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THE
(AVAL ATICHITECT'S AND SHIPBUILDER'S
POCKET-BOOK
ADVERTISEMKNTS
MARINE PATTERN
Duplex Steam Pumps
To Uoyd't Rtqairetnmntt.
VERTICAL or HORIZONTAL.
HAYWARD-TYLER&CO.,Lt(l
99 Queen Victoria Street,
LONDON, E.C.
THE
NiYAL ARCHITECn and SHIPBOILDER'S
POCKET-BOOK^
OF
AND
MABINE ENGINEER'S AND SURVEYOR'S
HANDY BOOK OF REFERENCE
BY
/a »U.
b'(^'-
CLEMENT MACKROW
lATB UBHBSB OF THK INSTITUTION OF NAVAL ABCHITECTB
I.ATB I.KCTUBSB ON NAVAL ABOHITEGTUBB AT THE BOW AND BBOMLET INSTITUTE
AND
LLOYD WOOLLARD
KOTAIi GOBPS OF NAVAL CONBTBUCTOB8
MRBCBRR OF THE INSTITUTION OF NAVAL AB0HITE0T8
INSTRUCTOB IN NAVAL ABOHITECTURE AT THE B.N. COLLEGE, ORBENWICH
lEIebeutij lEuiti0u
THOROUGHliY 3BVI8ED
WITH A SECTION ON AERONAUTICS
* ^ • «
NEW YORK
THE NO'RMAN W. HENLEY PUBLISHING CO.
132 NASSAU STREET
1916
PRINTED BY
STEPHEN AUSTIN AND SONS, LTD.
HERTFORD.
• » • •
• • •
• • • • » »
PREFACE
TO
THE ELEVENTH EDITION.
The need of a new edition of this Pocket-book has
arisen through the continual development of the
science of Naval Architecture, and the tendency
towards standardization and regulation of parts of the
structure and equipment of ships. Very many changes
have been introduced, and much of the book has been
rewritten, but where possible its form has been left
unaltered. Its object remains the same as that stated
in the Preface to the original edition, viz. to condense
into a compact form all data and formulae that are
ordinarily required by the Shipbuilder or Naval
A rehitect.
Amongst the new matter inserted, it is believed that
the section on Speed and Horse-power will be useful in
enabling ships of ordinary form to be approximately
powered from the data therein given ; a brief description
of modern methods of powering and determining forms
suitable from a propulsive standpoint has also been
included. The necessity for economizing weight where
possible without diminution of strength has led to the
sections on Strength of Materials, Riveted Joints, and
Stresses in Ships being considerably extended. In-
formation concerning British Standard Sections, Screws,
Keys, etc., has also been added, by permission of the
Engineering Standards Committee^. Finally, two new
sections on Aeronautical matters will be of service,
not only to those engaged in that modern and rapidly
35^437
VI PREFACE.
developing branch of engineering, but also to Naval
Architects on account Of the kindred nature of the
subjects, and of the direct application of many air data
to questions relating to the resistance of bodies in water.
The remainiiig subjects treated, -which were also
included in previous editions, have now been brought
completely up-to-date; the excision of obsolete data
has enabled the new matter to be inserted without
increase in the size of the book. The new tables of
logarithms, etc., it is trusted, will be found of great
practical convenience to those using them.
The scope and extent of the revision were arranged
in the first place with the original author ; although,
owing to his death before the completion of the work,
the absence of his advice and experience daring the
later stages has been felt and regretted, the reviser has
had the benefit of securing great assistance from many
sources during the preparation of the new edition.
Among those who kindly contributed, the reviser is
greatly indebted to Mr. A. W. Johns, the results of
whose valued experience have been embodied in various
parts of the book; the new sections 'Aerodynamics'
and ' Aeronautics ' are entirely due to him. Considerable
aid in the treatment of Speed and Propellers has been
rendered by Professor T. B. Abell, while Mr. E. F.
Atkinson has supplied useful data concerning small
craft and tugs. To these, and to many others to whom
reference is made in the course of the book, the reviser
tenders his cordial thanks. He also trusts that the
numerous correspondents who have offered suggestions
and pointed out errors in previous editions may be led
to take the same kindly interest in the present revision.
L. W.
Barnes : Jantuiry 1, 1916.
PREFAX:!E
TO
THE FIRST EDITION.
The object of this work is to supply the great want
which has long been experienced by neej*ly all who are
connected professionally with shipbuilding, of a Pockets
Book which should contain all the ordinary Formuke^
Bales, and Tables required when working out necessary
calculations, which up to the present time, as far as the
Author is aware, have never been collected and put into
so convenient a form, but have remained scattered
through a number of large works, entaihng, even in
referring to the most commonly used Formulae, much
waste of time and trouble. An effort has here been
made to gather all this valuable material, and to con-
dense it into as compact a form as possible, so that the
Naval Architect or the Shipbuilder may always have
ready to his hand reliable data from which he can solve
the numerous problems which daily come before him.
How far this object has been attained may best be
judged by those who have felt the need of such a work.
Several elementary subjects have been treated more
fully than may seem consistent with the character of the
book. This, however, has been done for the benefit of
those who have received a practical rather than a theo-
retical training, and to whom such a book ds this would
be but of small service were they not first enabled to
/
vm PREFACE.
gather a few elementary principles, by which means
they may learn to use and understand these Formulae.
In justice to those authors whose works have been
consulted, it must be added that most of the Eules and
■
Formulae lierei given are not original, although perhaps
appearing in a new shape with a view to making them
simpler.
There are many into whose hands this work will fall
who are well able to criticise it, both as to the usefulness
and the accuracy of the matter it contains. From such
critics the Author invites any corrections or fresh mate-
rial which may be useful for future editions.
SUMMARY OF CONTENTS.
— •<>« —
PAGES.
Signs and Symbols ....
1-3
Logarithms
4-7
Trigonometry . . . . .
7-18
Carves (Conic Sections, Catenary, Cycloid, ete.^
> 13-18
Differential and Integral Calculus
. 19-21
Practical Geometry .
22-86
Mensuration of Areas and Perimeters .
. 86-49
Mensuration of Solids
49-69
Centres, and Moments of Figures .
. 59-69
Moments of Inertia and Radii of Gyration
- 69-75
Mechanical Principles
76-79
Centre of <Travity
80
Motion . . . . * .
. 81-83
Dynamics
. 84-88
Hydrostatics .....
. 88-89
Displacement, etc. . . . . .
90-102
Weight and Centre of Gravity of Ships
. 102-109
Stability .......
110-143
Waves .......
143-149
Boiling
150-160
Speed and Horse-power . . . .
, 160-190
Propellers
190-197
Speed Tria.ls and Tables . . . .
197-207
Saihng, Force of Wind . . . .
208-211
X
SUMMARY OF CONTENTS.
PAGES.
Distances down Rivers
. 212-221
Weights and Dimensions of Materials .
. 222-287
Wire and Plat^ Gauges ....
. 238-240
British Standard Sections .
. 241-254
Notes on Materials ....
. 255-259
Weight and Strength of Materials
. 260-268
Admiralty Tests, etc., for Materials
. 268-284
Lloyds* Tests for Materials .
. 284-285
Riveted Joints and Rivets .
. 286-294
Braced Structures . . .
. 294-800
Shearing Forces and Bending Moments
of Beams .....
. 800-808
Strength of Materials and Stresses, etc. :
General
. 809-812
Bending
. 812-827
Compression ....
. 828-881
Shear
. 882-887
Miscellaneous ....
887-839, 844
Keys and Wheel Gearing
. 840-844
Longitudinal Stresses in Ships .
. 846-862
Mechanical Powers ....
. 858-868
Notes on Steering, Rudders, etc. .
. 864-871
Launching
. 872-878
Armour and Ordnance
. 878-888
Notes on Machinery ....
. 888-890
Notes on Design ....
. 891-894
Fans and Ventilation of Ships .
. 896-401
Hydraulics
. 402-404
Heat ...
. 406-406
Aerodynamics (Forces on Plates, etc.) *
. 406-481
f
SUMMARY OF CONTENTS. XI
Aeronautics (Notes on Airships, etc.) . 481-448
B. of T. Regulations for Marine Boilers, etc. 448-466
„ „ for Motor Passenger Vessels 466-469
„ for Ships .... 469-488
Strength of Bulkheads .... 488-487
International Regulations for Preventing
Collisions at Sea .... 487-490
Tonnage ....... 490-495
B. of T. Rules, etc., for Life-saving Appliances 496-502
„ for Emigrant Ships . 602-506
Lloyds' Rules for determining Size of Shafts 606
„ for Ships .... 607-519
„ „ for Yachts of the International
Rating Classes . . 520-526
Anchors and Cables ..... 527-688
British Standard Pipes and Screws . 688-687
Ship Fittings 588-656
Seasoning and Measuring Timber . 667-661
Miscellaneous Data . . . . 662-666
Dimensions and Weights of Blocks 667-671
Weight and Strength of Hemp and Steel-
wire Rope 672-688
Lloyds' Rules for Yards, Masts, Rigging, etc. 684-698
Distances of Foreign Ports from London . 694
Paints, Caulking Varnishes, Galvanizing, etc. 696-605
English and Foreign Weights and Measures 606-627
Decimal Equivalents 628-681
Foreign Money 682-688
Discount and Equivalent Price Tables . . 684-686
Useful Numbers and Ready Reckoners 686-638
Xll SUMMARY OF CONTENTS.
Tables of Circular Measure 639—64 J
Tables of Areas of and Circumferences of
Circles 642-65 ]|
Tables of Areas of Segments of Circles 652-654
Tables of Squares and Cubes and Boots of
Numbers ...... 665-69^
Tables of Logarithms and Antilogarithms . TOO-TCi
Tables of Exponential and Hyperbolic
Factions 708-7 IC
Tables of Hyperbolic Logarithms . . 711-715
Tables of Natural Sines, Tangents, etc. . 716-719
Tables of Logarithmic Sines, Tangents, etc. , 720-723
Index 725-742
■7
MACKROW AND WOOLLAED'S
POCKET BOOK
OF
FOEMULJ!, RULES, AND TABLES
FOB
NAVAL ARCHITECTS AND SHIP-BUILDERS.
SIGNS AHB SYKBOLS.
The following are some of the signs and symbols commonly
used in algebraical express! onii: —
= This is the sign of equality. It denotes that the quantities
so connected are equal to one another ; thus, 3 feet » I yard.
+ This is the sign of addition, and signifies plus or more ;
thus, 4 + 3 = 7.
^ This is the sign of subtraction, and signifies minus or less ;
thus, 4-3«l.
X This is the sign of multiplication, and signifies multiplied
by or into ; thus, 4 x 3 » 12.
-7- or / This is the sign of division, and signifies divided by ;
thus, 4^2»2 or 4/2=^2.
0 {} [] ^cs® signs are called brackets, and denote that the
quantities between them are to be treated as one quantity ; thus,
6{3(4 + 2)-6(3-2)}-5(18~6)«60.
This sign is called the bar or vinculum, and is sometimes
used instead of the brackets ; thus, 3(4 + 2) -6(3 -2) x 6 = 60.
Letters are often used to shorten or simplify a formula.
Thus, supposing we wish to express length x breadth x depth, we
might put the initial letteiis only, thus, / x } x <^, or, as is usual
when algebraical symbols are employed, leave out the sign x
between the factors and write the formula l.d,d.
When it is wished to express division in a simple form the
divisor is written under the dividend : thus, (a? + y) + « = ^^
z
2 "• SIGNS AN© 'symbols.
t y'y.f •• > TUese-ard 8igi|s ctf proportion; the sign : « is
to, the sign : : = as;* thuVl t '3" : : 5*: 9, lis to 3 as 3 is to 9.
< This sign denotes less than ; thus 2 < 4 signifies 2 is less
than 4,
> This sign denotes more than ; thus 4 > 2 signifies 4 is more
than 2.
'/ This sign signifies because.
/. This sign signifies therefore. J5». ; */ 9 is the sqnars of
3 /, 3 is the root of 9.
«^ This sign denotes difference, and is placed between two
quantities when it is not known which is the greater; thus
(a? f^ y) signifies the difference between .r and y.
^f ', These signs are used to express certain angles in
degrees, minutes, and seconds ; thus 25 degrees 4 minutes 21
se<;onds would be expressed 25° 4' 21".
Nate. — The two latter signs are often used to express feet and
inches; thus 2 feet 6 inches may be written 2' 6".
n/ This sign is called the radical sign, and placed before a
quantity indicates that some root of it is to be taken, and ft
small figure placed over the sign, called the exponent of the rootj
shows what root is to be extracted.
Thus :^a or Va means the square root of a.
^a „ cube „
/ya „ fourth „
^^ This denotes that the square root of a has to be taken
b
and divided by d.
This denotes that J has to be divided by the sc^^x^kie
root of a.
y
~ — - This denotes that the square root of a + h has to be
a + a
divided by the square root of a + d. It may also be written
thus, /±^,0T^^^^,
oc This is another sign of proportion. JSx,: accb; that is,
a varies as or is proportional to b,
00 This sign expresses infinity; that is, it denotes a qtumtit^r
greater than any finite quantity.
0 This sign denotes a quantity infinitely small, nought.
L This sign denotes an angle. Ex. : l^Ahc would be written,
the angle abc.
SIGNS AND SYMBOLS.
L This sign denotes a right angle.
X. This sig^ denotes a perpendicular ; as, abied, i.e. ab is
perpendicular to cd,
A This sign denotes a triangle ; thus, Aabo, i.e. the triangle
abo,
II This sign denotes parallel to. JEx. : ab i ed would be
written, ab is parallel to cd.
f or F These express a function ; as, a «=/(«) ; that is, a is
a function of x or depends on a;.
/ This is the sign of integration ; that is, it indicates that the
expression before which it is placed is to be integrated. When
the expression has to be integrated twice or three times the sign
is repeated (thus, //"^ J//) ; but if more than three times an index
is placed above it (thus,/").
JDoxd These are the signs of differentiation ; an index placed
above the sign (thus, d') indicates the result of the repetition
of the process denoted by that sign.
3 This sign (the Greek letter sigma) is used to denote that
the algebraical sum of a quantity is to be taken. It is commonly
used to indicate the sum of finite differences, just as the symbol/
is used for indefinitely small differences.
g This sign is used to denote the acceleration due to gravity
at any given latitude. Its value is about 32-2 in foot-second units
and 981 in C.G.S. units.
T The Greek letter pi is invariably used to denote 3*14159 ; that
is, the ratio borne by the diameter of a circle to its circumference.
« or € This letter is generally used to denote 2-7182^, which
is the base of hyperbolic or Napierian logarithms.
I n or It I termed * factorial n ', where n is a positive integer,
denotes the product of the series n (n~I) (n-2) . . . 2.1.
Thus, [^» 3 . 2 . 1 or 8 ; and [6^= 5.4.8.2. 1 = 120.
}£ denotes the midship section or ipiidship part of a vessel.
As the letters of the Greek alphabet are of constant recur-
rence in matbdmatidal fonnulsB it has been deemed advisable to
append the f (blowing table : —
ha
r7
E 6
Z f
Alpha.
Beta-
Gamma.
Delta.
Epsilon.
Zeta.
Eta.
Theta.
I (
K K
AX
M/i
N V
H i
O o
Iota.
Kappa.
Lambda.
Mu.
Nu.
Xi.
Oraicron.
Pi.
2 or f
Tt
T V
xx
Bho.
Sigma.
Tau.
Upsilon.
Phi.
Chi.
Psi.
Omega.
LOGARITHMS.
LOOABITHMS.
Definition. — ^The logarithm of a number to a given base is
the index of the power to which the base must be raised in
order to. become equal to the given number. Thus, if a* = N,
X is called the logarithm of N to base a.
The logarithms naturally occurring in analytical formula are
to the base e, which is equal to 2-718.. . . or to the sum of the
infinite series 1 + 1 + rs- + nr + rT + • • • ; tl^© values
Li LL LI
of the logarithms are obtained indirectly from the formula
log, (l-\-x)-x--^ "'"T''T'*' * ' * ^"^^ logarithms are
termed Napierian or hyperbolic logarithms ; their values are given
in the table on pp. 700-4.
When used to shorten arithmetical work, ' common
logarithms' are employed, having 10 as their base*
Note, — ^The logaritmn of 1 to any base is sero.
To Change the Base of a Logarithm.
KuLE. — ^To obtain the logarithm of a number to base h
from that to base a, multiply the latter logarithm by the
logarithm of a to base h, or, equally, divide it by the logarithm
of 6 to base a.
The logarithm of N to base a is denoted by loga K.
.*. logft N — loga N X log5 a = loga N -f logj 6.
Since log, 10 = 2-303 . . , = ;t5Tq » ^^^ hyperbolic logarithm
of a number is obtained by multiplying its common logarithm by
Note, — ^The integral part of a logarithm is termed its
characteristic, and the decimal part its mantissa.
To Find the Logarithm op a Number.
BuLE. — The oharacteristio is one less than the number of
digits in the integral part of the number ; when there is no
integral part, the characteristic is negative and is numerically
one more than the number of cyphers between the decimal
point and the first significant figure. In the latter case the
minus sign is placed over, instead of before, the characteristic.
The mantissa is invariably positive ; its value for numbers
of three or less significant figures is directly obtained from
the tables on pp. 7Q0-i ; for numbers having four significant
figures the tabnlar differences given in the colonuis on the
right are employed thas-^
Ex. 1.— Pind log of 42-63. Ex. 2.--Pind log of -7897.
log 42-60 » 1-6294 log •7890-1*8971
tab. diff. 8 =» 3 tab. diff. 7 = 4
log 42-63 = 1-6297 log -7897=111975
Note. — The tabular difference is placed under the extreme
right-hand fig^e or figures of the mantissa.
To FIin> THE AnTILOGABITHM, or the NuMBEB OOBBE8P02n)INO
TO ▲ GIVEN LoaABITHM.
BiTLE. — From tiie tables of antilogarithms, find the number
corresponding to the given logarithm, using the tabular
differences as before if four significant figures are required.
If the characteristic is positive^ the decimal point is so placed
that the number of digits to the left is one more than the
characteristic ; if nefi^ative, the number of ciphers betweem
the decimal point and the first significant figure is one less
than the characteristic. For tables v. pp. 705-8.
JBx. 1. — Find the number J3x. 2. — ^Find the numbei
whose Ic^arithm is 5*8178. whose logarithm is 3*1763
antilog 8170 « 6561 antilog 1760 » 1500
tab. diff. 8 = _J14 tab. diff. 3 = 1
antilog 8178 = 6675 antilog 1763 = 1501
Nilmber required is 657,500 to Number required is 001501.
lour significant figures.
To Multiply and Divide by LooAarrHMS.
BuLE. — ^Add together the logarithms of the numbers in the
numerator, and those of the numbers in the denominator ;
subtract the latter sum from the former. The antilogarithm
of the result is the number required.
IP t:. 1 * 2 17-63 2-052
Ex. : Evaluate r x ^_ x
35 -008176
log 2 « -3010 log 8 - .4771
log 17-63 = 1-2462 log 35 = 1-5441
log 2-052 = -3122 log -008176 = 3-9125
1-8594 1-9337
subtract 1-9337
antilog 1-9257 - 84-28-^the required result to
four significant figures.
Note.— 'It ia advisable to perform the operations of addi-
tion, multiplication, etc., on the mftnt'^sa and characteristio
separately.
Xkvolution asd Evolution bt Looarithms.
BuLE. — ^Multiply the logarithm of the number by the index
of the power to which it is to be raised. The antilogaritlim
of the result is the number required.
Ex, 1. — ^Find the cube and cube root of '9873.
log -9873 » i'9944 log -9873 ^ 1>9944
Multiply by 3 = - 3 + 2-9944
i-9832 Divided by 3)
Antilog 1-9832 = .9620 which 1-0981
is the cube of -9873. Antilog f -9981 = -9956, which
is the oabe coot of •9673.
Ex, 2.— Evaluate (20-4)1 ««.
log 20-4 t= 1-3036, say 1.310.
To multiply this by 1*83,
log 1-310 =1173
log . 1*83 = -2625
'3798
Antilog -3798 » 2-397 ; antilog 2-897 ^ 242-^, the re-
quired result.
AccuKACY OP Numerical Calculations.
In general, the accuracy of the result of a numerical
calculation is the same as that of the factor liable to the
greatest proportional error. Exceptional cases arise, viz.,
(a) when two nearly equal numbers are subtracted the per-
centage error in the result is usually greater than that' in
either of the numbers ; (6) when a large number of similar
quantities, such as the ordinates in a displaoesnent sheet, are
added, the individual errors of measurement tend to neutralize,
and the accuracy of the result is usually greater than that
of its component factors ; (c) the percentage error in the
nth power of a number is n times that of a number ; thusr
in the cube the error is trebled, but in the cube ro^t it is
divided by three. Subject to these qualifications a con-
siderable saving in the numerical labour of a calculation may
be effected by limiting the number of significant figures at
each stage to that appropriate to the accuracy of the result.
