li United States " are reproduced in this. The
peculiarities of the work are its great conciseness and the prominence given to the
chronological order of events. The margin marks each successive era with great dis-
tinctness, so that the pupil retains not only the event but its time, and thus fixes the
order of history firmly and usefully in his mind. Mrs. Willard's books are constantly
revised, and at all times written up to embrace important historical events of recent
date. Professor Arthur Gilman has edited the last twenty-five years to 1882.
Lancaster's English History.
By the Master of the Stoughton Grammar School, Boston. The most practical of the
"brief books." Though short, it is not a bare and uninteresting outline, but contains
enough of explanation and detail to make intelligible the cause mid effect of events.
Their relations to the history and development of the American people is made specially
prominent.
Willis's Historical Reader.
Being Collier's Great Events of History adapted to American schools. This rare
epitome of general history, remarkable for its charming style and judicious selection of
events on which the destinies of nations have turned, has been skilfully manipulated
by Professor Willis, with as few changes as would bring the United States into its ] in iper
position in the historical perspective. As reader or text-book it has few equals and no
superior.
Berard's History of England.
By an authoress well known for the success of her History of the United States.
• I life of the English people is felicitously interwoven, as in fact, with the civil
and military transactions of the realm.
Ricord's History of Rome.
Possesses the charm of an romance. The fables with which this history
abounds are introduced in such a way as not to deceive the inexperienced, while adding
materially to the value of the work as t reliable index to the character and institutions,
as well as the historv of the Roman people.
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
vwrte
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A Brief History of An-
cient Peoples.
With an account of their monuments,
literature, and manners. 340
12mo. Profusely illustrated.
In this work the political history,
which occupies nearly, if not all,
the ordinary school text, is con
to the salient and essential facts, in
order to give room for a clear outline
of the literature, religion, architecture,
character, habits, &c, of each nation.
Surely it is as important to know some-
thing aboui Plato as all about Caesar,
and to learn how the ancients wrote
their hooks as how they fought their
battles.
The chapters OH Manners and Cus-
toms ami the Scenes in Real Life repre-
sent the people of history as men and
women subject, tnthc same wants, hopes
ami fears as ourselves, and so brine the distant past near to us. The Scenes, which are
intended only for reading, are the result of a careful study of the unequalled collect ions oj
monuments in the London and Berlin Museums, of the ruins in Rome and Pompeii, and
of the latest authorities on the domestic life of ancient peoples. Though intei I
written in a semi-romantic style, they are accurate pictures of what might have occurred,
and some of them are simple transcriptions of the details sculptured in Assyrian
alabaster or painted on Egyptian walls.
36
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
HISTORY — Continued.
The extracts made from the sacred books of the East are not specimens of their style
and teachings, but only gems selected often from a mass of matter, much of which would
be absurd, meaningless, and even revolting. It has not seemed best to cumber a book
like this with selections conveying no moral lesson.
Ihe numerous cross-references, the abundant dates in parenthesis, the pronunciation
of the names in the Index, the choice reading references at the close of each general
subject, and the novel Historical Recreations in the Appendix, will be of service to
teacher and pupil alike.
Though designed primarily for a text-book, a large class of persons — general readers,
who desire to know something about the progress of historic criticism and Aie recent
discoveries made among the resurrected monuments of the East, but have no leisure to
read the ponderous volumes of Brugsch, Layard, Grote, Mommsen, and lime — will tiud
this volume just what they need.
From Homer B. Spkague, HeaJ Master
Girls' High School, If 'est Newton St. , Bos-
ton, Muss.
" I beg to recommend in strong terms
the adoption of Barnes's 'History of
Ancient Peoples' as a text-book. It is
about as nearly perfect as could be
hoped for. The adoption would give
great relish to the study of Ancient
History."
HE Brief History of France.
By the author of the " Bri f United States."
with all the attractive features of that popu-
u work (which see) and new ones of its own.
It is believed that the History of France
has never before been presented in such
brief compass, and this is effected without
sacrificing one particle of interest. The book
reads like a romance, and, while drawing tht
student by an irresistible fascination to his
task, impresses the great outlines indelibly upon the memory.
27
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DRAWING.
BARNES'S POPULAR DRAWING SERIES.
Based upon the experience of the most successful teachers of drawing in the United
States.
The Primary Course, consisting of a manual, ten cards, and three primary
Irawing I is, a. B. and U.
Intermediate Course. Four numbers and a manual.
Advanced Course. Four numbers and a manual.
Instrumental Course. Pour numbers and a manual.
iiie Intermediate, Advanced, and Instrumental Courses are furnished either in book
or <-ar>l lorm at the same prici s. The books contain tin-usual blanks, with the unusual
advantage of opening from the pupil, — placing the copy directly in front and above
the blank, thus occupying hut little desk-room. The cards are in the end more econom-
ical than the books, if used in connection with the patent blank folios that accompany
this series.
The cards are arranged to be bound (or tied) in the folios and removed at pleasure.