In calcula'oions affecting the weight, buoyancy, stability,
speed, strength, etc., of ships, a proportional error of ai:
least 0-1 per cent, i.e. one in a thousand, may generally be
expected ; three or, at most, four significant figures are suffi-
cient in such cases, any additional figures being meaningless
and redundant.
THIGONOMBTEIC&L BATI09. 7
The slide rale, whieh meohanieally perfonnB the operations
of maltiplication, division, evolution, etc., by the aid
(virtnally) of three-j^aoe Warithms, is nsnally snlRciently
accnrate lor the majority^ of mioh calculations ; tables of
logarithms^ trigonometrical fnnolions, eto., to four (or at
most five) places of decimals are sufficient to perform any
calculations in which rather greater acouraoj ia desired and
can be obtained.
TBI60N0KETET.
The. complemeni of an angfle is its defect from a right
angle ; thus if A denote the number of degrees contained in
ftny angle, 90® — A is the number of degrees contained in the
complement of that angle.
The iupplement of an angie Is its defect from two right
angles ; thus 180° — A is the number of degrees contained in
the supplement of that angle.
Trtgonometrical Batio8«
The trigonometrical ratios of an ans^Ie are
defined as follows :— Let bag (fig. 1) be any
angle ; take any point in either of the con-
taining sides and from it draw a perpen-
dicular to the other side ; let P be the point
in the side AC, and pm perpendicular to
Fia. 1, j^B J let A denote the angle bag. Then —
perpendicular PM
sine A = % *^ , = —
hypotenuse AP
eo-sine A =
base
AM
hypotenuse
AP
tangent A »=
perpendicular _
base
AH
co-tangent A =
base
AM
perpendicular
PM
secant A =
hypotenuse
: base
AP
AM
co-secant A »
hypotenuse _
perpendicular
AP
PM
versed sioe A t^
1 - COS A
CO- versed sine A »=
1 - sin A.
fhfm raiios depend only on tiie angle, and are independent
of the position of the point P.
8 waaaaxMSKT of akglbs.
Keasubehettt or Angles.
There are three modes of measuring^ ang^Ies, via.—
Ist. The sezagesimal or Engliah method.
2nd. The centesimal or Ftenoh method.
3rd. The circular measure.
The sexagesimal method and the circular measure only will
be dealt with here.
The Sexagesimal Method. — ^In this method a right angle is
supposed to be divided into 00 equal parts, each of which parts
is termed a degree ; each degree is diyided into 60 equal
parts called minutes, and each minute is divided into 60 equal
parts called seconds. One degree 16 minutes 15 seconds or
V 16' 15", is therefore equal, to 1 + ^ + gg^ or 1-271 degrees.
The Circular Measure. — ^The unit of circular measure is
an angle which is subtended at the centre of a circle by an
arc equal to the radius of that circle. It is called a radian.
Such an angle is equal to
The circular measare of an. angle is equal to a fraction
which has for its numerator the arc subtended by that angle
at the centre of any circle, and for its denozniuator the radius
of that circle.
Since the circumference of any circle is 2ir times the radius,
four right angles are equal to 2«' radians. Consequently one
right angle is equal to -^ radians.
Approximate values of » ajre 8*1416 and -^ and 7=-r>
7 Ho
To find the circular measure of any angle expreued in degrees^
minutes^ and seconds,
BULE. — Multiply the measure of the angle in degrees by «,
and divide by 180.
Ex. : Express 1° 16' 16" or 1-271® in circular measure.
1-271 X « ^^^^ .
— Yqq — " -0222 euro. meas.
To find the measure of any angle in degrees^ minutes, and
seconds, the circular measure being given,
BULE. — Multiply the circular measure of the angle by 180.
and divide by ir.
1
GSNESAL F0BMUL2. 9
Ex, 1. — ^Expresft in degrees, etc., an angle tht eixoolar measure
of which is "r~ « - «,/»
8 2wXl80 ^
dxx " ■^^*^-
Tables giving the circular measure of angles are on
pp. 639-41.
General Fobmulje.
sin" e + cos" tf = 1. sec'* tf = 1 + tan« tf.
cosee" • «= 1 + cot" tf.
sin (a + b) — sin A cos B + cos A sin B.
cos (a + b) = cos A COB B -^ Sin A sin B.
sin (a ~ b) a sin a cos b - cos A sin b.
cos (a - b) = cos A COS B + sin A sin B.
Sin A + sm B » 2 sm — r — cos — 5 —
.. ^« A + B A-B
COS A + COS B =■ 2 COS — 5 — COS — n —
« . A + B . A-B
sm A - sm B B 2 cos — 3— sm — —
. „, A + B. A-B
COS B - COS A "s 2 sin — ;r— sm — -—
ii 2
^ /. I «\ tan A + tan B
tan (a + b) = , — 7- 7— •—
> ' 1 - tan A tan B
. . tan A - tan B
tan (a - b) = i— r-T 1
^ ' 1 + tan A tan B
sin'^ « 2 sin A cos A. sin 8a « 3 tin A - 4 sin' A.
COS 2a = cos* A - sin" A. cos 3A = 4 cos' A - 3 cos A.
.A , * /I - cos A A , * /I + cos A
sm- = ± V 2 cos- = + V
If < « tan g, sm A = YTjl^i; cos A = ^^^ ; tan A = y^j.
And when a, b, c are the three angles of a triangle,
A + B + o « » radians or two right angles ;
and sin (a + b) ~ sin (it - c) =. sin c.
When A is any angle,
sin ( - a) s - sin A. cos ( - a) = cos A.
tan ( - a) = T. tan A.
sin (W* - a) = cos A. cos (90° - a) = sin A.
tan (90° - a) = cot A.
sin (90°+ a) = cos A. cos (90°+ a) = - sin a.
tan (90° + a) = - cot A.
sin (180° - a) = sin a. cos (180° - A) = - cos A.
tan (180°- a) = - tanA.
sin (180° + a) = - sin a. cos (180° + a) « - cos a.
tan (180° + a) = tan A.
10 FUNCTIONS^ PROPEBTIES OF TBIAK6LES.
The &lgebtftio formaln for the sine and oodne
smA - 2 - A - gj +gj - . . .
cos A - 2 ~ 2! 4! " • * • .
where A is in circular measure.
a"
Where A is smallj sin A » tan A == A ; cos A » 1 * — ;
sec A = 1 + ^
Tables of the trigonometrical functions are given on pp. 716-19.
Inverse Functions.
If sin o — X, then a = sin~^a;.
If cos a = y, then a -^ coar^.
And so on.
Note. — sin~^x is read * inverse sine a;\ etc. ■
LOQARITHMIC FlTNCTIONS.
The logarithms of the sines, cosines, etc., are denoted log
gin, log COS, etc., and th^r values are given on pp. 720-3.
For convenience the oharaeteristic is in each case increased by
the number 10. .
PBOIiERTIES OF TRIANOI^^.
Fig. 8.
Fia. 4.
Note. — The sides opposite the angles A, B, C respectively will
be denoted by the letters a, bf c. The angle BDA in figs. 2 and 3
is a right angle.
In fig. 2, where B and c are acute angles, we have —
sm B = — =
AB
C
AD
gm 0 s» -7- = ,
AC 0
sm B
fiin o
AD
c
AD
b
b^
0
THUNOLBS. 11
In fig. 3, where c is an obtuse angle, and in fig. 4, where c is
a right angle, the proof is similar.
And therefore in any triangle ^^ = ?^— « ^HLf .
a b c
Also COS A =
. A
Sin —
2
""V — Vo — * ''''^ 2" v ~~jr^'
A /(g-bVs-o) 2 ,
^2=V sis-a) > 8mA = j^^/,(,-«X«-JX*-^)
where 2g=^a + b + o.
a = b cos c + o cos B: tan ?J1? = *^ cot -.
2 ^-fo 2
6c .
Area of triangle =* -^ sin A =* Vs (« - «) (s - 6) (s - c)
Solution of Triangles.
Every triangle has six elements — three sides and three
angles. If any three of these be given (provided they be not
the three angles) the triangle can be completely determined.
Right-angled Triangles.
Let c be the right angle, and therefore c the hypotenuse.
(i.) Given hypotenuse (c) and one side (a).
b= \/c*-a\ tan B = -> and A = 90® - B.
a
(ii.) Given the two sides (a and b).
e= -v/a^n^ tan B= , and A = 90°-B.
a
(iii.) Given an angle (b) and one of the sides (a).
b=:a tan B, e^a sec B.
(iv.) Given an angle (b) and the hypotenuse (c).
« = c cos B, * = f» sin B, A *: 90° - B.
Any Triangles.
(i.) Given the three sides, a, h, and e.
tan4= . /5EME^, tan «= , /'^E^lSElS
C = 180<='-A-B,
where 2s-a + b + o,
12 MEASUREMENT OF HEIGHTS AND DISTANCES,
(ii.) Given two sides, 5 and <•, and the included angle A.
A
2*
tan — - — = r—— cot -.
2 b^o 2
?^"t? = 90°-^.
From and we can get B and c ; and a^b ■- — ,
2 2 sin B
(iii.) Given two sides, h and 0, and the angle B opposite to
one of them.
sin c = Y sin B. We thus obtain c ; and A « (180 — B — C).
b
Also
1^ sin A
a = o .
SUlB
As there are generally two angles between 0® and 180**
whose sine is ^ sin B, two values of G are often admissible, and
b
sometimes two triangles can be constructed.
(iv.) Given one side and two angles, a, B, and C.
1 oAo « « I. sin B sin c
A = 180° — B - c ; b = a — — ; <? = a - — .
sin A sm A
(v.) When the three angles only are given, the absolute
magnitude of the sides cannot be determined, but their ratios
■.a b c
are given by -; — = -; — = -;
sin A sm B sm c
Table giving the Signs
AND Values of the
Tbioonometeical Ratios
FOB Certain Angles.
Ratios
0«
Signs
80° Signs
46°
Signs
3
Signs
900
Signs
120°
Sine
0
+
1
3
+
1
+
+
1
+ ~
^8
2
Co-sine
1
+
^3
2
+
1
V2
+
1
2
+
0
—
1
2
Tangent
0
+
1
V3
+
1
+
V8
+
00
—
V3
Co-tangent
QO
+
V3
2
V3
2
+
1
+
1
V8
+
0
-
1
^8
Secant
Co-secant
1
00
+
+
+
+
V2
V2
+
+
2
2
V8
+
+
00
1
+
2
2
Ratios
Signs
+
136°
Signs
1C0°
Signs
180<>
0
Signs
270°
1 .
Signs
860°
0
Sine
1
V2
+
1
2
+
Conine
—
1
V2
—
V3
2
—
1
—
0
+
1
Tangent
—
1
—
1
-/3
+
0
+
00
—
0
Co-tangeut
—
1
—
+
00
+
0
—
00
Secant
—
V2
—
2
./a
—
1
—
00
+
1
lOo-aeoant
+
V2
1 -V "
t 1 I
-1- 00
—
1
-
00
HYPERBOLIC FUNCTIONS, PARABOLA.
13
Htpebbolic Functions.
The hyperbolio functions are nsed in oonneotion with thn
catenary ; they are six in number, and are represented by
affixing h to the symbols of the trigonometrioal functions.
They are determined by the following formulso :^
ginh^B s
ooshx =
2
a;' X*
tanhx —
seeha; =
Binh X
coshx
1
coshii;
IL IL
11. l±
coth X
cosh X
cosech X
sinh X tanh x
1
sinh x.
Note. — All formulsB connecting sin, oos, and tan can be
converted into the corresponding formula for sinh, cosh, tanh by
changing sin a; to V - 1 sinh a;, cos x to cosh a;, and tan x to
V - 1 tanh x ; thus cosh^ x - sinh^ a? =« 1 ; sech* x + tanh' x
« 1 ; etc.
The yalues of sinh a;, cosh a;, «*, and 6~* are given in the
taUes on pp. 708-10.
CTTBYES.
CONIC SECTIONS.
DEFZNinoN.->-The locus of a point which moves so that its
distance from a fixed point is always in a constant ratio to its
perpendicular distance from a fixed straight line is called a conio
section.
The fixed point is called the focus, the constant ratio the
eccentricity, and the fixed straight line the directrix.
The straight line passing through the focus and perpendicular
to the directrix is called the axis.
Parabola.
The conic section is called a parabola
when the eccentricity is equal to unity.
In fig. 5, F is the focus, ab the
directrix, AX tiie axis, o the intersection
of the curve with the axis, OY a line
perpendicular to AX, and P any point on
the curve ; then PQ = PF.
The e<|uation of the curve with OT
and ox as axes is
Fig. 5.
= 4 ox, where AO = OF = a.
A parabola may also be defined as the section of a cone cut
by a plane parallel to one of the slant sides.
14
ELLIPSE, HYPERBOLA.
Fio. 6.
Ellipse.
Th6 conic seddon is called an ellipse
when the eccentricity is less than unity.
In fig. 6 CD is the directrix, F the
focus, aa' the major axis, o the middle
point of aa', and bb' the minor axis, ^
down through 0*. perpendicular to the _
axis, and P any point on the curve so ^
PF
that — = the eccentricity e. The equa-
tion to the curve with oa', oy as axes is—
^ + 52 = 1.
Also OA = oa' = a ; ob «« ob' « 6 ; a* - 6" - a* •■ ; of =»
a
ae; od = -
e
An ellipse may also be defined as the intersection of a cone
by a plane passing through its slant sides, but not perpendicular
to the axis.
Hyperbola.
The conio section is called a hyperbola when the eccentricity
is greater than unity.
In fig. 7 ab is the directrix, F the focus, xx' the axis,
cc' the points where the curve intersects the axis, OY a line
Pig. 7.
y.
- t
O A
drawn through the middle point of co' perpendicular to the axis,
and p any point on either branch of the curve.
PF
Then « the eccentricity e,
PQ
Taking OX and OY as axes the equation to the curve is— >
1?" b^
1.
Also 00 = 00' s= a ; a* + 6' = a^ «■ ; of = a$ ; OA = -
GATENA&T.
15
If the Bides of a cone be prodaced beyond the rertez to as
to fonn a second cone with the saine axis as the first, and these
two coneS be eut by a plane, the section will be a hyperbola.
If & be made equal to a in the above equation, it becomes
x^-^ss a^, which is a rectangular hyperbola. By taming
Fia.8.
the axis through an angle of 45^ the equation becomes of the
form jcy as c' (fig. 8), where c^ ^^^a*
Oatenabt.
(See pp. 27 and 28 for meihod of constmction.)
If a uniform chain be freely suspended from two pointsi a
and B, the curve in which it will hang it termed a common
catenary ; the parameter 00 is equal to the length of a piece of
the chain whose weight is equal to the tension al the lowest
point 0 in the curve.
Fig. 9.
The directrix OX is a horizontal line drawn through the
extremity 0 of the parameter.
The tension at any point p in the carve is equal to the
weight of a piece of- the chain whose length is equal to the
ordinate FM.
Equations to the Catenary (see fig. 9).
Take ox (horizontal) and od (vertical through 0 the lowest
point) PM axes.
16 EQUATIONS TO THB CATENARY.
X — abscissa OM. y = ordinate PM. c = parameter 00.
8 = lengtli GP of chain, to = weight ol chain per linear unit run.
T « tension at P. 0 ~ angle to horizontal of chain at p.
e » base of hyperbolic logarithms = 2-718 ...
y = c eosh| = |(<j« + tf "*) = V (c" + »•).
« = c sinh- =-fe^ - 0 c\ =, ^ (y» _ c").
T = wc cosh— = wy. Dip (do in fig.) = c (cosh— _ i)
c c
tan ^ = sinh— = — sec 0 «= cosh — « ^
0 c c c
The valaes of the hyperbolic functions (stnh x, cosh x, etc.)
are tabulated tm pp. 708-10. Examples showing their
application to the eatoMiry are giren below.
A'pproximaie Equations for flat piece of chain, nearly
horizontal
^ = dip DC = y - c ; « = } span.
« = X +-S- ta= a? + "S" r
1 of _ , 2 »^
6 c^""^"^ 3 a;
Tension * =* i total weight x span -f sag at centre.
i^^o^.'^When the points of support are in the same
horizontal plane, the catenary is symmetrical aboat a vertical
line ^passing midway between them, and the preceding f ormulsB
can be directly employed to determine the particulars of
the curve.
Sx, 1. — ^A chain weighing 151b. per foot run is suspended
between two points at the same level and 100 feet apart. The
dip is observed to be 40 feet. Determine the length of chain,
the maximum tension, and the inclination a't the supports.
Dip = e (cosh- - l). Here dip = 40 ; a? = H^ = 60.
Hence 40 « c (cosh— - l). By trial, from the tables (p. 709),
c
50
c
c » 36 approximately.
Length of chain » 2s = 2c sinh— — 135 feet approximately.
* On sabatltnting 'pressure* for 'weight*, this is appUoable to a rope
or net under uniform pressure when 8a« is moderate.
EQUATIONS TO THB CATENABT.
17
Maximam tension occnrs at supports and is given by—
T = wc cosh — = 1150 lb.
c
SB
Angle at supports ar ^ ss seo~* cosh— s 62f.
c
Ex. 2. — ^Tlie chain in the preceding example is tightened
u-stil the length suspended is reduced to 120 feet. To determine
the dip—
« = c sinfa— . Here 5 == 60 ; sb s 50.
c
50
Hence 60 = c sinh — . By trial, from the table?, c = 47
approximately.
Dip = c (cosh - - l) = 29-2 ft.
Fx, 8. — If the chain in example 2 is tightened further
nntll the dip is reduced to 9 feet; determine the Icngtii, and
Uic tension.
Using the approximate formulsB for a flat chun-—
4 8^
Length = 2s = 2aj + r —
o x
Here x » 50, S = 9. Hence length ~ 102 feet approximately.
Tension = wc = w (| ^ "^'l) ~ ^'^^^ ^^•
FormulcB for the Catenary between two points not in the same
horizontal plane.
Take axes as before (fig. 10),
let A, B be the points of support
8 =s total length of chain ACB.
5 = vertical distance am between
A and B.
a = horizontal distance mb be-
tween A and B.
- X t? = height of A above axis ox
a
Vs' ^ 6» = 2c sinh
6 s 2 c sinh r- sinh
2c
2c
aj-_2tt
2c
8 » 2 c sinh -- cosh —^ —
2c 2c
n
« =s c cosh-
c
Ex. 1. — A chain of length 100 feet is suspended between
supports distant 60 feet horizontally and 60 feet vertically.
Determine the position of its lowest point, and the maximum
tension (weight 10 lb. per foot run).
18 CICLOIDAL CUfiTXS, ITOLUTES AND HTVOIUIVS.
Hen s - 100; a » 50 ; 6 » 60.
By trial from the above foimulaa, nsing the tables.
e = 14-2 ; u = 15-2 ; v = 23.
A ~- §£
Maximum tension (at b) = w c sinh = 820 lb.
c
Note. — ^If a is negative^ the lowest point of the catenary
occora outside the point of support A ; in that oase no pai%
of ihe chain is horizontal.
Cycloidal Cubves.
Dbfinitiox. — If a circle be mitde to roll without slip^ng^
on a straight linCi the locus of a point P (fig. 11) on its circum-
ference is the cycloid mnm', and that of any point Q inside
the circle is the trochoid BSB^.
The cycloid meets the straight line ab at a series of onspa
hm' • • • > corresponding to the positions when the point P is
vertically above the centre o of the rolling oircle ; on the
trochoid these become ' crests ' similar to those in the section
of a wAve-sorfaoe, which this curve is found closely to
resemble ; intermediate between M and m' is the ' trough ' s
which has a smaller curvature than the 'erest'. With co-
ordinates ox, oy, as shown, the equations to the cycloid
X =^ "R {9 - sin 9)t y =^ "R cos 9^ and to the trochoid is —
a; = R tf - r sin ^, y = r cos 9 ;
where 9 is the angle POL, R » PC, and r = QC.
Fio. 11.
The curve described by a point on the OLrcumferenee of
a oircle rollings on the exterior of another oircle is termed aa
epicycloid ; when rolling on the interior of the second circle
it is termed a hypocycloid.
E VOLUTES AND INVOLUTES. '
Definitiok. — ^If a curve be drawn passing through the
centres of curvature at various points at a curve, the new curve
is said to be the evolute of the original curve ; . conversely
the original curve is termed the involute of the derived curve.
The involute of a curve is also derived by wrapping^
a thread around the circumference of the curve ; the path
described by a point on the thread as it is wound or unwound
is the icvolute.
DIFFBRBNTIAL CALCULUS.
19
DIFFEBEHTIAL AND IVTSaBAL CAUULUS.
DlFFBBENTIAL CaLCULUS.
Definition. — ^A quantity is said to be a function of anoUicr
when its yaloe depends upon the yalue of tlio other.
Thus, x^, sin op, e^ are funotiona of « ; dry is a function
X and of y ; and so on. A function of x is denoted by /(x).
The differential coefficient of a function (y) with rospoot to
a yariable {x) is the rate of increase of y corresponding to an
indefinitely small increment of x. It is denoted by ^ or /'(:r).
Thus, the speed of a ship is the differential coefficient of
the distance travelled with respect to the time occupied.
Values of the differential coefficients with respect to ^ of
some functions of x are g^ven in the table below :—
V or fix)
ftin«
eo&x
tM.nx
eoix
sec as
coseoas
/(tt)
sin"'jc or-=^ooB"'al
tan"'«or— cot-*a8
Beo'^x or— cosec-'«
vers ■ 'a Of — covers '}x
dv , du
COB 9
— Mn«
see^dS
WIVUU mm
see X . tan x
— co$ecfle.cota;
dt^^dfKn)
d» du
_J
l+ac«
1
1
y or fix)
a*
u
V
tUnhx
eoaha
tanbas
coth'
■echflB
coseck X
log^aB
8lnh"'a5or )
log(a;+V«2+i);
cosh'^acor i
logix+^/x^-l)\
tanh'^apor )
coth"'a5or
ilog
^:t
a^orfix)
a' log^a
du_^ dv
dx ds^
ooshfli
Blnhx
8ech*ae
— cosech*«
— sech X . tanh x
— coBe^hac.eethx
X
1
1
1
dy
Note, — The differential coefficient of -^ with respect to sc is
denoted ^, and is termed the second differential coefficient of y.
Application to Cubves;
The equation of the tangent at a point («, y) on a curve is
T-y = |{x-.).
That of the normal to the curve is Z - a; + (T - y) ^ =s o.
20
ArrUCATJON TO CURVES, INTEGRAL CALCULUS.
Tlic angle ^ made by the tangent with the x axis, and the
perimeter s measored from any fixed point on a curve are
connected by —
dy . ^ dy dx /dx\^ /dy\^ -
The radius of curvature (p) at a point is given by^
dx^ da^
= +
d^
d^
When the curve is tangential to the x axis at the origin, then
p is equal to r- when x and y are very small.
Integral Calculus.
Di^FiNiTiON. — ^If y is the differential coefficient of a functioQ
z T«ith respect to x, then, conversely, z is termed the integral
o6 y with respect to x, being denoted Jy dx.
A tabU of integrals frequently required is appended.
dx
cos a;
cot a;
1
1
1
■ecic
1_
Aor
5-
doe
n+1
sinflB
log sin «
a
-sec * —
a a
Binh-J -
a
2 +¥**" a
—2 — sr"°"^ '«
logtan(f+f)
JLi SL±*
2a '^ o-«
*«i
jror
i-
d«
2.
••
sins
tan«
_1
1
1
Va2+^
cosec »
1
a*— a*
log X
— oos»
log 1600
1 * -1 *
~ tan ' "•
a a
vers 1-
cosh"*—
aeVa^+ae«^a«
2
log tan a*
2i»^
2a '^ «+a.
INTEORAL CALCULUS. 21
Definite Integrals. — On evalaating the integral \y »dx At
two constant quantities a and h are separately sabstituted for x^
and the former result substraoted from the latter, the difference
is termed a definite integral between limits a and h^ being denoted
y . dx. Thus f a:* . <to = -jjand / «' dx = — 7 — =16J.
The following definite integrals are of frequent ocourrenoe:—
cos 0 . c20 = 1
o 0
y sine.<?e = i y'
/ ^ Bin» e . <?e = / ^ COS" e,d$ = ^^ • ^ . . . ?
/
n n-2 • • -^^l^en
o 0
H is an odd integer.
41 — 1 91 — ^ ftlv
~ "»" ' n^ ' * '4*2*2 '^^^^ ** ^^ ""^ ®^®° integer.
w
2
sin «tf . cos *•• . do (m and n bemg integral)
0
_ (^-l)(w-3) . . . (»~l)(yt-~3) . . .
"" (m + n)(m + n- — 2)(w + 7i-4) . . .
n is odd.
when either vi or
__ (7n-l)(w-3) . . . (n — l)(w — 3) . . . w
"" (m + n)(m + n-2)(m+.» 4). . . ' "2 '^^^'^ "'^ ^^^
n are both even.
22
GEOMETRY.
Fm. 12.
FlQ. 14.
PBACTIOAIi OEOMSTST.
1. F^vm any given point in a ttraight line
to erect a perpendicular, (Fig. 12.)
On each side of the point a in the line from
which the perpendicular is to he erected set off
equal distanced Ab, Ac ; and from b and o as
centres, with any radius greater than Ab or Ac,
describe arcs cutting each other at d, d' ; a line
drawn through dd' will pass through the point
A, and aA will be perpendicular to he,
2. To erect a perpendicula/r at or near the
end of a line. (Fig. 13.)
With any convenient radios, and at any
distance from the given line AB, describe an
arc, as bag, cutting the given point in A;
through the centre of the circle N draw the
line BNC : a line drawn from the point A,
cutting the intersection at c, will be the
required perpendicular.
3. To divide a line into any number of equal parti, (Fig. 14.)
Let AB be the given straight
line to be divided into a number of
equal parts; through the points
A and B draw two parallel lines AG
and DB, forming any convenient
angle with ab ; upon AC and db set
off the number of equal parts re* A
quired, asA-l,l-2,<fec.,B-l, 1-2, 4c. j ^
join A and D, 1 and 3, 2 and 2, 3 and 1, c and B, cutting AB
in ay b, and c, which will thus be divided into four equal parts.
4. To find the length -of any given arc of a circle, (Fig. 16.)
With the radius Ad, equal to one- -piQ. 16.
fourth of the length of the chord of the
arc AB, and from A as a centra, cut the
arc in e ; also from B as a centre with
the same radius cut the chord in b;
draw the line cbf and twice the length
of the line cb is the length of the arc nearly.
6. To draw from or to the cir- Fig 16.
cumference of a circle Una tend-
ing towards the centre, when the
centre is inaccessible. (Fig. 16.)
Divide the given portion of
the circumference into the
desired number of parts; then
with any radius greater than the
distance of two parts, describe arcs cutting each other as Al, cl , &c. •
^^"^
'■^-'1
^ -^-V"
/^-i"^
ePOMKTIlT.
23
draw the lines Bl, o2, etc. . which will lead to the centre, as required.
To draw the end lines hr\ rr, irom B and E with the same radii
as before describe the arc r\ r, and with the radius Bl. from A as
centre, eut the former arcs at r', r ; lines then drawn from Ar'
and Tr will tend towards the centre, as required.
6. To describe an arc of a circle cf large radius, (Fig. 17.)
Fio. 17. Let A,B, C be the three points through
B which the arc is to be drawn ; join ba
atMl BO ; then construct a flat trian-
gular mould, having two of its edges
perfectly straight and ma.king with
each other an angle equal to abc.
Each of the edges should be a little
longer than the chord AC. In the points A, c fix pins ; and fix a
pencil to the mould at b, and move the mould so as to keep its
edges touching the pins at A and c, when the pencil will describe
the required arc.
7. Another method. (Fig. 18.)
Fi»l8. Draw the chord ado, and
draw bbf parallel to it; bisect
the chord in d and draw db per-
pendicular to AC; join ab and
BC; draw AB perpendicular to
AB and or perpendicular to BC ;
also draw An and en perpendicular to Ao ; divide AO and ef
into the same number of equal parts, and Aw, en into half that
number of equal parts ; join 1 and 1, 2 and 2, also B and s, s,
and B, and /, t ; through the points where they intersect
describe a curve, which will be the arc required.
8. To describe an ellipse, the major and minor axes being
given.
(Fig. 19.)
Fig. 19.
AB
be the major and CD the
axis, bisecting each other at
Let
minor
right angles in the centre E ; from c as
a centre, with ea as radius, describe arcs
cutting AB in p and f', which will be the
foci of the ellipse ; between s and f
set off any number of points, as 1, 2 (it
is advisable that these points should
be closer as they approach P).
From P and p', with radius Bl, describe the arcs G, G', G", G'".
From P and p', with radius Al, describe the arcs H, H', h", h'",
intersecting the arcs G, o', g", G"'in the points i, i, i, i, which will
be four points in the curve.
Then strike arcs from f, p' first with A2, then with b2 ;
these radii intersecting will give four more points. Proceed
in this way with all the points between E and p; the curve of
the ellipse must then be traced through these points by hand.
24
OBOlfETBT.
Fig. 20. 9. Another method. (Fig. 20.)
Let AB and CD be -the axes; find
P, p', the two foci as before ; join CP,
CP' ; make an endless thread equal in
length to the perimeter of the triangle
_j- CPP', and passing it round two draw-
^' ing-pins at P and p', draw it taut by
means of a pencil-point P, so as to
make a triangle ppf' equal in peri-
meter to CPP'; move the pencil-point
P along, keeping the thread taut, and the required curve will be
described.
10. Another method. (Fig. 21.) (Ap-
proximate.)
At 0, the intersection of the two dia-
meters, as a centre, with a radius equal
to the difference of the semi-diameters,
describe the arc ab^ and from & as a
centre with half the chord boa describe
the arc od ; from o as centre with the
distance od cut the diameters in dr, dt ;
draw the lines rs^ rs, fo, tSy then from r and t describe the arcs
SDS, scs ; also from d and d describe the smaller arcs HAS, SBS,
which will complete the ellipse required.
This method is applicable when the minor axis is at least
J the major.
11. To draw a tangent and a normal to an ellipse at
any point, (Fig. 22.)
Let 0 be the point; from F, f', the
two foci of the ellipse, draw straight
lines through G and produce them ;
bisect the angle made by the pro-
duced parts, by GH, then GU is normal
to the curve ; at G bisect the angle
formed by FG and f'g produced, by
IJ^ then ij will be the tangent to the
curve at G.
12. 2^0 describe an eUiptio arc, the spin and height being
given. (Fig. 23.) (Approximate.)
Bisect with a line at right angles the
chord or span AB ; erect the perpendicular
AQ, and draw the line qd equal and parallel
to AC ; bisect AC in c, and aq in 91 ; make
CL equal to CD, and draw the line Loq ;
draw also the line «SD, and bisect SD with
a line KG at right angles to it, and meeting
the line LD in G ; draw the line gkq, and
make cp equal to CK, and draw the line
Gp2 ; then from G as centre with the radius
Fio. 22.
Fig. 28.
GEOM ETUT.
25
GD describe the arc sd2, and from K and p as ccntroa with the
radius ak describe the arcs as and 2b^ which complete the arc,
as required. This method gives good resulii for ellipse? of
all proportions.
13. Another method, (Fig. 24.)
Bisect the major axis ab, and fix at
right angles to it a straight guide, as
bo ; prepare of any material a rod or
staff, def ; at / fix a pencil or tracer so
that df is one-half ab, and at E fix
a pin so that ef is one-half the minor
axis ; move the staff, keeping its end
d to the guide, and the pin e to ab,
and the tracer will describe a half of the arc required.
14. To describe a parabolic aro ichen its height and base
are given. (Fig. 25.)
Let CD be the base and AB the height; set them off as
shown in the figure, so that CB = CD, and complete the rect-
angle CDFE ; divide EC and fd into
any number of equal parts, say three,
at a, by c, and d ; join Aa, a&, A0,
Ad ; divide ae, af, bc, and bd into
the same number of equal parts at
e,g, &, w,/, h, I, n; join ef, gh, U,
mn, cutting A J, Aa^ Ac, Ad at q, p^ r,
and *. A line drawn through c qp A r s D will ce the parabola
required.
15. Another method^ . when the
directrix and focus are given. (Fig.
26.)
Place a straight-edge to the direc-
trix AB, and apply to it a square CDE ;
to the end c of the square fasten a
thread, and pin the other end to s the
focus, making the length of the thread
equal to CE ; slide the square along the
sti-aight-edge, holding the thread taut
against the edge of the square by a
pencil P, by which the curve is de-
scribed.
16. To describe a hyperbola, the diameter, abscissa, amd d&ktble
ordinate being given, (Fig. 27.)
Let AB be the diameter, bc its abscissa, and DB its double
ordinate ; then through B draw gf parallel and equal to DB ;
draw also dg and ef parallel to the abscissa bc.
Fig. 26.
26
GEOMETRY.
Divide DC and CB into the same number of equal parts, as
1,2, &c., and from the points of division draw lines meeting in A.
Divide GD and EF each into the same number of parts as DC
or CE, and from the points of division 1', 2', &c., draw lines
meeting in B.
Pic 27.
4
//Mll\\\
o ///ft4\\\ r
/MM
7 / M M
I I I \ \ \
Fig
The points of intersection of the lines 1 and 1', 2 and 2', &c.,
thus found, will be points in the required curve.
17. Another method^ when the foot a/n,d a paint on the owroe
are given. (Fig. 28.)
A hyperbola is a curve such that the difference of the
distances of any point in the curve from the two foci is equal to
the transverse axis ; and this pro-
perty suggests the following me-
chanical construction : —
Ijet P (fig. 28) be any point on
the curve, and F and P, its foci;
join PF, and produce it, making
XX' the axis ; draw PM perpendicu-
lar to XX', and produce it to Q,
making mq equal to PM ; bisect FFi
at c, and produce PC, QC, to CP*,
cq', making them equal to CP or
CQ. The P, P', Q, q' are four points on the curve. From one of
tbem, say P, stretch two pieces of string PF and pf„ fastening
them to the paper at F and Fj, and simply knotting them at P ;
slip a small bead over them at p, and taking hold of P and
keeping the thread taut, slide the bead along the threads, and
the bead will describe the curve as far as the axis. Repeat this
process at p', Q, and 9'.
Let 0)t, or be the '
oiymptotes, and I Ihe
given point. Drsw rM
parallel to Ot, and F3
psT<el to OK ; Bet off an^
ordinateafgcnerallj eqni-
diitant lor conrenience)
II, 22, 33, 44, 55, 66, and
join 0 to the intenections
of these ordinates with
PS, cutting PM at 1', 2',
3', eto.; through 1' draw I'l parallel to OX, onttinj 11 in
throosh 2', 2'ii cutting 22 in u, and so on (or m, iv, v, and vi ;
then P, I, II, III, etc., are points on tho required curve.
19. Siveti five pointt on ani/ conic to obtata any number of
additional pointi derired. (Fig, 30.)
Danota the given pointi b; a, i. o, d, b. Draw anj lino
Pit' tbrongh a, od which it is required to find a sixth point.
Fid. W.
Join IB, Dl, cutting at x, and CD outting pap' at T. Join BC,
cutting K1 in z; juin ez, cutting faf' in f. f ia the required
point on the curve ; bj drawing additional lines through A,
any Dumber of points on tho curve maj be obtained,
20. To eonttrvct a eatenary approximately. (Fig. 30a.)
Let 1 be the lowest point in the curve, oti its pirametcr,
and ox ita dirootrjx. Make AE ei^ual to oe; then witli a as
28
OEQMETBY.
centre and ae as radius describe the small arc bp. Join fa
and produce it to U and to b, making bf equal to vu; then
with B as centre and bf as radius describe the small arc fo.
Fia. 80a.
Join BO and produce it to n and to c, making: oa equal to
ON ; then with 0 as centre and ca as radius describe the small
arc on. Proceed in a similar manner till the curve is of the
required length.
21. To obtain by measurement the length of any direct
line, though intercepted by some material object, (Fig. 31.) ,
Suppose the distance between ^^- ^^'
A and B is required, but the
straight line is intercepted by
the object G. On the point d,
with any convenient radius,
describe the arc eo\ and make
the arc twice the radius do in
length ; through e' draw the
line do'e, and on o describe another arc -jf equal in length to
the radius do ; draw the line efr equal to efd ; from r describe
the arc g'g^ equal in length to twice tha radius rg\ oontinue the
line through r^ to B : then A and B will make a right line,
and de or er will equal the distance between dr, by which the
distance between ab is obtained, as required.
22. To ascertain the distance geomeifically of an in-
accessible objeot on a level plane, (Fig. 32.)
Let it be required to find the distance between a and
B, A being inaooessible. Produce ab to any point d, and
GSOHBTRT.
29
FlO. 89.
Fio. 88.
Fio. 84.
Mseef n> in 0 ; throti^h d draw
Da, making any angle with DA, and
take DC and db respectively and set
them off on Da ad J>b and De ; join
Be, cb, and Aft ; throoffh b, the inter-
section of Be and cb, draw DEF meet- •
ing A& in F ; join BF and prodnoe it ^
tm it meets Da in a : then ab will be
eqoal to ab, the distance required.
i 23. Another method, (Fig. 33.)
Produce ab to any point d ; draw the
line J)d at any angle to the line ab * bisect
the line Dd in c, through which draw the
line Bb, and make cb equal to BC; join ao
and db, and produce them till they meet
at a : then ba will equal ba, the distance
required.
24. To measure the distance between two
objects, both being inaccessible, (Fig. 34.)
Let it be required to find the distanoo
between the points a and B, both being in-
accessible. From any point c draw any line
Ce, and bisect it in d ; produce ac and Be,
and prolong them to B ana F ; take the point
E in the prolongation of ko, and draw thei
line ED^, making De equal to de.
In like manner take the point F in the
prolongation of Be, and make d/ equal to df ;
produce ad and ec till they meet in a, and
also produce bd and /d till they meet In b:
then the distance between the points a and
b equals the distance between the inaccessible
points A and B.
25. To cut a beam of the strongest section
from any round piece of timber, (Fig. 35.)
Divide any diameter CB of the circle into
three equal parts ; from d or e, the two points
of division in CB, erect a perpendicular cutting
the circumference of the circle in D or a ;
draw CD And db, also ac equal to db and ab
equal to CD : the rectangle abcd will be the
section of the beam required.
Note, — ^To get the stiff est beam make Cd = i cb and
proceed as before.
Fia. 35.
GEOMETBT.
26. Ta detcribt the proper foi-m
of a flat plata by which to efinalmet
any giren fvuiium of a cone. (Fig, 36.)
Let ABCD represent the reqnired
frostam of a cone ; continae tJie linea
AC and Bl) till Uiey meet in E ; from B
tta « oentre, irith ed m radjaa, desoribe
the are du, and from the aame oentM,
as radius, deacribe tho arc ci;
equal in length to twlee A OB,
the ciToamfereuM of tile base
of the cone : draw the line EI : then
DDHI is the form of the pla(« required.
Let knm> represent the reqnired
frustum of tho cone ; oontinne the lines
ic and BD till the; meet at b g divide
tho circumfcrenca of the base of the
cone into any number of equal parte —
t&j 12— in the points 1, 2, 3, etc. )
join the projections of tl^eae pointa to
E ; next find the development of the
s base of tJie ooae, as shown in the pre-
ceding example, and on it set o& the
' same number of point* — vii. 12— and
draw lines from them to E ; project
the points of intersection of each of the
linea e1, e2, £3, etc., with the line cd^
horizontally on to either of the slant
sides (say ed) ; then from £ as oentre
measure the distance down along eb to
the foot of each projection and set it
o9 on the corrGsponding numbers
(measuring from E) in the develop-
ment : a line drawn through these
tho curve of the top of the eection, aa
mBMMJSlUI*
ox
aad draw lines ihrongli those points on the cylinder parallel
bo ic and bd ; draw a line efq eqoal in length to the circom-
fctrence of the cylinder, and divide it infco the same number
ol parte ; on each point of division set up perpendicalars to
Fig. 88.
if h !» U jjal ! !
A. 1 I 1/*
a a ♦ « Q
»-i'%
it, making EH c^nd OK equal in length to bd, and make Fi equal
in length to ac; then take the height at 1 and set it up on
the corresponding number on each side of fi, and so on with
each number : a line traced through the points thus obtained
will be the curve of the required development.
Fio. 41.
29.. To find the approxinuxte development of any given
portion of a segment of a
sphere, (Figs. 39, 40, and Fio. 89.
41.)
Let ABC (fig. 39) be the
middle section of the seg-
ment, and CFQ in the plan
(fig. 40) the 3»ortion to* be
developed; bisect ab (fig. 39)
in E, and set up the perpen-
dicular EC ) divide the arc
AC into any given number
of equal parts— say, four —
and through the points of
division draw the lines 1 1,
2 2, etc., parallel to ab ; on
the plan (fig. 40) from c
as a centre, with the radius 1 1 taken from fig, 39, draw tlio
arcs 1 1 cutting FC and CQ in 1 and 1, and so on with 2 2 and
3 3; draw any line bc (fig. 41), making it equal in length to
BC (fig. 39), and on it set off the same number of equal parts;
at each point of division draw lines perpendicular to BO, and
number them the same as on fig. 39.
Fio. 40.
82
GEOMETRY.