The pupil at the end of each number has a complete book, containing only his own
work, while the copies are preserved and inserted in another fol.o ready for use in the
next class.
Patent Blank Folios. No. 1. Adapted to Intermediate Course. No. 2. Adapted
to Advanced and Instrumental Courses.
ADVANTAGES OF THIS SERIES.
The Plan and Arrangement. — The examples are so arranged that teachers and
pupils can see, at a glance, how they are to be treated and where they are to be copied.
In this system, copying and designing do not receive all the attention. The plan is
broader in its aims, dealing with drawing as a branch of common-school instruction,
asd giving it a wide educational value.
Correct Methods. — In this system the pupil is led to rely upon himself, and not
upon delusive mechanical aids, as printed guide-marks, &c.
One of the principal objects of any good course in freehand drawing is to educate the
eye to estimate, location, form, and size. A system which weakens the motive or re-
moves the necessity Of thinking is false in theory and ruinous in practice. The object
should be to educate, not cram ; to develop the intelligence, not teach tricks.
Artistic Effect —The beautj of the examples is not destroyed by crowding the
pages with useless and badly printed text. The Manuals contain all necessary
instruction.
Stages of Development. —Many of the examples are accompanied by diagrams,
showing the different stag* - of development
Lithographed Examples. — The examples are. printed in imitation of pencil
drawing (uoi in hard, blach lines) that the pupil's work may resemble them.
One Term's Work. — Each book contains what can be accomplished in an average
term, and no more. Thus a pupil finishes one book before beginning another.
Quality — not Quantity. —.Success in drawing depends upon the amount of thought
exercised by the pupil, and not upon the large number of examples drawn.
Designing. — Elementary design is more skilfully taught in this system than by
any other, m addition to the instruction given in the books, the pupil will and printed
on the inside- of the covers a variety of beautiful patterns.
Enlargement and Reduction*. — The practice of enlarging and reducing from
cop.es is not commenced mini the pupil is well advanced in the course and therefore
better able to cope with this difficult feature in drawing.
Natural Forms. -This is the only course that gives at convenient intervals easy
and progressive exercises in the drawing of natural forms.
Economy. — By the patent binding described above, the copies need no! be tin-own
at de when a book is filled out, but arc preserved in perfeel condition for future use.
'ihe blank books, only, will have to De purchased after tlu first introduction, th
ing of nmre than half in the usual cost of drawing-books.
Manuals for Teachers. — The Manuals accompanying this series contain pra< tical
g in the class-room, with definite directions for draw-
ing each of the examples in the books, instructions lor designing, model and object
tlrawing, drawing from natural forms, &c.
28
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DRAWING — Continued.
Chapman's American £)rawing-Book.
The standard American text-book and authority in all branches of art. A compilation
of art principles. A manual for the amateur, and basis of study for the professional
artist. Adapted for schools and private instruction.
Contents. — "Any one who can Learn to Write can Learn to Draw." — Primary In-
struction in Drawing. — Rudiments of Drawing the Human ' Head. — Rudiments in
Drawing the Human Figure. — Rudiments of Drawing. — The f Geometry. -
Ferspective. — Of Studying and Sketching from Nature. —Of Painting. — Etching and
En-raving. — Of Modelling. — Of Composition. — Advice to the American Art-Student.
The work is of course magnificently illustrated with all the original designs.
Chapman's Elementary Drawing-Book.
A progressive course of practical exercises, or a text-book for the training of the
eye and hand. It contains the elements from the larger work, and a copy should be in
the hands of every pupil ; while a copy of the " American Drawing- Book," named above,
should be at hand for reference by the class.
Clark's Elements of Drawing.
A complete course in this graceful art, from the first rudiments of outline to the
finished sketches of landscape and scenery.
Allen's Map-Drawing and Scale.
This method introduces a new era in map-drawing, for the following reasons : 1. It
is a system. This is its greatest merit. — 2. It is easily understood and taught. —
3. The eye is trained to exact measurement by the use of a scale. — 4. By no special
effort of the memory, distance and comparative size are fixed in the mind. — 5. It dis-
cards useless construction of lines. — 6. It can be taught by any teacher, even though
there may have been no previous practice in map-drawing. — 7. Any pupil old enough
to study geography can learn by this system, in a short time, to draw accurate maps.
— 8. The system is not the result of theory, but comes directly from the school-room.
It has been thoroughly and successfully tested there, with all grades of pupils. — 9. It
is economical, as it requires no mapping plates. It gives the pupil the ability of rapidly
drawing accurate maps.
FINE ARTS.
Hamerton's Art Essays (Atlas Series) : —
No. 1. The Practical Work of Painting.
With portrait of Rubens. Svo. Paper covers.
No. 2. Modern Schools of Art-
Including American, English, and Continental Painting. Svo. Paper covers.
Huntington's Manual of the Fine Arts.
A careful manual of instruction in the history of art, up to the present time.
Boyd's Karnes' Elements of Criticism.