Measure the length of the arc 1 1 in fig. 40, and set off
half of it on each side of BC on line 1 1, and so on with each
arc, indading TQ ; a line traced through the points thns
obtained will give the curyo of the sides of the given portion
of the segment when it is developed. To describe the curve at
the bottom of the figure, take one-fourth of the circumference
of the base as a radius, and from F and o as centres describe
arcs cutting bo in s ; then from s as centre, with the same
radius, describe the arc fbq, which w;;U be Uie curve of the
bottom of the figure, as required.
Should the top of the figure be cut off at the line 1 1
(fig. 39), from 8 as a centre in fig. 41 describe the are 1h1,
which will be the curve of the top of the figure, as required.
30. To find the approximate development of a%y given
portion of a paraboloid, (Figs. 42, 43, and 44.)
Fio 41.
The development is found
in the same manner as that
of a portion of a segment
of a sphere, as described in
the last example (No. 29),
with but one exception —
that is, the length of the
, radius for describing the
^FiQ. 43. bottom curve of the figure,
^ which instead of being equal
to one-fourth of the circum-
ference, as in example
No. 29, is equal to one-half
the length of the arc acb
(fig. 42) in this example.
31. 70 find the development of an entablature plate.
Let fig. 45 be the side elevation, fig. 46 the front elevation,
fig. 47 the plan, and fig. 48 the development of the figure ;
divide adg (fig. 46) into ei^ht equal parts, and from the
points of intersection draw lines parallel to ABC, cutting CD
(fig. 45) in the points 1, 2, etc.; on BD (fig. 45) erect a perpen-
dicular EC, and from the points on CD draw lines parallel to
BED. From fig. 46 take the points 1, 2, etc., on abc and set
them off on afc (fig. 47), and erect perpendiculars from apo at
these points. From c (fig. 45) alon^ CE measure the points c, 1,
c, 2, etc., and set them off on their corresponding lines from
AFC in fig. 47 ; draw a line through those points, then measure
it with its divisions and set it off in fig. 48 as a straight line
GEOMETRY.
88
««^'- *''*** <^.® P«?1^ 0^ division erect perpendiculars, con-
fanning them either side of the Une aec ; zJearnre tiie distant
1, 1 ; 2, 2, etc. (%. 45), on either side of ce, and set them ^
Fig. 45.
af^ F-
from AEO (^. 48) on their corresponding lines, and on their
respective sides of aec. These will give the development.
32. To describe a cycloid, the generating circle beina aiven.
(Fig. 49.) "^ •"
Let B% bo the generating circle ; draw a line abc, equal to
the circnmferenoe of the generating circle, by dividing the
circle into any number of given p^s, as 1, 2, 3, ebo., and
setting off half that number of parts on each side of B; draw
lines from the intersections of the circle 1, 2, 3, eto., 7, 8, 9,
34
GEOMETRY
etc., parallel to AO ; set of! one division of the circle outwards
on the first lines 5 and 7, two divisions on the next lines 4
and 8, then three on the next, and so on : then the intersection
of those points on the lines 1, 2, 3, etc., will be points in the
curve.
33. To draw a trochoid or wave-form, the height and length
being given, (Fig. 50.)
Draw AB equal to the length ; with centre c on ab produced
describe a circle whose diameter is equal to the height.
Divide the circumference into a convenient number (say 12)
of equal parts 0, 1, 2, 3, . . . , CO being vertical. Divide ab
into the same number of equal parts f, o, H, . . . From
A, F, 0, . . . B, draw aa, f/, og^ . . . B&, parallel and equal
c 0, 0 1, c 2, ... 0 0, respectively. A curve drawn through
the points a, f, g, . , , b is the required trochoid.
Fio. 60.
i4. To describe an epicycloid, the generating eireU and the
directing circle being given. (Fig. 61.)
Let BD be the generating circle, and ab the directing circle ;
divide the generating circle into any number of equal parts
(say 10) as 1, 2, 3, etc., and set off the same distances round
the directing circle ; draw radial lines from a through tiiese last
points, and produce them to an arc drawn with a as centre and
AB as radius, as shown by cccc and c'c'c'c' on the diagram ; draw
35
concentric bme also throagfa ail the paintg on the gecerating
circle, with k aa centre; then taking c, c, c, c and c',e', e', e' u
centres, and BB as radius, describe arcs cutting the conceotrio
circles at 1', 2', etc. : the points thug foand will be points in tbs
required curve.
35. To deteribe a hypo- rio.ei.
cycloid, the gtnemting
mrcte <tnd tht direetinif
circle being given. (F^.
52.)
Proceed a» ii
cfcloid, the
Ijeing tJkat the conrtniction
llnsg are drawn within the
direoting circle iustead of
outside, Bi in the epicy'
cloid. A
36. To deicribs the involute of a circle. (Pig. 53.)
Let AD ba the given circle, which divide iuto any eqnol
number of parts (say 12) u I, 2, 3, etc. ; from the centre
draw radii to theao points ; then draw linos (tangents) at right
angles to these radii. On iba tangent to radius No. 1 set off
from the cirole a distance eqnal to one part, and on each of the
1 the epl-
tangents iet off the number of parts correaponding to the
nnmb«f of iti radius, so that No. 12 has twefve divisions set
off on it (that is, equal to the oironmferenoe of the circle) ;
a line traced throogh the ends of these lines will be the curve
reqaired.
86
MENSURATION OF AREAS AND PERIMETERS.
37. To find the dip of the horizon. (Fig. 53a.)
r^t o denote the centre of the earth, pb a
tangent from the eye of an observer looking
from a height ap to the earth*8 surface at b:
then B is a point on the horizon: draw PC at
right angles to PC ; then the angle bpo is
called the dip of the horizon.
Let op cut the earth's snrface at A, and let
the angle bpc be denoted by 9 ; with distances in miles,
and
AP
9 in
Fio. 64.
PB = \^ 2 . AP . AD approximately = V 8,000 x
degrees = 1'28 Vap.
MENSUBAtlOir.
I. Mensuratiok of Areas and Periicbt£B8.
1. To find the area of any parallelogram, (Fig. 54.)
Bdle. — Multiply th© length by
the perpendicular height, and the
product will be the area. Thus, if
A = the area, a = the length, and
b «s the perpendicular height, tiieo
<■— <^ * A = ah,
2. To find the area of a trapezoid, (Fig. 55.)
BuLC.—Multiply the sum of the parallel
sides by the perpendicular distance between
them; half the product will be the area. Thus,
if A = the area, b and a = the parallel sides,
and 0 = the perpendicular distance between
Fia. 55.
^ tiiem, then
^ _ (g + h)c
8. To find the area Of any triangle, (Fig. 56.)
Fig. 66.
BuLE. — Multiply the base by the per-
pendicular height ; half the product will
be the area. Thus, if a = the area, b == the
base, and <2=x:the perpendicular height,
then A = —
4. Or, if the lengths of the 3 sides a, 6, and e are given, then
A = Vsis-a) (s-6) (s-c) where 2s =s a + 6 + c.
5. To find the area of any regular polygon. (Fig. 66a.)
BuLE. — Multiply the sum of its sides by a
perpendicular drawn from the centre of the
polygon to one of its sides ; half the produet will
be the area. Thus if A =; the area, c s the number
of sides, b = the length of one side, and a s the
-^ .^...; perpendicular, then A = —
MENSU&AXION OF ABRAS AND PERIMETERS.
87
Table of Regulab Polygons.
A a* the angle contained between aiiy two sides.
K = the radius of the circumscribed circle.
r = the radius of the inscribed circle.
8= the side of the polygon.
'.1
3
Name
it
B=8X
r=sx
8=RX
sssrx
A«a=8;
Trigon
60<»
•57735
•28868 1-73206 3-464101 ^43301
4
Tetragon ,
90°
'70711
•50000ll-4142ll2-00000 100000
5
Pentagon .
108°
•85065
•68819 l-17557|l-45309 1-72048
6
Hexagon .
120°
1-00000
•86603
1-00000115470 2-59808]
7
Heptj^gon .
128$°
1-15238
103826
•86777
-96316 3-63391
8
Octagon .
135° 1-30656
1-20711
•76537
•82843 4-82843
9
Nonagon .
140° 1-4^19(>
1-37374
•68404
•72794 6-18182
10
Decagon .
144° 1-61803
1-53884
-61803
•64984, 7-69421
11
Undecagon
147^° 1-77473
1-70284
-56347
•58725! 9^36564
12
Duodecagon
150° '1-93185 1-86603
-51764
•63690 11-19615
6. To find the area of a quadrilateral.
Rule. — Multiply the diagonal d by the
sum of the two perpendiculars a and h let
fail upon it from the opposite angles ; half
the product will be the area. Thus if A =
the area, a and h = the perpendiculars, and
d = the diagonal* then
. (er + 5) rf
(Fig. 57.)
Fig. 57.
7. To find the eiToumferenee of a cvrele., the diameter being
given ; or to find the diameter of a eitrde^ the eircumference being
given,.
KULE. — Multiply the diameter by 3*1416, the product will
be the circumference; or divide the circumference by 3-1416,
the quotient will be the diameter.
8. To find the length of any arc of a circle, (Fig. 68.)
Rule (I). — ^From eight times the chord
of half the arc subtract the- ol^ord of the
whole arc : on€- third of the remainder will
be the length of the arc, nearly. Thus if
L = length of the arc, c ss chord of the
whole aro, e =3a chord of half the aroy then
8c - c
Fig. 58.
88
MENSURATION OF AREAS AND PERIMBTERS.
Rule (II). — ^The radius being known, mnltiplv tog^ether
the number of degrees in thB arc, the radius, and the number
'01745 ; the product will be the length of the arc. Thus if
h s= length of the «rc, d = diegrees in the arc, B = radius,
then
L = D X R X -01745.
BuLE (III). — (Applicable to any fairly flat curve.) Add
to the chord eight-thirds the square of the maximum height
(or versed sine) divided by the chord. The sum is the length
of the curve, very nearly. Thus if c = chord, and v =*
8 y'
greatest height of arc above chord, length = o -|- - —
8 O
9. To find the diameter of a eirole, the
chord and versed sine being given.
(Fig. 69.)
RtJLE. — Divide the square of half the
chord by the versed sine, to the quotient
add the versed sine, and the sum will be
the diameter. Thus if D = the diameter,
c = the chord, and v = the versed sine,
then
2
Fia. 59.
-m
Pig. C3.
" 10. To find the length of any ordinate of a segment of a
circle. (Fig. 60.)
BuLE. — ^Find the radius of the arc of
the segment (if not given) by the pre-
^*^v, ceding formula; and from the square root
^ \ of the difference of the squares of the
-t.x-=> ^ radius and distance of the ordinate frola
the centre of the segment, sut>tract the
radius ; and to the result add the height of the segment, and
the sum will be the required ordinate. Thus if R = the radius,
X = the distance of the ordinate from the centre of the
segment, v s= the height of tiie segment, and T =s the required
ordinate, then
ysVr^-x'^-R+v.
11. ^o find the area of a circle,
BuLE (I). — ^Multiply the square of the diameter by *7854,
and the product will equal the area, nearly. Thus if a = the
area, D = the diameter, then a =» D^ X '7854.
Bulb (II). — ^Multiply the square of the circumferenoo by
'07958, ai^ the product will be the area. Thus if a «= area,
c ^= circumferenoo, then a = c* X '07958.
MENSURATION" OF SUPERFICIES.
89
Table of Properties of the Circle.
» = 3-14159266358979323846
2 « 1-57079632679439661933
IT
i"
ir
v^2 =- 1-41421356237309504880
n/J « -70710678118664762440
2>/ir «3-64490770181103206460
r-« -785398163397448309620 /I , .«oo-«
4 oovi^u^ 2^ i = 1-12837916709561257390
l^ •52359877559829887308 iv{« -88622692545275801365
n/2 «4-44288293815836624702Wi« .07052369794346963587
VA = 2-2214414690791831 23S1 1 ^ «•
2ir =6-28318530717958647693
2
- = -63661977236758134308
Vi =2-22144146907918312351
n/t =1-77245385090551602730
^1 -= -66418958354775628696'
ISO = '^^7^5
- = -3183
IT
«« = 9*870
^^?- = 67-3
IT
60
2ir
9-5493
In the foUowing formulae a = area, c = circumference, D = diameter,
s = side of square.
•=Dxir = ltx27r= a/a x 2 v^'ir
Circumference
Diameter
Radius
Area
Side of equal square
Side of inscribed square
Diameter of equal circle
= Cx-= \/Ax2 /-
» ^^ IT
= B«yir=D'x^ = J IIC
«Ex >/» = Dx J>/»«Cx J^/-
=Dx>/J«Cxi>/f=: v/AX^I
=sx2yi
Diameter of circumscribing circlets x \/2
Circumference of circumscribing circle«s x »>/2
Circumference of equal circle —s x 2\/ir
2
Area of inscribed square = A x -
40 MENSURATION OF AREAS AND PERIMETERS.
12. To find the ar$a of q, sector of a circU,
Rule (I.) — Multiply the length of the arc hy the mdias of the
sector, and half the product will equal the area.
A = area of sector^ L = length of arc, B » radios.
Rule (II). — ]!!l[ultiply the number of degrees in the arc
by the area of the circle, and 9)77 of the product will equal
the area. Thus, if a =; area, d =3 number of degrees in the arc
a = area of circle, then
A = i^
360
13. To find the area of the segment of a circle.
Rule (I). — Find the area of a sector having the same arc
as the segment ; then deduct the area of the triangle con-
tained between the chord of the segment and the radii of the
sector. The remainder will be the area of the segment. Thus,
if A =s the area of the segment, c » the chord, and H ^ the
height, then
Rule (II). — To two-thirds of the product of the chord and
height of the segment, add the cube of the height divided by
twice the chord ; the sum will be the area of the segment^
nearly. Thus,
/2cH , h'\
PiQ Qi \i. To find the area of a circular zone,
...:.,; (Fig. 61.)
«' ^N^ Rule.— Find the area of the circle of
which the zone forms a part, and from it
subtract the sum of the two segments of the
circle formed by the zone ; the remainder
will be the area. Thus, if A = area of the
\^ ^ / zone, a and b = the area of the two seg-
ments respectively, and c = area of the
circle, then A = c — (fl + ft).
15. To fmd the area of a flat circular ring. (Fig. 62.)
Rule. — Multiply the sum of the inside and outside
diameters by their difference, and the result by '7854 ;
MENSURATION OP AREAS AND PERIMETERS.
41
the product last obtained will be the area.
Thus, 11 A = a.rea ol ring, d s=s diameter of
large oiiele, and d ^ diameter of small
circle, then
A= -78541(0 + ^) (D-d)}
16. To find the area of an ellipse. (J^ig, 63.)
BuLE. — Maltiply together the trans- Fia. 68.
verse and conjaga^e diameters ol theeUipse^
and the result by ^ ^' *7854 ; the product
will be the area. Thus, if a = area of
ellipse, a =■ the conjugate diameter, and
b =s the transverse diameter, then
A = •a6 X •7854.
17. To find the area bounded by a rectangular hyperbola,
two ordinaie9 and the base, (Fig. 8, p. 15.)
BULE. — Multiply the product of either ordinate and the corre-
sponding abscissa by the hyperbolic logarithm of the ratio between
the two abscisssB. Thus the area o| abcd is equal to ab x OB
1^ 00
*«0B
18. To find the area bounded by a cycloid and the line
joining the cusps, (Fig. 11, p. 18.)
BULE. — Multiply the area of its generating circle by 3; or
maltiply the product of its length and height by }.
19. To fi>nd the area bounded by a trochoid and a line
joining the crests, (Fig. 50, p. 34.)
Bulk. — II r be the radius of the rolling circle (or the length
divided by 2ir)y and r the radius of the tracing. <»rele (or one-half
the height from crest to trough), the required area is equal to t r
(2r +r).
Note. — The area between the curve and the line joining the
troughs is «* r (2 B ~ r) .
20. To find the area of a segment of a parabola.
BuLE. — Multiply the base by } of the maziwum height.
42
MENSURATION OF AREAS AND PERIMETERS.
21. To find a general expression for the area of any plane
curve*
Using cartesian co-ordinateS| the area intercepted between the
curve, the x axis, and two ordinates distant a and h from the origin,
is equal to the definite integral / y . dx,
J a
Using polar co-ordinates, the area intercepted between the
curve and two radial lines making angles a and fi with ox, is
equal to ^ I r^ , d9.
to} fr^.
Fio. 64.
Remark.— A, curve whose equation is given by
y=a-hbx-\-cx^+dx*+ . . . Kar"
is said to be a parabolic curve of the n^ order. Thus a parabolic
of the first .order is a straight line ; of the second order a common
parabola. Bules for the area of a parabola of any order are
applicable also to curves of a lower order, but not in general to
those of a higher order.
22. To find the area of a parabola of the third order when
three ordinates are given, . (Fig. 64.)
BuLE. — To the sum of the two endmost
ordinates add four times the intermediate
ordinate ; multiply the final sum by i of the
common interval between the ordinates. The
^^ result will be the area. Thus, if yu y2$ and yt
be the ordinates, ^x the common interval, and
\ydx the area, then
= -2-(yi + 4yj+yt).
Note, — This is termed Simpson's first rule.
23. to find the area of a parabola of the third order when
four ordinates are given,
Pio. es. IUJZ4E. — To the sum of the two end-
most ordinates add three times the
intermediate ordinates ; multiply the
final sum by J of the common interval
yy y^ y^ P^ between the ordinates : the result will be
the area. Thus, if (ydossthe area> then
^1/^= -g-Cyi +3y«+ 8^8+^4).
Note. — This is termed Simpson's second rule.
\ydx
MENSURATICnr OF AREAS AND PEEIMSTBB8. 48
Tablb skowthq thb Mcltiplibrs fob THB FOBEOOINO
and some otheb bulbs.
Vu |/9f Vtf etc. = the ordinatQS, and Ax s the eommon
interval or abscissa between the ordinates.
1. Trapezoidal rule.
Area = — (yi + yj).
2. Bule for parabola of the third order with three ordinates.
Area = — (y, + iy^ + y,), (Sunpson'g first rule.)
3. Bole for parabola of the third order with foar ordinates.
Area = — g- (yi + dy% + Sys + y*) . (Simpson * s second rule . )
4. Bole for parabola of the fifth order with five ordinates.
Area = -j^ (7yi + 32y2 + 12y8 + 32y4 + Tys) .
5. Bole for parabola of the fifth order with six ordinates.
Area = -^ (19yi + 75y% + 50y8 + SOy* + 75y« + 19y«) .
6. Bale for parabola of the seventh order with seven ordinates .
Area = ^(41yi + 216y8 + 27yi + 272y4 + 27y6+216y« + 41y7).
7. Approximation for curve with six ordinates.
Area = -2j^(0-4yi + y.2 + y$+y4 + y5 + 0'4ya).
8. Weddle*s approximation for curve with seven ordinates.
Area = -^(yi+6yi+y8 + 6y4 + y5 + 6y« + y7).
24. To measure any curvilinear area by means of the ira-'
pezoidal rule,
BuLE. — ^To the sum of half the two endmost ordinates add
all the other ordinates, and multiply the sum by the common,
interval ; the result will be the area. Thus,
Jyi«=Aa:(^^^-^+y8+y« . . - . + yn-i)
Remark, — In shipbuilding work it is very often convenient
to perform the additions in the above rule mechanically, by
' measuring off the ordinates continuously on a long strip of
paper> and measuring the total length on the proper scale.
This rule is only approximate, but it is especially useful for
getting the areas of the transverse sections in the first rougb
calculations of trim, ddsplacement, etc.
25. To measure any curvilinear area by means of Simpson* s
first rule,
Bule. — ^To the sum of the first and last ordinates add four,
times the intermediate ordinates and twice all the dividing
44 MENSUBATIOK OF CURVILINEAR AREAS.
ordinates ; multiply the iinal sum by |, the common interval : the
result will be the area. Thus
/.
Bema/rk. — The number of intervals in this rule must be
etven. The ordinates which separate the parabolas into whloh
the figure . is conceived to be divided, are called dividing ordi-
oateSf and all the other ordinates except the two endmost ones
£^j^e called intermediate ordinates.
26. To measure any curvilinear area hy means of Simpson' f
second rule,
RuLE.^To the sum of the two endmost ordinates add thr^e
times the intermediate ordinates and twice all the dividing
ordinates ; multiply the final sum by |, the common interval, and
the result will be the area. Thus
/.
OAn,
3^ = __(y, + 3y,+ 3y,42y4 + 3y4 , . . . + 3y,_i + y,).
The number of intervals in this case must be a multiple of three.
Rema/rk. — The sequence of the multipliers in the two fore-
going rules is obvious. Thus in the first rule the simple multi-
pliers ar^ 1.4. 1, but they are combined thus : —
1.4.1
1.4.1
1.4.1
1.4.1
1.4.1
1.4.1
1.4.2.4.2.4 4.2.4.2.4.1
In the second rule the multipliers are 1 .3.3.1.
1.3.3.1
1.3.3.1
X.o.o.x . . ^. •
1.3 3.1
1.3.3.1
And ib the same way the multipliers to measure any ounii-
linear area may be obtained from the table on p. 43.
Simpson's first rule is superior to the second rale in
accuracy as well as simplicity.
27. To measttre any curvilinear area when subdivided
intervals are used.
\st. When SimpsorCs first mile is used.
Rule. — Diminish the multiplier of each ordinate belonging
to a set of subdivided intervals in the same proportion in which
MENSURATION OF CURVILINEAR AREAS.
45
the intervals are subdivided. Multiply each ordinate by its
respective multiplier as thus found, and treat the sum of their
products as if they were whole intervals ; that is, multiply the
sum thus found by } of a whole interval, and the product wfll
be the area.
2nd. When Simpson's second nile is iised.
RULV. — Proceed as in the first rule, but multiply by f of a
whole interval for the area.
Exmnple to SiiwpsovCs First Ride. — ^The series of multipliens
for whole intervals being 1 . 4.2.4.2, &c., those for half-
intervals will be J, . 2 . 1 . 2 . 1, &c., and for quarter-intervals
JRemarh. — When an ordinate stands between a larger and
a smaller interval, its multiplier will be the sum of the two
multipliers which it would have had as an end ordinate for each
interval. Thus for an ordinate between a whole and a half
interval the multiplier is J + 1 « 1 J, and between a half and a
quarter interval i + J '^^ f .
Table of Multipliers when Subdivided Intervals
Alts UBEt>.
Simpson's First Bnle.
Ordinates
0
1
2
li
1
1
2
'n
2
2
21
s
2
T
3
f
8 3J
4
1
1
u
6
4
6
4
6}
2
1
6
6
1
64
2
I
7
1
i
i
74
2
61
1
1
8
i
7
J
4
5
Ji
Multipliers
1
0
0
1 Hlfl
1
4
J
1
4
li
i
Ordinates
3*1 3
MoltipUers
2
li
Ordinates
H\H
Multipliers
1
*
Simpson's Second Rule. 1
Ordinates
0
T
T
i
1
T
1
X
J
2
3
i
1
ttaft.
I
5
3
i*
5
1
J
4
_*i6 6i;6
6J
J
i
J
8
[1
1
8i
i
6f
i
i
J
61
4
J
7
J
4
J
6
J,
Multipliers
1
li
'*!
»
(^ditaftteft
ll!3
2i«t
Multipliers
1
i
% f i
<^ai96atM
2J'S IgiSf 4
Multipliers
i±
J_
1
1
J
Nate. — ^The ordinates in this table are numbered the same
as if they were the number of intervals from the origin.
46
MENSURATION OF AREAS AND PERIMETERS.
Fig. 66.
Thomson's Bute may be used when subdivided intervals are
used at each end ; the advantage being that all multipliers except
the three end ones are unity ; so also in the common multiplier.
Thus the ordicates should be multiplied by ^, }, U, 1, 1, 1, ...
...,1,1,}},^,^; the spacing of the three ordinates at each end
being one half that elsewhere.
28. To calculate the area separately of one of the two
divisions of a parabolic figure of the second order, (Fig. 66.)
BuLE. — ^To eight times the middle ordinate add five times
the near end ordinate, and subtract the far end ordinate ;
multiply the remainder by ^ the common interval : the result
will be the area.
Note, — ^The near end ordinate is the ordinate at the end of
the division of which the area is to found.
Bx. : In figure abcd let it be required
to find the area of the division acef. Let
yi ^ the near end ordinate, y^ = ^^ middle
j^ ordinate, and ^3 = thp far end ordinate ;
then ^ydx = -^(^Vi + Sj/a - ys).
29. To 'measure an area bounded by an arc of a 'plane curve
find two radii, (Fig. 67.)
Fio. 67. Rule. — Divide the angle subtended by
the arc into any number of equal angular
intervals by means of radii. Measure these
radii and compute their half -squares.
Treat those half-squares as if they were
ordinates of a curve by Simpson's first or
second rule, as the number of intervals
c may require.
' Note, — ^The common interval must be taken in circular
measure. (See pp. 8 and 9.)
Ex.: In the figure abc let n, rg, rs, r^, rs = the radii,
A^ = the common angular interval, and I -d9 =s the area; then
2 6
^ 30. To measure any curvi-lmeai' area by means of Tcheby^
eheff*s rule.
BuLE. — ^Find the middle point of base, and frosftit set off,
along the bases &i^d (n both directions, distimces equal to- the
half length, of base multiplied by the • constants- given in the
Schedule below. Erect ordinates at the points so obtained
and measure them. Their sum, divided by the number of
ordinates, and multiplied by the length of base Is the area
required.
/:
MENSURATION OF AEEA8 AND PERIMETERS.
47
Schedule. 1
Nnmber of
Ordinates osed.
Distance of Ordinates from Middle of Base In
Fiaetiant of Half the Base Length.
2
3
4
5
6
7
9
•5773
0, -7071
•1876, -7947
0, -3745, -8325
•2666, -4225, -8662
0. -3239, ^5297, -8839
O, •1679. •6288, •6010. -9116
Note. — Aa evident from the Schedule, there is an ordinate
at the middle of base^ only when an odd number of ordinates-
is employed.
Examples. — With four ordinates. (Fig. 68.)
Let ABGD be the figure. Bisect
the base AB at E. Calling the half
length of base 6, set off l&F and £F'
equal to •1876 b and EG and eg' equal
to ^7947 h. Erect ordinates 6L, FK,
¥'k\ o'l' at o, 7. f'. q' ; and call
them yu 2/9> ^s. and Va.
Then area of figure abcd = y> •*• y« -^ y» + V^ y^ 26.
With five ordinates. (Fig. 69.)
As before, let abcd be the
E' — ^ — -t p figure, E the middle of base, and h
its half-length. Set off ef and ef'
equal to -3745 h and EG and eg'
equal to •8325 6, and erect ordinates
B at G, F, E, f', and g', calling them
yu yiy ytt ytt y«*
Then area of figure abcd
yt -f ya + ys + y< + ys ^ gft.
6
Note. — ^This rule can be used for calculating displacements,
and fewer ordinates are required for the same degree of
aocnracy than if Simpson's rale is used. Ten ordinates are
osaally employed instead of twenty-one, the rule for five
ordinates h^ng applied separately to each half of the ship, tt
is Rilso of great assistance in preparing^ cross curves of stability.
If eight ordinates are used (four repeated), the following
"Simpeon'^ sections, assumed numbered from 1 to 21, can
be utilized : 2, 6, 7, 10, 12, 15, 17, 20.
48
MENSURATION OF AREAS AND PERIMETERS.
31. To measure any curvilinear area hy three ordinate^
irregularly spaced,
BuLE. — Let ABODEF (Fig. 70) be the carvilinear area, whose
ordinates AB, FC, ED, are ^i, j/a* and ^t* Let AF=/^, and FK^MA;,
where /c is a ratio.
^^|f^^[yi7.(2-fc) + ya (fc + l)' + y3(2fe-l)]
Note.^lt AF = 2fe, so that A; = J, '
Area = -^(yi + ^Vi)
9k
32. To find the area between the first two ordinates of
a curvilinear area given three ordinates irregularly spaced.
Rule. — ^The area included between the ordinates ab and cf.
•If
= gj^ (fe ^ 1) [Vi Ti (3/fi + 2) + t/a (3fc + 1) (& + 1) - 2/.]
Note.—U AF = 2fe, 80 that * = J,
Fig. 70
AF
Area = jg (7 2/i + I5y% - iys)
If AF = Jfe, so that * =s 2, ^
Area = ^ (16l^i + 21ya - y,)
33. To obtain a general expression for the length of any
plane curve.
Using cartesian co-ordinates the length intercepted between
two points whose * x * co-ordinates are a and h is equal to the
definite integral / ^ 1 + ij^) » dx. This may be obtained
by Sin^)8on*B rules in a similar way to the area ; the * ordinate *
2
or
in th^s case being replaced by the ralue of ^ ^ "^ \d)
sec. ^, and the common interval being measured along ox.
84. To find approximately the length of any plane curve,
(Ti^. 71.)
If the eurve ii rather irregular, divide it by the eye into
any niunber of fairly itat area ; join tha extr^nities of each
of these arcs by chords. The sum of the length of each of
these arcs found by the following rule will be the total length
of the curved line.
MENSURATION OP SOLIDS. 49
Rule. — 'Drttw a tangent to the curve at each of its ex*
tremities ; then take the sum of the two distances from the point
of intersection of the two tangents to the extremities of the
curve, together with tmce the length of the chord ; divide the
result by 3 for the len^g^th of the arc.
Bx. Q&g. 71): Let aob be one of
the arcs, and ab a diord joining
the two extremities, and at, bt^
tangents to the ouvve at its extremi-
ties, cutting each other in D ; then
the length of the curve
AOB = } (ad + db + 2ab).
Alternatively, for a flat curve see Rule III, p. 38.
35. To find the perimeter of an ellipse of moderate
eccentricity.
Bulb. — If 2a is the major and 26 the minor axis, where - is
not very smaU, then the perimeter is equal to — (3a' + b^) very
nearly.
36. To find the length of the evolute of a curve,
BuLE. — ^The length of the arc of the evolute is equal to the
cBfferenoe between the lengths of the tangents drawn from
either extremity of the arc to the involute.
II. Mensuration of Solids.
1. To find the volume of any parallelepiped , prism, or
eylinder, (Fig. 720
BuLE. — Multiply the area of the base by the perpendicular
height ; the result will foe the volume.
Fio. 79.
I
^
''^^^
2. To find the volume and slant surface of a cone or
pyramid. (Fig. 73.)
Pig. 73.
Rule. — ^Mul^ply the area of the base by J the perpen-
dicular height ; the product will be the volume. The slant
£
50 MENSURATION OF SOLIDS.
surface is equal to the perimelfcetr of the base multlpJUied
by half the slant height,
3. To find the volume and slant surface of the frustum
of a cone or pyramid, (Fig. 74.)
EuLE. — ^To the sum of the areas of the two ends add the
square root of their product ; this final sum being multi-^
plied by J of the perpendicular height will give the volume.
The slant surface is the product of the sum of the perimeter
of the two ends and half the slant height.
Pig. 74.
^ s ^
4. To find the volume of a wedge whose base is a
parallelogram, (Fig. 75.)
Pig. 76. Rule. — Add. the length of the
^. I edge to twice the length of the base ;
fi vv A n multiply the sum by the width of
k^ y i^ — ^ the base and the product by \ of the
perpendicular height : the result will be the volume.
5. To find the volume of a prismoid, (Fig. 76.)
KuLE. — To the sum of the areas of the two ends
Fig. 76. ^^^^ f qqj. times the area of a section parallel to the
A \ base and equally distant from both ends ; the sum
/yCZiA being multiplied by ) the perpendicular height will
^ '^ give the volume.
6. To find the volume and surfoxe of a sp}tere or globe.
(Fig. 77.)
FIG. 77. RuLE.-Multiply the cube of the diameter by
g- or -5236 ; the product will be the volume. To
obtain the surface, multiply the square of the diameter
by IT or 3-1416.
7. To find the volume and stir face of the segment of a sphere*
(Fig. 78.)
EuLE. — Add the square of the height to S tunes
the square of the radius of the base ; that sum
Fig. 78. ir
j0l^^ multiplied by the height and that product by ^ or
j ] '6236 will give the volume. To obtain the surface,
'\^ / multiply the diameter of the whole sphere by the
height of the segment and that product by v or
31416.
MENSURATION OF SOLIDS.
61
8i To find the volume and surface of a mom of a ephere.
(Fig. 79.)
BuLE.^-To the Sam of the squares of tiie radii of
the two ends add' } the square of the height ;
multiply the sum by the height and that resoit by ^°'
^ or 1*5708 : the result will be the volume. To
obtain the surface, multiply the diameter of the
whole sphere by the height of the zone, and that
product by ir or 3' 1416.
^
9. To find the volume and surface of a cylindrical ring.
Rule. — ^To the thickness of the ring add the inner dia-
meter ; multiply that sum by the square of the thickness, and
the product by -^ or 2*4674 : the result will be the volume.
To obtain the surface^ multiply the sum of the inner diameter
and thickness by the thickness, and that product by v^ or 9*87.
Table to find thb Volume and Subfacb of ant
Begulab Polyhedbon.
T s= volume. A s area. L = linear edge,
r = radius of inscribed circle.
No. of
Sides
4
6
8
13
20
No. of
Edges in
each side
Name
3
4
3
5
3
Tetrahedron
Hexahednm ^
Octahedron
Dodecahedron
Icosahedron
A-L«X
1*732051
6000000
3-464102
20-646729
8-660254
V«L»X
•117851
1-000000
•471405
7-663119
2181695
r-LX
•204124
•500000
•408248
1113616
•765750
* Or cube.
10. To find the volume of an ellipsoid. (Fig. 80.)
Rule. — ^Multiply the product of the three fig. 80.
principal axes by ~ or '5236 : the result wUl be
the volume.
11. jTo find the volume of the segment of an ellipsoid of
revolution when the base is circular, (Fig. 81.)
BuLE. — ^Take double the height of the segment from
i
i
5 a MENSURATION OP SOLIDS.
Fm. 61. ^bi*Ge' times ike lengiiU of the fixed axis ; multiply
the difference by the square of the heigfht^ an^
thai ' product by — or '5236 : then that zefloU
6 •
multiplied bj the square of the rerolving axis and
the product divided by the square of the fiised axis
will ^ive the rolurae.
12. To find the volume of the segment of an eUipioid af
revolution when the base is elliptical, (Fig* 82.)
BuLE. — Take doable the height of the segment from tiiree
times the length of the revolving axis ; multiply
FTff.83. the difference by the square of the height^ and
If
that product by ^ or '5236 : then that resiilt
multiplied by the fixed axis, and the prodaot
divided by the revolving axis, will give the
volume .
13. Ta find the volume of the middle frustum of am
ellipsoid of revolution when the ends are circular, (Fig. 88.)
Fig. 83. Rule. — Multiply the sum of the square of the
middle diameter and one-half the square of the
diameter of one end by the length of the frustum^
and that product by ~ or '5236 for the volume.
6
14. To find the volume of the middle frustum of «m
ellipsoid of revolution when the ends are elliptical, (Fig. 84.)
Rule. — To" twice the product of the transverse and con-
jugate diameters of the middle section, add the
Fio. 84. product of the transverse and conjugate diameters
><~^^^ of one end ; multiply the sum by the height of
'ISffllilH the frustum, and that product by — or '2618 :
the result will be the volume.
15. To find the volume of a paraboloid, (Fig. 85.)
Fia. 85. Rule. — ^Multiply the square of the diameter
of the base by the perpendicular height, and the
result by ■- or '3927 ; the product will be tke
8
volume.
MENSURATION OF SOLIDS. 58
16. To find ths volumB of the frustum of a paraboloid
when its ends are perpendundar f» its suHs, (Fig. 86.)
Rule, — ^Multiply the Bnm of the squares of Fio.86.
the diameters of ^e two ends by the height of
tiie fmstom ; the product mnlUpUed by - or
*3927 will be the volume.
/
17. To find the volume of anf solid cff revohtUon,
»
(I) The volume is represented by the definite integral
h
V y^ . dx, where OX is the axis oi symmetry.
o
Rule. — ^Divide the length of the sxii Into a contenient
number, of equal parts. Measure the ofdinates, and treat
their squaioa as if th^ were the oardinates of a plane curve)
of the sam« length as the solid ; the area of this curv»
multiplied by r or 3*1416 will be the volume rflquirefl.
(II) If the |>osition of the centre of gravity *f the
generating area is known, the following method is applicable.
BuLE. — ^Multiply the area of the generating section by the
distance of its centre of gravity from the axis ; 2r or 6*283
times Hke product will be the volume required.
Ewttmple. — ^Find the volume of the soUd ^eircrated by the
revolution of an equilateral triangle about its base.
If 9a be the side of the triangle, its area is V8 . o^ ; and its
a
centre of gravity is -j^ from the base. Henoe the volume
— fl
sequised is 2r X ^^ ^ vl ** ^'^*
18. To Pleasure th^ volume of an^ solid.
(I) To measure the volume in slices.
Rule. — ^Take one of tho plane surfaces as the base, and
divide tlie mass into siloes parallel to that base and sufficiently
thin as to be able either to neglect or account separately for
the curvature.
Then take f&e volume of each slice separately^ and add
them together for the whole volume, taking account of the.
curvature in this addition if necessary.
54
MENSUUATION 07 SOLIDS.
(II) To meawra the volume bf the rtOei applicabU to the
area of a plane curve. (Fig. 87.)
Fio.87. BuLE.— Take a rtraiglit Ime in
the figoie as a base line, or line of
abscissa, and divide the fignre alon?
that line into any number of equal
parts, and measure the areas of the
plane sections at those points of dividon by the rulee
applicable to the area of a plane ourre.
Then treat the areas thus found as if they were the
oidinatee of a plane onr?e of the same length as the figure,
and the area of this will be the volume of i£e solid.
ExamffU. (See A^. 87.)
Number of
Seotloiui
AreM of SeetUnui
la savaKa f eJM
Multipliers
Frodnots
1
2
8
4
5
5
10
15
20
25
1
4
2
4
1
5
40
80
80
25
180
2
3 '
AxeaasSGO cubio feet.
Benuirk, — ^The yolume is above assumed to be repveiented by
rh
the definite integral / A . da;, where ▲ is the area ol any seotion
J 0
perpendicular to the base line 09. The volume may also be
represented by the double integral U « . da; . dy taken orer the
area of the base, the m axis being ^u^osed perpendicular to
the base.
(ill) To measure the volume hy Dr, Woolley*% method,
(Fig. 88.)
BuLE.— -Take a straight line in tiie figure as a base line,
and divide the figure along that line by an odd number of
parallel^ and equidistant planes perpendicular to the base.
Then divide the figure horizontally in the same way by a
number of plane sections parallel to the. base. Then take
ordinates lot the intersections of the horizontal with the vertical
plane sections In their consecutive order, and treat them as
foUowf :•—
MBNStJRATION OF SOLIDS.
55
(1) Neglect absolutely all ordinatea wJuch are odd In both
planes of seetion.
(2) Neglecting the ontside rows of ordinatea, doable every
ordinate which la even in either or both planes of section, and
add them together.
(3) Add to this the simple earn of all the even ordinates in
the outside rows.
(4) Multi;^ly this final sum by | of the product of the
c(Mnmon vertical interval, by the
common horizontal interval, and the
result will be the volume.
JBx, : In the acoompanying figure
the multiplier for each orduuite is
shown above it, so that if 8 =s the
sum of the products of the ordinates
by their respective multipliers, v =
the volume, and Av' = the common vertical interval, and
Ax = the common horizontal interval, then
Fzo.88.
V=s
8(8 X Aa?' X Ag)
3
Mem4irk. — ^This method is inferior in accuracy to that
obtained by a double application of Simpson's rules.
Fia. 69.
19. To meaeure the volume of a wedge-shaped solid
bounded on one eide by a curved surface, (Fig. 89.)
The volume is represented by the double integral J])^*
dx • dBf where r is a radius from the edge. 9 is the angle between
the radius and the plane of the base, and oxis parallel to the edge.
Rule. — ^Divide the figure longitudinally by a number of
planes radiating from the edge at equal angular intervals, and
also divide the length of figure into
a number of equal intervals for
ordinates, and treat each of the^
radiating planes as follows : —
(I) Jieasure the ordinates as if
for taking the areas of the several
planes, but instead of the ordinates
themselves compute their half-
squares, and treat them as if they
were the ordinatee of a plane curve of the same length as
the figure. The result of this calculation is called the moment
of the radiating plane.
(Jl) Treat the moments of the radiating planes as if they
were the ordinates of a curve, but taking the common angular
interval in circular measure.
56
MENSUHATION OF 80IJBS.
ExampU, (See fig. 89.)
No. of Planes
Momenta of fhe
Badiating Planes
MoltipUers
Prodaets
1
2
3
4
5
105
110
115
120
125
1
4
2
4
1
105
440
230
480
125
9
8
1380
angular interval ^.^^
® 1380
12420
2760
Yolnme s 40-1580
20. To find the mean Motional areu of a solid.
BuLE. — ^Divide the volume of the solid by its length ; the
result will be the mean sectional area.
21. To set off the correct form of a mean cross-seciion,
BuLE. — ^Divide the figure longitudinally by a number of
horizontal planes ; take the mean breadth of each of the
horizontal planes and set them off perpendicular to a fixed
«traight line, and at the asax^ height as their corresponding^
planes in the solid : a line passing through the ends of Idiese
mean breadths will be the correct form of the mean sectional
area of the solid.