The best edition of the best work on art and literary criticism ever produced in
English.
Benedict's Tour Through Europe.
A valuable companion for anyone wishing to visit the galleries and sights of the
continent of Europe, as well as a charming book of travels.
Dwight's Mythology.
A knowledge of mythology is necessary to an appreciation of ancient art.
Walker's World's Fair.
The industrial and artistic display at the Centennial Exhibition.
29
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DR. STEELE'S OJME-TERM SERIES,
IN ALL THE SCIENCES.
Steele's 14-Weeks Course in Chemistry.
Steele's 14-Weeks Course in Astronomy.
Steele's 14-Weeks Course in Physics.
Steele's 14-Weeks Course in Geology.
Steele's 14-Weeks Course in Physiology.
Steele's 14-Weeks Course in Zoology.
Steele's 14-Weeks Course in Botany.
Our text-books in these studies are, as a general thing, dull and uninteresting.
They contain from 400 to GOO pages of dry facts and unconnected details. They abound
in that which the student cannot learn, much less remember. The pupil commences
the study, is confused by the hue print and coarse print, and neither knowing exactly
what to learn nor what to hasten over, is crowded through the single term generally
assigned to each branch, and frequently comes to the close without a definite and exact
idea of a single scientific principle.
Steele's '• Fourteen. Weeks Courses" contain only that which every well-informed per-
son should know, while all that which concerns only the professional scientist is omitted.
The language is clear, simple, and interesting, and the illustrations bring the subject
within the range of home life and daily experience. They give such of the general
principles and the prominent facts as a pupil can make familiar as household words
within a single term. The type is large and open; there is no fine print to annoy ;
the cuts are copies of genuine experiments or natural phenomena, and are of tine
execution.
In line, by a system of condensation peculiarly his own. the author reduces each
branch to the limits of a single term of study, while sacrificing nothing that is essential,
and nothing that is usually retained from the study of the larger manuals in common
use. Thus the student has rare opportunity to economize his time, or rather to employ
that which he has to the best advantage.
A notable feature is the author's charming "style," fortified by an enthusiasm over
his subject in which the student will not fail to partake. Believing that Natural
Science is full of fascination, he has moulded it into a form that attracts the attention
and kindles the enthusiasm of the pupil.
The recent editions contain the author's " Practical Questions" on a plan never
before attempted in scientific text-books. These are questions as to the nature and
cause of common phenomena, and are not directly answered in the text, the design
being to test and promote an intelligent use of the student's knowledge of the foregoiDg
principles.
Steele's Key to all His Works.
This work is mainly composed of answers to the Practical Questions, and solutions of the
problems, In the author's celebrated " Pourteen-Weeks Courses " in the several sciences,
with many hints to teachers, minor tables, fee. Should he on every teacher's desk.
Prof. J. Dorman Steele is an indefatigable student, as well as author, and his books
have reached a fabulous circulation. It is sate to say of his hooks that they have
accomplished more tangible and better results in the class-room than any other ever
offered to American schools, and have been translated into more languages for foreign
schools. They are even produced in raised type for the blind.
32
\
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
NATURAL SCIENCE — Continued.
BOTANY.
Wood's Object-Lessons in Botany.
Wood's American Botanist and Florist.
Wood's New Class-Book of Botany.
The standard text-hooks of the United States in this department. In style they are
simple, popular, and lively ; in arrangement, easy and natural ; in description, graphic
and scientific. The Tables for Analysis are reduced to a perfect system. They include
the flora of the whole Unitr Worman? C'est un veritable
tresor, merveilleusement sdapte au devel-
oppement de la conversation familiere et
pratique, telle qu'on la vent aujourd'hui.
silent livre met successive!]
scene, d'une maniere vive et inter
toutes les circonstances possibles de la vie
ordinaire. Voyez l'immense avantage
il vous transporte en Frame ; do premier
mot, je m'imagine, et mes eleves avec moi,
que nous sommes a Paris, dans la rue, sur
une place, dans une gare,dans un salon,
dans une chambiv, voire meine a Is cui-
sine ; je parte comme avec des Prancais ;
les eleves ne songent pas a tradnire de
1'anglais pour me repondre ; ils pensent
en franc. us ; ils sont Francais pour le
moment paries yeux. par l'oreille. par la
pens, e Quel autre livre pourrait produire
cette illusion ? . . ."
Votre tout uevoue,
A. DE KOUCEMONT.
Illustrated Language Primers.
French and English. German and English.
Spanish and English.
The names of common objects properly illustrated and arranged in easy lessons.
Pujol's Complete French Class-Book.
Offers in one volume, methodically arranged, a complete French course — usually
embraced in series of from five x>> twelve books, including the bulky and expensive
lexicon. Here are grammar, conversation, and choice literature, selected from the
best French authors. Each branch is thoroughly handled ; and the student, having
diligently completed the course as prescribed, may consider himself, without further
application, au fait in the most polite and elegant language of modern times.
45
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