Note. — ^The mean breadth of a plane curve is found by
dividing the area of the curve by its length.
Via. 90.
t 22. To find the volume of a
I fourway 'piece of piping.
Let r C^. 90) be the radius
7« of the piping and I and I' the
lengths.
Then volume mrr* (Z + J' - |r).
J <
28. To find the surface of any soiid of revolution.
(I) The surface is represented by the definite integrsl
"^y . dSt where ds is an element of arc of the generating curve.
MENSURATION OF 80LII>S. 67
BuLE. — ^Divide the perimeter of the g^eaeratiMg oarre into
a ooiXTNiieiit number of eqaal aTOs. Moagnro the ordinatea
at the points of division^ and treat them as if they were the
equidistant ordinates of a oanre, with the .oommon interral
equal to the length of the arcs. The area of this oorvo
multiplied by Sir or 6*288 wall be the area of the mirfaee.
(O) If the (position of the oentre of gravity of the peri-
phery of the generating curre is known^ the following method
is applicable : —
BuLE. — 3fnltiply the length of the arc of the generating
curre by the distance of the oentre of gravity of the arc
from the axis ; 2t or 6*288 times the produot wUl be tiie area
of the surface.
Example. — ^Find the snrfaoe of the solid generated by the
revolution of an equilateral triangle about its base.
If 2a be the side of the trianglsi its perimet6r» exclu8i?e of the
axis, is 4a ; and the centre of gravity of the two sides is VSa/2
from the base.
Henoe the surface required is 2t x 4a x V3 a/2 s 4 VS ra*.
24. To find the area of any iurfac&»
(I) Exact method, — The area is given by the double integral
y^BQti^ .dx , dy, wh^e 9 is the angle made by the surface with
the wy plane. This is equal to J J V (1 + tan' ^ + tan" ^) .
dx • dfy, where ^ and ^ are the angles made with thjs s azia by
the sections of the surface with the OMt and ys planes.
BuLE. — ^Take a straight line in the figure as base line and
divide the figure along that line by a convenient nun^ber of
parallel and equidistant planes perpendicular to the base ;
call these the vertical sections. Then divide the figure
honsontaUy in the same way by a number of plane sectional
parallel to the base. At the intersections of the two sets of
sections measure tan ^ and tan ^, ^ and ^ being the angles made
in the sectJona by the tangents to the curves witii the base.
Evaluate Vl + tan'^ + tan^V ^ M^h intersection. Treat this
as tlie ordinate of a 8olid» and proeeed to find its volume by any
of the rules given above. The result is the area required. It. Is
desirable that no part of the surface should be approximately
perpendicolsr to the base.
Swample. — ^A portion of a ehip^s sBde is bounded by two
sections 40 feet apart, and two wateriiiniee 8 feet apart. 1^
i»ngetits of the angles (jp) made with the middle line at the
sections and at <me sitiiated midway between themi and th»
58
MENGURATI-ON OF SOUDS,
(lUDgenia of the anglos (f) made with the middle Uno at
iJie two waterliiias, and ^t one midway, are as follows:—
W.I*.
BeolioBl
Section a
Section 8 1
tan'^
tanV'
tan^
tanW'
tan<^
t&n^
1
2
8
•30
•21
•10
•21
•22
•22
•35
•31
•14
•16
•17
•18
•48
•44
•25
•06
•07
•09
ifind VI + tail' ^ + tan* 4' in each case, and proceed as in the table below :
5»
1
4
1
Section 1
Vl+tan«*+tan«V'
106
1-04
1^08
o
2
106
416
103
7-25
1
7-25
Sections
Vl+tan«^+tan«4'
107
106
1-03
o
107
4-24
103
6-84
4
Sections
Vl+tan«^+tan«^
111
1-10
102
o
o
111
4-40
102
6-53
1
25*86
6-68
= 39-14
20 4
The area of the onrred surface is 39-14 x -r x - = 848 square
feet.
(II) Approximate method.
Rule. — Take girths along (say) the vertioal sections at
eqnidistant intervals. For each section in the half-breadth
plani note the angle at wfiich. the various waterlines croas,
and estimate the mean slope of the waterlines snrronndingi
the onrfaoe under question. The secant of this mean angle
with the middle lino is termed the modifying factor, and is
multiplied by the girth eonoemed. These modified girtha
are then regarded as the ordinates of a curve, whose area 19
the surface required.
Example. — ^In the previous example, the girths at Sectlionf
1, 2, and 8 are 8'2, 8'3, und 8*7 feet res^tively. The
respeotivd mean angles of the waterlines with the middle
line are '22, -17, and '07. Find the surface.
The first modifying factor is ^/l 4- (•22)« or 1'02 ;
similarly the others are 1*015 and I'O approximately.
cnnnm ai?d uohemts of nouBss.
69
No.
Olrtli
Modllyine
Faotor
Modified
Girth
Bimpoon*!
Multiplier
Prodvcl
1
2
3
8*9
6-8
8*7
1*09
1*016
1*0
8*4
8*46
8*7
1
4
1
8-4
88*8
8*7
60*0
20
Area of surface s 60*9 )c ^ ^s 840 square feet approximately.
CSHTBX8 A9D MOtfEHTS OF FIOITBtt.
To FIND THE 0E5TBBS OF fi^VlTTOF A FEW SPBCUJ:« FlGUBES.
1. Triangle. (Fig. 01.)
BuLE. — ^From the middle points of any
two aides draw lines to the opposite angle;
the point of interaeotion Dot these lines is
the required centre.
Or, trisect the line joining the middle
point of one side with the opposite vertex ; the point of
triseotion ^ear^ to the base is the required centre.
Fio.lKL
2. Trapezoid. (Fig. 02.)
BuLE. — ^Biseot ab in b and OD in
p and join ef. Produce ab beyond
B to H, making bh = CD, and pro- 1.
duce CD beyond c to i, making ci
B3 AB ; then join hi, ai^ where thia line intersects ef is the
centre of gravity o.
Note.—EQ is to OF as 2oD *f AB is to 2ab + cD. If the
angles at A and o are right angleSi the distance of o from ao
, . AB^ + AB.CD + OD*
" "^"^ ^ 8(AB + 0D>
8. QuadrUatertA. (Fig. 98.)
BuLE.-*Draw the diaji^onals
AD and CB intersecting in b ;
along OB set off OF equal to eb«
and join fa and fd ; the centre
of the triangle afd will be the
centre of the qufidrilateral.
Fto. M.
60
CENTRES AND MOlkfENTS OJ" FIGURES.
Fig. gSA..
Fio. 94.
Or, luBect the diagonal bd (fig. 93a)
-ai R ; join £a, Be; Make ef =s Jba tind
EG =s)bo. Join FQ, cutting bo at h.
Make kg r=s hf. Then K is the required
centre.
4. Circular €tre, (Fig. 94.)
EuLE. — Let ADB foe isho oireular are
and c the centre of the circle of wliich
^ it is a part : bisect the arc ab in D, and
join DO and ab .; multiply the radius CD
by the chord AB, and divide by the
length of tjie arc adb ;. lay oft the
Quotient CE upon CD ; then E is the
centre required.
I
S. Very fU:t curved Hfw
'(npproziPMtiv'). (Fig; 95.)
KuLE.—Let ADB !)e the
arc ; draw the chord ab,
and Insect it in c ; draw cd
erpeudicular to AB ; make CE equal to 2 ^^ ^ > ^^^^ ^ ^^U
e the centre required.
6. Sccti>r tif 0 i^fvle, (Fig. 96.)
Rule.— Let abo be the sector, E its
cen^^e; multiply the chord Ati by | of the
^ radius CA ; divide the product by the length
of tiie arc: the quotient equals the distance
CE set along, the line CD, D being at the
bisection of the arc ab.
7. Sector of a plane circular ring. (Fig. .97.) ' ' ' "
Pio.97. Rule.— Let oa be the outer and cb
the inner radius of the ring ; divide twice
^ the difference of the cubes of the inner
and outer radii by three times the
difference of their squares;- the quotient
will be an intermediate radifis CF^ with
whi^h describe tiie ate PF, EfnfbtendSng the
same angl^ with the sector : tiie centre H of the circular are
IT, found by Rule 4, will be the centre required.
GBNTRBS AND MOMENTS OF FIGUUBB.
8. Circular Begment, (Fig. 98.)
EoLE, — ^Let c be the centre of tiha
circle of which it is a i»art ; biaeci the are
AB in D, and join OD ; divide the cube of ^
half the chord ab by three times the area
of half the segment add ; set off the
qaotient ce along co, and B wiU be the
centre required.
61
9. FarahoUe hdlf'9egmeni, (Fig. 99.)
HuLE. — ^Let ABD be a half-segment of
a parabola, bd being part of a diameter
parallel to the axis and ad an ordinate con-
j agate to that diameter— that is, parallel
to a tangent at B. Make be equal to | bd«
and draw ef parallel to ad and equal to
f AD. Then F will be the centre of the
half -segment.
Pxo.99.
10. ffeight of centre if setnicirele or iemi-eRipie from
its base,
BuLE.— -Multiply the radius of the semicircle (or that
scml-axis of the ellipse which is perpendicular to the base)
by 4, and divide the product by Sir,
11. Height of centre of parabola from its base,
BuLE. — ^Multiply its vertical heighb by 2, and dzvi^ the
product by 5.
12. Prism or cylinder with plane pcfrallel ends,
BuLE. — ^Find the centres ol the ends ; a straight line
joining them will be the axis of the prism or cylinder, and the
middle point of that line w^ be the oenfcre required.
13. Cone or pyramid,
BuLE. — Find the centre of the base, from which draw a line
to the summit ; this will be the axis of the cone or pyramid,
and the point at I from! the base along that line will be the
centre.
14. Hemisphere or hemi- ellipsoid,
BuLE. — ^The distance of the centre from the circular or
elliptic base is | of the radius of the sphere, or of that semi-
axis of the ellipsoid which is perpendicular to the base.
62
CENTRES AND MOMENTS OF FIGVBBS.
15. Paraboloid,
Rule. — ^The distance of its centre from the base along its
axis is J of the height from the base. .
FiQ. 100.
Ifi. To find the centre of ffrai>iiy
o/ any continuous curved tine. (Fig.
100.)
Ex. : Let abc be the giyen curve;
bisect it at B ; join ad and BC^ and
fuBect those chords at the points D
and E respectively ; set off FD per-
pendicular to AB^ and EO perpen-
dicular to BO ; make fH = Jdf and
gk=}ge, and join H& ; bisect he
at the point L, which will be a close
approximation to the position of the
centre of gravity of the curved line
ABC.
Remark, — If the line is too irregular to permit its two
parts to bo rogarded as flat regular curves, it should be divided
into four or eight parts as required. The points corresponding
to L in the above figure are found separately for each pair of
parts, joined in pairs and bisected ; this process is repeated
until only one point remains, this being the required centre
of gravity.
BCTLES rOB FINDING THE MOMENTS AND CENTRES OF
FiGUBES.
The geometrical moment of a figure, whether, a line, an
area, or a solid, relatively to a given plane or axis is the
product of the magnitude of that figure, into the perpendicular
distance of its centre from the given plane or axis, and is
equal to the sum of the moments of all its parts relatively to
the same plane.
The centre of an area is determined when its distance from
two axes in the plane of the figure is known.
The centre of a figure of three dimensions is determined
CENTBES AND MOMBNTS OF FIOUaES.
68
when its distance from three planes not parallel to one another
is known.
1. To find tlve mome-nt of an irregular figure reUUivelg to a
given plane m* axis.
Rule. — Divide the figure into parts whose centres are known;
multiply the magnitude of each of its parts into the perpendi-
cular distance of its centre from the given plane or axis ; dis-
tinguish the moments into positive and negative, according as
the centres of the parts lie to one side or the other of the plane :
the difference of the two sums will be the resultant moment of
the figure relatively to the given plane or axis, and is to be
regarded as positive or negative, according as the sum of the
positive or negative moments is the greater.
2. To find the perpendicular distance of the centre of an irre-
gular figure from a given plane or aans*
Rule.— Divide the moment of that figure relatively to the
given plane or axis by its magnitude ; the quotient will be the
perpendicular distance of its centre from the given plane or axis.
3. To find the centre cf a figure consisting of two parts whose
centres a/re known. (Fig. 101.)
Rule. — Multiply the distance between the two known cen-
tres by the magnitude of either of the parts, and divide the
product by the magnitude of the whole figure ; the quotient
will be the distance of the centre of the whole figure from the
centre of the other part, the centre of the whole figure being
in the straight line joining the centres of the two parts.
JSx.: Let abcd be such a figure, M and m
the magnitude of its two respective parts, m + m
the magnitude of the whole fig^e, D the dis-
tance between the centres M and m of the two
parts, and c the centre of the whole figure.
WXD MxD
FiQ. 101.
M+}»
M + W»
Fig. 102,
4. To find the centre of any plane area hy means of ordinates.
(Fig. 102.)
Let ABC, the quadrant of a circle, be such
an area ; CB the base line, divided into a
number of equal parts by ordinates ; AC the
transverse axis traversing its origin.
\st. Determine the perpendicular distance
of the centre cf the quadrant from the trans-
verse axis in the following manner: —
Rule. — Multiply each ordinate by its dis-
tance from the transverse axis; consider the
products as ordinates of a new curve of the same length as the
given figure : the area of that curve, found by the proper i-ule,
will be the moment of the figure relatively to the transverse
N
1 *\
1 1 \
\ i
i
43
uL^
64
CENTRES AND MOMENTS OF FIGUHES.
a^^is ; this moment, divided by the whole area of the figure, will
give the perpendicular distance of its centre from the transTerse
axis.
In algebraical symbols the moment of a plane figure rela-
tively to its transverse axis, and found by the above rule, is
expressed thus : —
fxydx.
JVate, — ^In practice it is better to proceed as follows : — Multiply
the ordinates first by their multipliers, and then those products
by the number of intervals from the origin ; take the simi of
those products and multiply it by Jrd of a whole interval
squared, if Simpson's first rule is used, by Jths of a whole inter-
val squared, if Simpson's second rule is used, and so on for the
other rules.
^xavijfle.
No. of
Interyala
Ordinate?
16-0000
16-4919
13-8564
12-4900
10-5830
9'3274
7-7460
5-5678
0-0000
Mnlti-
pUera
1
4
H
2
i
4
Products
160000
61-9676
20-7846
24-9800
7-93725
9-3274
3-«730
5-5678
0-0000
Products x No. of Intervals
from Origiii
•00000
61-9676
41-5692
62*4500
23-81175
30-31405
13-5555
2087925
•00000
Interval lgO'^3765 Interval^
254-54735
i«
3
Approximate area = 200' 68353 Approx. moment = 1357- 685
Moment 1357-585 ^ g.^gg f approximate perpendicular distance
Area 200-5835 * * \ of centre from the transverse axis.
2nd, Find the perpendi&ular distance of its centre from the
base line.
Rule. — Square each ordinate, and take the half -squares as
ordinates for a new curve of the same length as the figure ; the
area of that curve, found by the proper rule, will be the moment
of the figure relatively to the base line : this moment, divided
by the whole area of the figure, will give the perpendicular
distance of its centre from the base line.
In algebraical symbols the moment of a plane figure rela-
tively to its base line, found by the above rule, is expressed
thus: —
2
A
CBNTBBS ANP MOMENTS OF FIGURES.
Example.
66
No. of iQteirvals
Oidinates
Half-squares
Multiplicn
I^odocts
0
1
2
3
4
16-0000
16-4919
13-8564
10-5830
0-0000
128-0000
119-9996
95-9999
66-9999
00000
1
4
2
4
1
Interval
3
128'0000
479-9980
191-9998
223-9996
00000
1023-9974
3
Approximate moment— 1366*3298 |
Moment 1365-3298 /..^q/» / approximate perpendicular* dis-
Area 201*0624 " ' \ tance of centre from base.
Actual moment =1365-3
~Actu^~area == 201-0624
5. To find tJte centre of a plane area hounded hy a eurre and
tfva radii hy means of jwlar eo-ordinates. (See fig. 68.)
Igt. Determine tlie perpendieiflar distatiee of its centre from a
plane traversing the pole and at riy/it angles to one of the hound-
ing radiit called tJie first radius, in thefolhwing manner: —
Rule.— Divide the angle subtended by the arc into a conve-
nient number of equiangular intervals by means of radii ; mea-
sure the lengths of the radii from the pole to the arc, and
multiply the third part of the cube of each of them by the
cosine of the angle which they respectively make with the first
radius ; treat these products by one of the rules applicable to
finding' the area of a plane curve (the only difference being that
the common interval is taken in circular measure) ; the result
will be the moment of the figure relatively to the plane tra-
versing the pole : this moment, divided by the area of the
figure, will g^ve the perpendicular distance of its centre from
the plane traversing the pole.
Example,
No.
of
Badii
Radii
1
12
2
12
3
12
4
12
5
12
Angles
Cabes of Badii
with
3
First
Radius
576
0°
576
5'^
576
10=*
576
15*^
576
20°
Ooaines
Prcclacts
10000 576-0000
-9962573-81 12
-9848 567-2448
•9659 556-3584
•9397.541-2672
Simpson'e
Mnlti-
pliers
1
4
2
4
1
Products
5760000
2296-2448
1134-4896
2225-4336
541-2672
Interval in circular measure
3
Moment relatively to plane traversing pole =
F
6772-4352
•0291
: 197-077864
66
CENTRES AND MOMENTS OF FIGURES.
Moment 197 '07 7864 ^ 7.041 /perpendicular distance of centre
Area 25-1327 ** \ from plane traversing pole.
In algebraical symbols the moment, as here found, is ex-
pressed thus : —
cos Odd,
f
2nd, Determine the mom-ent of tits figure relatively to the first
radius precisely i?i the same way as in ilie foregoirvg rule, with the
exception tJiat sines must he used in the place of cosines; this
moment, divided by the area of the figu/re, will give the perpen-
dicular distance of its centre from the first radius.
Note, — It is usual, in practice, to defer the division of the
cubes of the radii by 3 until after the addition of the products.
Example,
No.
of
Radii
12
Radii
1
2
12
3
12
4
12
5
12
Cubes of Radii
3
Angles
with
First
Radius
676
676
676
676
676
0°
5°
10°
16°
20°
Sines
of
Angles
•0000
•0872
•1736
•2688
•3420
FrodnctB
•0000
60-2272
99-9936
149-0688
19a-9920
Simpson's
Molti-
plien
1
4
2
4
1
Frodncts
•0000
200^9088
199^9972
696-2762
196-9920
Interval in circular measure
1194-1732
•0291
Moment relatively to first radius =» 34-760440|
Moment 34-75044 T perpendicular distance of centre from
Area 26-1327 " \ first radius.
In algebtaical symbols the moment as here found is ex-
pressed thus : —
f
-s" sin BdB,
6. To find the perpefidicular distance of the centre of a solids
hounded on one side by a curved surface (figs. 87 and 88), from
a plane perpendicular to a given axis at a given point,
BuLE. — Proceed as in Bule 4, p. 63, to find the moment
relatively to the plane, substituting sectional areas for breadths :
then divide the moment by the volume (as found by Bule 2, p. 54) ;
the quotient will be the required distance. To determine the
centre completely, find its distance from three planes no two of
which are parallel.
CBNTSB8.AirD llOMENTS OF FIGUEB8. 67
7. HAving the moment and centre of a figure relatively to a
given plane^ to find the new moment and centre tff the fi'gute rela*
tivelg to the same plane when a part cf the figure is shifted,
(Fig, 103.)
In the iignre wlk let c be its Fio. lOs.
centre, and zz' a plane with respect
to which the moment of the figure is
known ; suppose the part WSM to
be transferred to the new position
SNL, so as to alter the shape of the
figure from wlk to hnk ; let i
be the original and H the new cen-'
tre of the shifted part: then the
moment cf the figture mkk reloHvelg
to the plane zz' is found as.foll4nes : —
Rule. — Measure the distance, perpendicular to the plane of
moments, between the centres of the original and new position
of the shifted part, as hd, and multiply it by the magnitude
of the shifted part ; the product will be the moment required.
The newpositioti cfthe entire figure is thenfov/nd hy the following
rule : —
BuLE. — Multiply the distance between the centres of the
original and new position of the shifted part by the magnitude
of that part ; that product, divided by the magnitude of the
whole figure, will give the distance the centre has traversed in
the direction in which the part has been shifted, and in a plane
parallel to a line joining the centres of the original and new
position of the shifted part, as from c to c' in fig. 103.
8. To fifid the centre of a wedge-shaped solid (fig. 104) hy
means of polar co-ordinates.
1st, Determine the perpendicular distance ff its centre rela*
Uvely to a tranjmerse sectional plane, as pab.
Rule. — Divide the I^q, 104 -
solid by a number of
parallel and equidistant
planes, as pab, PiA,b„
PjA^,, &c.; then mul-
tiply each sectional area
by its distance from the
plane fab; treat the
products as though they
were the ordixiates of a curve of the same length as the figure ;
the area of that curve, found by the proper rule, will be the
moment of the figure relatively to the pdane pab : that moment,
divided by the volume of the figure, will be the distance
required.
/
m
CENTRES AND MOMENTS OF FIGURES.
2nd. DgUrmine the perpsndieular dUtanee qf iU eanih'e r0-
laiivaltf to a longitndiMLL plane pamn^ through its edge, as MPM|
perpend/tcular to the first radius, PB.
Rule. — Divide the figure by a nnmber of longitndiiial
planes radiating from the edg« mpm at equiangular intervals
(as PP4AA4, PP4OC4, PP4BB4) ; also divide the length of the figure
into a number of equal intervals by ordinates, and treat each
of the longitudinal planes as follows :— Measure its ordinOktes,
take the third part of their cubes, and treat thoae quanti-
ties as if they were ordinates of a new curve ; that is, find its
area by one of Simpson's rules : the area of that new curve is
termed the moment of iilertia of the longitudinal plane in
question. Then multiply each moment of inertia of the several
planes by the cosine of the angle made by the plane to which it
belongs with the plane pb, and treat these products by a pioper
set of Simpson's multipliers ; add together the products, and
multiply the sum by J of the common angular interval in cir-
cular measure if Simpson's first rule is used, and by | if Simp-
son's second rule is used. The result will be the moment of
the figure relatively to the plane mpm. This moment, divided by
the volume of the figure, will be the distance required.
The algebraical expression for the moment as found in this
rule is
J'J
^ cos $datd9.
Srd. Determine the perpendicular distance of its centre re-
latirebf to a longitudinal plane passing/ through its edge^ and a
radius as pp*bb*, by the foregoing rule, with the exception of
muUiplging by sines instead of cosines.
Note. — In practice it is usual to defer the division of ih^
cubes of the r^dii by 3 until afteir the addition of the ptrodviqts.
9. To find the centre of gramiy of a plane area containiCd
between two oonsecutitfc ordinates^ with retpeot to ths^ near end
ordinate.
Rule. — To the sum of three times the near end Ordinatot and
ten times the middle ordinate, subtract the far end ordiBate^ wid
maltiply the remainder bv the Wia.v^
square of the common interval. The
product, divided by 24, will be the
moment about the near end ordinate. t>^
On dividing this by the area, the
longitudinal position of the centre of
gravity is obtained.
Ex. I In fif;, 105 let ABO be the
base, and ad, be, and of the ordi-
nates. Call them y^, y^, and ^3
respectively, and let the common interval be denoted
b^ A. Then the moment of ^-Jia area abed about the
y,
*""""*ifr*-""^ ^•••-^-■-
y.
y^
B
MOMENTS OF INERTIA Am> KADII OF GTRATIOlf. 69
near end ordinate ad is equal to Q^.l±l^lZy»l'i^ . If thfa bt
24
divided by the arei of abed (see p. 46), the quotient will be the
distance of the C.G. from ad.
For an example, let the ordinates be 62, 85, and 9*4 feet,
and the common interval 12 feet.
mmm
No. of
Ordl-
nates
1
2
3
Ozdiniktes
62
8-6
9-4
Multipliers
for Area
5
8
-1
Products
OzdiMtes
310
68 0
^9 4
6-2
8-6
9*4
MidtipUers
lor
Momoate
3
10
-1
PfoduotB '
18^
85-0
-9-4 I
89-6
(Interval) _ ,
l2
Area of portion included \ _ gg.g
between 1 and 2 • j ""
94-2
(Interval)* _ g
51
Moment ab6ut 1 = 565*2
Moment 565*2
Area
89-6
= 6*308
Perpendicular distance of centre of
portion included between Nos. 1
and 2, from No. 1 ordinate.
Noie,-*-^WheA thft iKKnnent ef the area is required about the
middle wdinaie, tlM above multipliers sbotild be changed to
7, 6, - 1 ; so that moment = "^^ +^y^-P» ^ }^\
24
MoMElifTg OF InERTII AND BaDH OF GtKATTON.
!• To find the moment cf inertin of « bodff about « ffiven
axis.
Rule. — Conceive the body to be divided into an indefinitely
great number of small parts; multiply the mass (or area) of
each of these small parts into the square of its perpendicular
dislauioe frtfrn the given axis : Ihe suia of all these products as
obtained will be the moment of the body about the given aicis.
2. To find the square of the radius of gyration of a body
about a given axis,
KuLB. — ^Divide tbe mohient of inertia of the body relatively
to the given «xls by the mass (or area) of the body.
70 MOMENTS OF INBRTIA.
3. 6Hven tJie moment of inertia of a body about an axis
traversing its centre of gravity in a given direction, tc find its
moment of inertia about another axis parallel to the first.
Rule. — ^Multiply the mass (or area) of the body by the
square of the perpendicular distance between the two axes, and
to the product add the given moment of inertia.
4. Given the $eparat6 momMts 4if in»rtia^-ef « mt ef bedies
abovt parallel axes traversing their several centres of gravity ^ to
find the moment of inertia of these bodies about a common axis
parallel to their separate aaes.
Rule. — Multiply the mass (or area) of each body by the
'square of the perpendicular distance of its centre of gravity
^rom the common axis ; the sum of all these products, together
with all the separate moments of inertia, will be the combined
moment of inertia.
5. Given the sqiiare of tJie radius of gyration of a body about
an axis traversing its centre in a given direction, to find the
'Square cf the radius of gyration about another axis parallel to tlie
first.
Rule. — Square the perpendicular distance between the two
axes, and add the product to the given square, of the radius of
gyration.
6. To find the m^fment of inertia of a plane area, bounded an
on^ side by a mirve (see fig. 102), relatively to its base Une,
RULE.-*r-Divide the base line into a suitable number of equal
intervals, and' measure ordinates at the points of division ; take
the third part of the cube of each of these ordinates, and treat
those quantities so computed as the ordinates of a new curve :
the area of that new curve, found by the proper rule, will be the
moment of inertia required. In algebraical symbols the above
rule is expressed thus : —
fvl,..
yate,^When the moment of inertia is required as a whole,
and not in separate pa^rts, it is usual to postpone the division of
the cubes till the end of the calculation.
7. To find the moment qf inertia of a plane area, bounded on
one side by a curve, relatively to one of its lordinates.
RULB. — Multiply each ordinate by its proper multiplier, ac-
cording to one of the rules for finding the area of such figures ; then
multiply each of the products by the square of the number of
whole intervals that the ordinate in question is distant from the
MOMENTS OF INSBTIA.
71
ordinate taken as the axis ol moments : the som of these pro-
ducts, multiplied by J or | the cube of a whole interval, accord-
ing as Simpson's first or second rule Is used, will be the moment
of inertia required.
In algebraical symbols this rule is expressed thus :—
Example I,
Calculation of Moment of Inertia of the Quadrant
OF A Circle Relatively to the Base Line.
No. of Intervola
Ordinatw
Cubes tA Ordinates
Hnltlplien
Products
8
0
1
2
3
H
H
3|
4
16-00
15-49
13-86
12-49
10-58
9-33
7-75
5-67
000
1365-33
1238-89
887-50
649-48
394-76
270-72
155-16
57-29
000
1
4
2
I
1
i
1
i
Int-erval
1365-33
4956-56
1331*25
1298-96
296-07
270-72
77-68
57-29
0-00
9662-76
a
12870-34
Example II,
Calculation of the Moment of Inertia of the Quadrant
OF A Circle Relatively to the £ndmo8t Ordinate.
No. of
Intervals
Oidinates Multipliers
i
ProdactR
Squares of Noe*
of Intervals
Prodticts
0
160000
1
16-0000
ooo
000
1
15-4919
4
61-9676
1-00
61-9679
2
13-8564
H
20-7846
4-00.
83-1384
^
12-4900
2
24-9800
6-25
166-1250
3
10-5830
1
7-93725
900
71-4363
^4
9-3274
1
9-3274
10-5625
98-5207
3
7-7460
\
3-8730
12*2500
47-4443
3
5-5678
1
5-5678
140625
78-2972
4
1 ^-0000
\
00000
160000
00000
-
A
pproxima
te momei
Interval'
3
it of incTtia*
, 596-9288
12734'4810
72
MOMENTS OP INERTIA.
Definition. — If a body be oonoeived divided into an infinite
tt umber of parts, and the mass (or area) of each part be
multiplied by the square of its distance from a fixed point,
the sum of all these products is termed the polar moment of
inertia about the point.
^, To find the polar moment of inertia of a plane area
about a point,
I (I) EuLE. — At equal angular intervals sufficient to in-
clude the whole area, draw radii, from the point to the peri-
meter. Treat the fourth power of these radii as the ordina(«8
of a new curve having a common interval equal to the angular
interval between consecutive r^Klii expressed in oircular
measure. One qnarter of the area of this curve, found by the
proper rule, is the polar moment of inertia required.
Example. — ^Find the polar moment of inertia of a semi-
circle of 5 feet radius about one end of the diameter. '
I The polar radii at an angular interval of 15^ are 10'00«
9-66, 8-66, 707, d'OO, 2'59 feet.
No.
Radius
(Radius)*
Multiplier
Prodnct
1
1000
10,000
1
10,000
2
966
8,735
4
34,940
3
8-66
6,624
2
11,248
4
7-07
2,498
4
9,992
5
5-00
625
2
1,250
6
7
2-59
45
4
1
.8
180
Common i
nterval = -26]
67,610
X i^ X ^ X >aftl8
Polar moment of inertia s 1,475
I <II) Bole. — ^If the moments oMnertlaabottlrtwd perpen-
dicular axes through the point are known, their sum is equal
to the polar moment of inertia about the point.
JOefinitions. — The^produci of inertia of an airca about two
perpendicular axes is the algebraic sum of each element of area-
multiplied by the product of its co-ordinates with reference
to the two axes. In the first tfmd third <]uadiatit8 the product
of inertia is positive ; in the second and fourth quadrants
it is negative. . '
The principal axes of inertia through a point are ihose
axes about which the product of inertia is Sero.
9. Given the momenta and products of inertia about two
perpendicular axes, to find the corresponding quantities about-
any two other perpendicular axes,
BuLE. — If Go;, oy (fig. 106) are the axes, X and Y the moments
of inertia about them, and V their product, the moments and
M0MBNT8 OP INERTIA.
78
product €£ ineftia About 0«, Oy' (denoted by X',t', and p'), making
a positive angle B with the original axes, are given by the
following formulae : —
X' = Xcos-^9 + Ysin^a-apsln^cosO, '
y' = xsin'e + Yco8*tf + 2Pftinacos«,
80 that x' + y' = X + Y ; and
p' = P cos 2a - i(Y - X) sin 2tf.
Note. — If ox and oy are principal axes, P s O, and the
formulae become
x' = xcos'^a + YBfai'«; Y' = xsin'tf + Ycos'd;
p'a- J(Y-X)8in2#.
FlO. 106.
If an ellipse (fig. 106) be drawn having its prin(^pal
axes Od;, Ojf along the principal axes of inertia, and of
magnitude Da, ob eqtud to ntdii of gyration abovt oy and ox
respectively, me radius of gyiation about any other axis 0^
is represented by the perpendicular cm drawn to that tangent
tff the ellipse whtch is parallel to ihe AxLs Oaf ; the moment
of Inertia about Oaf is proportional to the square of cm, or
foully inversely proportional to the square of the radius op
along Ox', The product of inertia about Ox', oy' is similarly
represented by the product of OM and mq^ where OQ is <{on-
jugate to OP.
10. Given the moments and product of inertia ahout two
perpendioukfr axee, to find the principal mommitB and uxes
of inertia, •
BuLE. — If \, Y, and P are the moments and product of
ineribili respectively for the axes ox, oy, the angle 9 (reckoned
positively) made by the principal axes Ox^, Oy', with the
original axes is given by the formula—
2p
tan 2t> 5= — £
74 MOMBNTS OF INERTIA.
The magnitudes of the principal momenta of inertia x', T'
about ox', oy% are given by—
assuming x' to be the least and t' the greatest moment of
inertia.
11. Given the momenU of inertia about three axes, two
perpendioular and one bisecting the angle between them, to
find the principal moments and axes of inertia*
BuLE. — If X, T, are the moments of inertia about the axes
Ox, og, and z that about an axis bisecting the angle gox, tlie
angle $ (reckoned positively) made by the principal axoe
Ox', oy', with the original axes is g^ven by the formula —
tan 29 =
Y -X
The magnitudes of the principal moments of inertia x% T'
about Ox^, og', are given by —
, Y + x a/« /<. %.y' + x'
x'= -2--Vz'-z(Y.+ x)+— g-
, Y + x . . / , . . . , Y^ + X*
Y' = — 2--+Vz -Z(^ + ») + — 2^
Note. — Since X -f Y a= x^ + T^ the sum of the moments of
inertia about any two perpendicular axes is constant.
If the area has an axis of symmetry, the principal axes
are along and perpendicular to this axis.
Ex, — An nnequal-sided parallelogram is formed of two ri^bt-
angled Isosceles triangles of 1 inch side. Find the principal
moments and axes of inertia.
Take ox parallel to the shorter sides, and oz perpendioalar
to the longer sides. Then.x — ^; T*-}; Z^^.
By the formulis above tan 2tf a - 2 ;
6 a - 82^ or 58^, the former corresponding to the least moment
of inertia.
Greafcost iorY'«=i + -2^» -218.
V6
Xicast I or x' - i - "jj - -032.
TABia oy squAKES oy Radh oy Qybation oy a fbw Bpboiai. Pioui«b,
Boay
Reetangle; sides a and &
Sonare; side a
Triangle; rides a» h, e;
heights a'.b'.o'
Equilateral triangle ; height d
Trapezoid : height 7^, parallel
sides a and h
Trapezoid with two right
angles ; parallel sides a and
h, perpendionlar side h
Circle; diameter a
Ellipse ; ffiameten a, h
Ck>mmon parabola; height a,
base h perpendienlar to axis
Sphere ; radius r
{
{
Axte
ildea
axis through CQ. parallel
to side a
any axis tl^ongh C.G.
side a
(
axis through aO* parallel
to side a
any axis through aO.
side a
(
{
{
}
Spherical shell; external and
internal radii n and n
Ellipsoid of reyolntion ; trans- 1 1
Terse semi-axis r i
Ellipsdid ; semi-axes a, &, o
Circular cylinder ; radius r,
length Sa
Hollow circnlar cylinder ;
radina— external n» internal
rs; length Sa
Blliptic cylinder; teml-axes
b. 0. length 2a
axis throQgh 0.a. parallel
to side a
side^
axis through O.G. parallel
to side %
diameter
centre (polar)
diameter 0
axis of parabola
base b
axis through O.O. parallel
tobaseb
diameter
centre {s^Xax)
diameter
jols of reyolutifla
axisSa
Cone; height h, radlns of
baser
r longitudinal axis
l transverse diameter
^ through C.G.
( longitadinal axis
I transverse diameter
V. through C.a.
C longitudinal axis
1 transverse axis 2& through
I CG.
longitudinal axis
transverse axis through
C.G.
transverse axis through
base
plane of base
8
IS
11
fl?
6
«?
18
4!
IS
6 ' a+b
*? a»+4ft5+b«
18* (a+W«
iCoa-t-bS)
a^+aa»b+aab»-l-b
18(a+b)S
a^/ie
a^/B
6«
18
8a*/»
19a«/178
5
8tf
6
2 (ri*— rg*)
6 (ri'-rrf)
5
b«+£l
6
r!
8
4 8
8
4 '*"8
i
4'*'8
10 80^
10
Moment of laartlaM gatuttn ol mdlm ol gyimMwi x m— (or area) of the llwaw.
76
MECHANICAL PRINCIPLES
MECHAHICAJL PSIHCIPLE8.
Resultant and Rbsoltjtion op Fobcbs.
1. To find the reniUant of two forces acting throvgh one paint
hut 9iot in the sam-e direction. (Fig. 107.)
Let AB, AC represent the two Fia. lOT.
forces p and Q acting through the
point a; complete the parallelo-
gram ABCD : then its diagonal AD
will represent in magnitude and
direction the resultant of the two
forces p and q. ^ q
R = resultant. « * angle P makes with q.
a «= angle R makes with Q. ^ wangle R makes with p.
Fi«. IflB.
R« Vp* + Q* + 2.P.Q. cos«;
P Q
sin /i = sin 6-^; sin/3 = sin*-«
R R
2. To find the remltant of any number offerees acting in the
same plane and titrongh one paint but not in the Htme dirsetiam,
(Fi|. 108.)
Let p, p„ Pj, p, be the forces
acting through the point of
application o; commence at o
and construct a chain of lines
OP, PA, AB, BC, representing the
forces in magnitude and paral-
lel to them ; let c be the end
of the chain : then a line R
joining oc will represent in
magnitude and direction the 0^
resultant of the forces p, p„ Pj,
and p,.
yike. — This geometrical pro-
blem is true whether the forces
act in the same or in different planes.
R= resultant.
1^ = angle made by R with a fixed axis OX,
a, «„ ^2, &c. = angles made by the forces p, p„ p^ Slc^ with OX,
2x B sum of the series of P . cos a + P, . cos a, + p, • ocmei t^ too,
36y = sum of the series of P . sin a + p, . sin Oj + P* • sin a„ &c.
R . cos a = 5x. R = y/ixKy + i^y*
B . sin 0 = 2v.
tan e =
cos 9=s
2y
5x
2x
R
sin 6=.?!.
MECHANICAL JPItlKCIFLE9
77
Via. 109.
o4
)
f
.1^
i
»
3.. To Jind the retuUant cf three foroe% custrng through ene
paint and vtakin^ right anklet withfone emother. (Fig. 109.)
Let OA, OB, oc represent in magnitude
aikd direotion the forces x, T, Z acting through
one point o ; complete the rectangular solid
AEFB: then its diagonal OG will represent
in magnitude and direction the resultant
of the forces x, Y, z.
K« resultant.
a,/3, yaethe angles B makes with x, T, z»
respectivelj.
Y =sR . cos j8. R= A/'X« + Y« + Z».
Z = R.C0S7. X = R. cos a.
4. To find the resukamt of any mmber of forces acting through
onepoixt in different directions and not in the same plane.
Let p, p„ Pg, &c., be the farces a, 3,7 ; o„ /9„ 7i ; «2» ^2, 7j»the
angles their directions make with three axes passing through
the point of application and making right angles with one
another.
R=» resultant.
2X-P . cos O+P, . cos O, + P, . cos O2 + &C
2y = p . cos^ + P, . cos /3, + P2 . cos 3, + &;c.
2z»p . cos 7+P, . cos 7, + P2 . cos 72 + &C.
B « >v/(Sx)» + (2Y)» + (SZ)*
cos o« —
Sx
B
C0S^= ^
B
COS 7 = ~.
B
N.B« Cosines qf obtuse %nglcs are negative.
Note, — P cos o, p COS 3, and P cos 7 are termed the components
of the forces in the directiouiiB of x, Y, and Z respectively. The
components of the resultant are obtained by adding (allowing for
sign) the components of the several fences in their respective
directions.
y4BAH.EL FO^ICES.
J. eoupie conaiais of two equal forcesy as p and Q (see
^. 110)y acting in parallel and opposite direcUons to one
another, and is termed a righi* or leftrhanded
couple, according to whether the forces tend to turn
in a clockwise direction or the reverse.
The moment of a cowple is the product of either
of the forces into the perpendicular distance ab
between the lines of direction of the forces. The
distance ab is termed the anp or lever of the
couple.
Fig. ho,
4 L
Fig. 111.
Fia.lia.
78 MfiCH\NIOAL PRINCIPLES.
5. To find the resultant moment of any number of couples
acting upon a body in the same or parallel planes,
BuLE. — ^Add togetiher iihe moments of the rigbt-^ and left-
Landed couples separately } the difference between the two
soma will be the resultant moment, which will be right- or
left-handed, according to which sum is the greater.
6. To find the resultant of two partUlel forces, (Fig. Ill
and 112.)
The magnitude of the resultant of two parallel forces is
their sum of difference, according .to whether they act in the
same or contrary direotions.
Let fig. Ill represent a
case in which the two forces
act in the same direction, and
fig. 112 a case in which the
components act in opposite
directicms.
Let AB and CD represent
two forces ; join ad and CB,
cutting each other in £{ in da
(produced in fig. 112) take df
c=BA ; through F draw a line
parallel to the components ;
this will be the line of the
resultant, and if two lines bo
and AH be drawn parallel to
BC, cutting the line of action
of the resultant in a and H,
OH will represent the magni-
tude of the resultant.
Or, numerically, the lios of action of the resultant is
obtained by adding (allowing for sign) the moments of the
two forces about any jpoint, this being equal to the moment
of the resultant ; the perpendicular distance of the line of
action from the point is obtained by dividing this moment by
the magnitude of the resultant.
AF =
DO. AD
OH
DF =
AB.AD
OH
7. To find the resultant of any number of parallel forces.
BuLE. — Take the sum of all those forces which act in onf
direction, and distinguish them as positive ; then take the sum
of all the other forces which act in the contrary direction, and
distinguish them as negative. The direction of the r^mitant
(positive or negative) will be in that of the greater of these
two sums, and its magnitude will be the difference between
them. . .
8. To find the position of the resultant of any number of
parallel forces when they act in two contrary directions.
BuLE. — 1st. Multiply each force by its perpendicular dis-
tance from an assumed axis in a plane perpendicular to the
MECHANICAL PRINCIPLES. 79
lines of aotion of tbe forces ; disiiiifni^ those moments into
right- and left-handed^ and take their resultant, which divide
by the resultant force : the quotient will be the perpendicular
distance of that force from the assumed axis.
2iid. Find by a similar process the perpendlcalar distance
of the resultant force from another axis perpendicular to tiie
first and in the same plane.
9. To find the resultant of any number of couples not
necessarily in a plane.
Two couples of equal moments in the same or in parallel
planes are equivalent to one another, whatever the magnitudes
and positions of the forces composing the couples may be.
A couple is therefore conveniently represented by a lino
perpendicular to its plane, and of length ]proportional to
its moment ; usually tne direction of the line is taken so that
its relation to the direction of the couple is the same as that
between the travel and the rotation of a right-handed screw.
Note that any two parallel lines of the same magnitude and
sense represent the same couple.
BxTLE. — Replace the couples by lines as above, giving them
their correct magnitudes and mrection, and treat these as
forces through a point by Bule 4. The resultant gives the
magnitude and direction of the resultant couple.
10. To find the resultant of any number of forces in a
plane,
Bule. — ^Treat thorn as forces through any fixed point by
Bule 2y and find their resultant. Calculate also the moment of
each force aboutHhe point, and add them together allowing
for the sign of each. The resultant moment divided by the
magnitude of the resultant force gives the perpendicular
distance of its line of action from the point.
Definition, — ^The moment of a force about a line that it
does not m«eit is the product of the component of the force
perpendicular to the line with the shortest distance between
the line and the line of action of the force.
11. To find the resultant of any number of forces, not in
one place,
Bule. — ^Resolve the forces parallel to three perpendicular
axes as in Bule 4^ and find the magnitude and direction of thear
resultant B. Calculate the moments of each oompouient about
the three axes, and treating these as couples find the resultant
couple F by Bule 9. Kesolve this couple into couples Q
parallel to tne force B, and H perpendicular to B. Besolve the
couple H into 2 foroee Bxi B2> of wnich B^ is equal and opposite
to B, while B2 is equal and parallel to B ; find the position of
Bg (not in plane of figure). Then the final resultant is equal
to the force B^ combined with the couple Q (since b and B|^
neutralize). The combination of a force in and a couple
ifbovt the same line is termed a wrench.
80 CENTRE OF ORAVITY OF BODIES.
CENTBE OF 6BAVITY.
1. To find the moment of a hody''* weight relatively to a given
plane.
Rule. — Multiply the weight of the body by the perpen-
dicular distance of its centre of gravity from the given plane.
2. To find the common centre of gravity of a set of detached
l^odies relatively to a given plane.
KuLE. — Find their several moments .relatively to a fixed
plane ; take the algebraical* sum or resultant of those moments
and divide it by the total sum of all the weights : the quotient
will be the perpendicular distance of the common centre of
gravity from the given plane.
Note. — When the moments of some of the weights lie on
one side of the plane, and some on the other, they must be dis-
tinguished into positive and negative moments, according to the
side of the plane on which they lie, and the difference betw een
the two sums of the positive and negative moments will be the
resultant moment. The sign of the resultant will show on which
side the common centre of gravity lies.
Let w, w', w^, &c. = the several weights.
d, d\ d\ &c. = the several perpendicular distances of the
centres of gravity of w, ?^•^ w^, &c., from the plane of moments.
D = the perpendicular distance of their common centre of
gravity from the plane of moments.
wd + w^d* + w^d^ + &c.
3. 'Jb find the centre of gramty of a hod^f 'coymsting of parts
ofnneqval heavitiess.
RuirB.-T-Pind separately the centre of gravity of these several
parts, and then treat them as detached weights by the foregoing
rule.
4. To find the distance through 7vhic7i the common centre of
gramty of a set of detached weights moves when one of those weights
is shifted ijito a new position.
Rule.— multiply the weight moved by the distance through
which its centre of gravity is shifted ; divide the product by t^e
sum total of the weights: the quotient will be the distance
through which the common centre of gravity has moved in a
line parallel to that in which the weight was shifted.
Let w = weight shifted.
<f- distance through which w was moved.
Ws=8um total of weights.
Ds; distance through which the common centre of gravity
has moved in a line parallel to that in which the shifted weight
was moved.
^ wd , DW
W w
MOTION. 81
XOTIOV.
Veloctpt.
The speed of a body or of a point within a body lb iihe distance
tia?elled in an in£niteidinal space of tune aivlded by that
tune.^ The velocity of the body takep also into acooont the
direction in which the body is movii^ and is completely
represented by a line drawn in the direction of motion, whose
length represents to scale the speed.
Contpotition of velocities, — ^To combine several velocities
impressed simultaneously upon a body, if op, oP]^, dp^, op^
(fig. 108, p. 76) repres^it the component velocities, draw
PA parallel and equal to op, ab, and bc parallel and equal
respioctively to op* and 0P3. 00 is the resultant velocity of
the body. Similarly the resultant velocity 00 may be resolved
into two component velooities in any required directions x and
T by drawing lines from od^ do piaraHel to x and T ; the
lengths od, do represent the magnitudes of the component
velocities.
Example, — ^If a boat is propelled at a speed and in a
direction rcfpzesented by ao (fig. 107, p. 76) in a stream whoso
velocity is represented by ab, the resultant velocity of the
boat is represented by ad. To combine any number of
velooities analytioallyy resolve each idong three axes at right
angles (or two If all the velocities are in one plane) by
multiplyii^ each velocity by tiie cosine of the angle which it
makes with the axis ; add, allowing for sign, the components
along each direction. The sums are the components of tho
resoltant velocity in the three directions, wliich may bo com-
pounded as above. B.g., if v^, v^^ f^s* • • • are the velocities
making angles a|, a^, c^, . . . with the axis 02;, /9i, /32, iSs . . . with
the axis O]^, and yit Tit 7ii« • • • with the axis 02^ the components
p, Q, B, of the resultant along oo;, oy^ os, are given by —
P B t7i cos oi -I- v% cos oa + vs cos 03 -h ...
Q = Vi cos /9i -I- ra cos iSa + Vk cos iSs + • • •
B = i7i cos 7i + Vs cos 711 + Vk 80S 78 4- • * •
The resultant 8 is given by £^ = P* + Q* + B^ ; and it makes
angles A, B, 0, with the axis, given bj—
P Q R
cos A = - ; cos B = - ; cos 0 = -
Velocity diagram for a linJted nfieohanism. — ^To find the
velocity of any part of a linked mechanism, a velocity diagram
may be drawn as illustrated in the following example. AOB
represents diagrammatically (fig. 113) the crosshead of a
screw-steering gear, AC^ bd, the connecting links, and 0 and
D are forced by guides to follow the axis of the frame 00.
If the velocity of A is known, that of 0 (or any other part)
can be found ; and conversely.
82
ANGULAR VELOCITY.
Draw oa to represent the velocity of A, oa being perpen-
dicular to OA. ob in the opposite direction represents that
of B. The velocity of c relative to A is necessarily perpen-
dicular to AC, while relative to the frame it is parallel to the
axis. Therefore, draw ea perpendicular to ca, and oe parallel
to the axis ; this gives c. Similarly the point d is obtained.
00 and od are the velocities of the points 0 and D. The
velocity of any other point, say E in the connecting link AO,
is obtained by dividing ao at e so that ae : eo ^^f A.'B : ^o^
Join oe, which is the velocity of the point E.
Fig. 113.
Pig. 114.
If / be the middle point of cd, of is the mean velocity of
0 and p, I.e. the velocity of the screw shaft as a whole, to
allow for which a small amount of play haa to be given.
Ijlote that the shaft is movinig towards the crosahead, and
that the velocities of c and D relative to the shaft are iriven
by fo, df. *
Angular Velocity.
The angular velocity of a body about an axis is the ansrle
terned through about the axis in an infinitesimal space of time
divided hv that time. It is usually expressed in radians
per second or m jflevolutions par minute, the unit in the former
Art
case being ~ or 9-55 times that in the latter.
Composition and resolution of angular veloeiticg.^'FhQ
angular velocity about an axis may be represented by a line
ACCFXERATION. 83
drawn parallel to the axis, and of length proportional to
the magnitude of tha angular velocity. The direction of the
lino usually bears the eame relation to the direction of rotation
83 that existing between the trarel and rotation of a right-
handed screw. When so represent ed, angular velocities are
combined and resolved in the same way as linear velocitiea
(see p. 81).
ACCELEBATION.
The acceleration of a body is the rate of change of its velocity
or the change of velocity in an infinitesimal space of time divided
by that time. The velocity of a body comprises both its speed
and its direction ; hence the acceleration may generally be
divided into two parts — (a) that due to increase oi speed, which is
i^resented by -^ • where i; is the iqpeed» and is tangential to the
direotioii of motion ; (6) that due to alteration of direotion, which
is directed normally towards the centre of curvature and is equal
to tl^lfi where ^ is the radius of curvature of the path.
Anguhir acceleration is the rate of increase of angular
velocity; and, for a body revolving about a fixed axis, is
represented by -z- where w is the anguUur velocity.
Composition of acceleraiions, — Accelerations, linear and
angular, are combined and resolved similarly to velocities.
Acceleration diagram for a lihlced mechanism, — To find the
acceleration of any part of a linked mechanism, a velocity
diagram is first constructed as in fig. IIB, p. 82. To find the
accelerations of the steering gear shown, that of A, assuming the
crosshead to revolve uniformly, is equal to — or — , and may be
represented by o'a' parallel to AO. The acceleration o*b' of OB is
equal and opposite. The normal aoceleiation of o relative to A is
represented by a*c^ equal to — and drawn parallel to CA ; the
tangential acceleration of 0 relative to A is represented by c^c'
drawn perpendicular to <^a', the length of <?c' being at present
unknown. The acceleration of 0 relative to the frame 00 is
parallel to the axis ; so that o*c* is drawn paiallel to the axis
meeting (^o' at c', giving the length of o^e'. Similarly the
acceleration of D is found by drawmg b'tfi, and d^d', giving o'd\
That of any point B in the link AG is obtained by joining ac' and
dividing ft at «% so that a'e : e'c' ^ ab : EC. Join o'e\ which is
the aoeeleration of the point B. The accelerations of C and D
rda^ve to the screw are equal to } c'd'.
84 DTNAMIOS.
2>TKAMIG8.
Belations between Forob and Motion.
Ot, 0|/, Oz = Z perpendicular axes.
p, Q, B = component forces along Go;, oy^ oz acting on a body.
«, v, u; = component yelocities along Ox, Oy, oz*
m = mass of body.
Mx, My, Ml = momenta parallel to ox, oy, oz respectively =
mu, mv, mw.
f, g,h =^ component accelerations parallel io ox, oy, oz,
g = acceleration due to gravity = 32-2 in foot-second anits.
^ . du dUg €p» ^ d/f> duy ^y
Note, — In the above, if mi is in poonids^ T, Q, b are in.
poundals, one poondal being equal - pound or about half an
ounce weight ; if, on the other hand, the forces are expressed
in pounds, the mass m must be expressed in terms of the
gravitational unit equal to g (about 32) pounds.
Example. — A force of 2 lb. acts upon a mass of J lb. . To
find the acceleration. * *"
ft
The mass in gravitational units s —
• P 2 2^
.'. Acceleration's: — = ^r- = -^ ~21J ft. per second per second.
tn olg o
Anqulab Motion.
I = mass moment of inertia about axis of revolution.
( = angular acceleration.
Of = angular velocity.
9 = angle turned through.
M s angular momentum ^ l«.
G = moments of forces about axis*
N6U, — If G is expressed in foot-pounds, i must be expreescd
in the gravitational unit, or is ilg of the density of the
material multiplied by the volume moment of inertia pf the
body (eee pp. 69-76).
WOBK AND EnEBGT.
Th6 worJc done by a force on a body is the product of the
force by the distance moved resolved along the direction of the
force. The work done by a couple is the moment of the ooaple
DYNAMTC8. 85
multiplied by the an^le ttuned through resolved along the plane
of the couple. With the preyions notation, if the body runs
through distances x, y, 8 parallel to the axes, and rotates through
an angle 0, the work done is F» + QV + &r + o^, allowing for sign.
The energy of a body is its capacity for doing work.
Kinetic energy is energy due to motion. With the preceding
1 1
notation, its amount is ^i *»(«*+ v*+ to*) + s^ l«'> m and i being
expressed in pounds and the result in foot-pounds.
Potential energy is energy due to position, and is measured
from an arbitrarily fixed datum. A body of height h feel above
the sea-level has potential energy of mh foot-pounds. A ship
has potential energy due both to the height of its centre of gravity
and the depth of its centre of buoyancy.
Molecular energy, due to heat, electrical state, magnetism,
vibration, etc., is frequently waste energy as far as its capacity of
doing useful work is concerned.
Conservation of energy. The work done on a bodv (other than
that involved in a change in the potential energy) in a c^ven
interval of time is equal to the increase of its totaJ energy.
Power is the rate of doing work. It is equal to Pu + Qv +
Wio + 0«», allowing for sign. This is equal to the rate of increase
of energy. The practical unit of power is tiie horse-power,
equivalent to 550 foot-pounds per second, or 88,000 foot-pounds
per minute. Another unit is the waU, 746 of which are equivalent
to one H.P.
UiriFOBU FoBCB IN Line dp Motiov,
p s uniform force in pounds weight.
m = mass in pounds.
/ = uniform acceleration = Fglm,
y = initial velocity in feet per second,
f =s distance travelled.
t = time occupied.
V = final velocity.
t; = V -^/^ ; v* = V* -r- 2/a ; s = Yt + Iff?,
For retarded motion change / to -/.
For motion vertically under gravity / to ^ or ^g, according as
the initial motion is downwards or upwards. In that
^ ease P =t ±m,
¥<xe 'motlbn down an incline of angle a to the horizontal,
replace / by y tan a.
F6r angular rotation with the notation above, A being the
initial angular velocity, » ^ a + ^t ; «' = aV-h 2^0 ;
Gravity.
g ss acceleration due to gravity in feet/second*.
X = latitude of the place.
h ^ height above sea-level#
86 DYNAMICS.
radios of earth in feet » 20,900,000.
R
9
2k
32 088 (1 + -005302. ein* ^ - -000007 ein* 2^ - — )•
Usually g is taken as 82-2, or 981 in centimetres/second*.
Simple Vibbatior.
H s mass in pounds.
a = semi-amplitude of vUlration.
n s frequency or number of double vibrattona per second.
E a modulus or force in pounds required to prodnce nifit
extension.
t a= time,
a = displacement at time ' ^*
/ s acceleration at time * <•'
a =s a constant.
g = 82-2. _
tt = g^V^i » * a sin {2init + a) ; / » - ^ «.
Simple Pendulum.
L s length of pendulum in feet.
T s« time of a single small vibration in seconds.
g 33 acceleration due to gravity » 82-3.
V
Table giving the Lengths of Pendulume
{ IN Inches I
THAT Vibrate Seconds
IN vABious Latitudes.
Sierra Leone
3901997
New York
89-10120
Trinidad
39-01888
Bordeaux
8911290
Madras
3902630
Paris
8912877
Jamaica
39-03508
London
8913907
Bio Janeiro
39-04850
Edinburgh
39-15504
Table oiviNa the Times of Vibration fob Pendulum
SWINGING through LARGE ABCS.
Angle Bwxmg on eftch
side of vertical
8QP
&f 1
Wf
laop
wr
mf
Actual time of
vibration -f Time for
infinitely small angle
1017
1-078
1*188
1-878
1-762
Infinite
DTNAMICS. 87
CoMPOujiD Pendulum.
K s radius of g;fr»tion of body abotit aus of rotation.
h = centre of grayifty b«low axis.
I = length of equivalenti pendnlam.
I = If^lh.
The eetftre of pereu»9i6n, or point at which a blow struck
perpendicularly to the axis will oause no stress at the axis, is
situated at a distance I (determined by the above formula)
below the axis.
CfiNTBIFUGAL FOBCZ.
F = centrifugal force of body revolving in a circle at a uniform
speed, or apparent force required to balance that necessary to
produce the requisite normal acceleration.
w = weight of body.
N = number of revolutions per minute.
n = number of revolutions per second.
V = linear velocity in feet per second.
m s angular velocity in circular measure per geco'^d.
r = radius of circle in feet.
g = acceleration due to gravity = 82'2 nearly.
P =
gr g g -8154 2935
Gyeoscopio Action.
If the axis of a revolving body is made to rotate into a new
position, resistance is experienced due to the 'gyroscopic
action ' of ihe revolving mass. Let ab represent in the usual
way the angular momentum iod of a body having a moment
of inertia i about the axis of revolution, and an angular
velocity ». If this axis is forced to occupy after a short time
the position ac, bc represents the chamgo of angular
momentum. This is «qual to i« x Lbao. If this change is
effected by turnii^ the axis uniformly with angular velocity
a\ the rate of change of angular momentum is ic0<»', which is
equal to the moment o of the applied couple. Note that tlio
plane of G is perpendicular to that of shaft rotation, and of
the direction of movement. If I is in weight units (lbs. X
feet^), and n land n' are the number of revolutions per minute
of shaft rotation, and of bodily rotation,
I 4ir^ NN^ ^ I.n.n'
g^ 3600 "^ 2935
In tho case of a ship going ahead with a right-handed
screw, the forces required on the shaft When turning to
starboard are downward aft and upward forward ; the re-
action on the hull is j^hen such 9^% tp cause a slight trim by
the bov\r.
88
HYDROSTATICS.
Vl
Impact.
Uu Ui = the velocities of two bodies before impact (if moving
in opposite directions make u% negative).
Vit Vi = the velocities after impact.
nil, ma= the masses.
e = coefficient of restitution = ratio of velocity of separa-
tion to that of approach.
For direct impact,
_ Ui (wi -enh) -f fWj tfa (1 -f e)
Ui wi (1 + «) + «j (wij - enti)
Kinetic energy lost 2g(rm + nt,)
Total momentum is unchanged, or nti ui -\- niiUi = mi vi + tits Va-
For oblique impact, resolve the velocities along and perpen-
dicular to the line of impact ; treat the components along^
the line by the above formulae ; the. latter are unaltered hj
the imx>act.
The value of the coefficient e depends to some extent on
the shape of the bodies and the velocity of impact, as wM
as on the material. Approximate values for the impact of
like materials are g^ven in the following table : —
Material
Cast Iron
Mild Steel
Soft^BrasB
Iiead
Elm
Glass
Ivory
0
•70
•67
•38
•20
•60
•94
•81
HYDBOSTATIOS.
The densitftf of a fluid is the weight of a unit volume..
Generally it is stated in pounds ^er oubio foot, or inversely
as the number of cubic feet required to weigh 1 ton. (See
tables on p. 262.)
The specific gravity of a fluid is the ratio of its density to
that of wat^.
Density of a Mixturb op Two Liquids.
Wu W2 = densities of the two liquids.
w = density of the mixture,
mi, ma = proportion of the two liquids in the mixture by
volume.
nu fta =» proportion of the two liquids in the mixture by
weight.
Wi/na == mi wi/wia Wi] to = —^ ~ — - = ^ ■
mi + mj wi ttj
Vl 10»
hydrostatics. 89
Pbessube in a Liquid.
w = density of liquid in pounds per oabic foot.
B = depth below free suHace.
p = intensity of pressure in pounds per square inch,
p = intensity of pressure in pounds per square foot.
4
In salt water, w = 64, p = Biz, p = g.
In fresh water, w = 62-5, p = 62*5f , p » •488«.
■ If the absolute pressure bo required P and p must be
increased by 2120 and 14*7 respectively, in order to allow for
the pressure of the atmosphere.
Note.^-The centre of pressure of an immersed plane
sorfaoe is that point on the surfaoe through which the
resultant pressure acts.
PnESSUBB ON Immersed Plane Subfaob.
If surface be vertical find the centre of gravity Q and
take axes Qx horizontal and Qy vertically downwards. Let
A == area of plane.
h = depth of centre of gravity below^free surface.
w as density of fluid.
T =: total thrust or pressure on plane.
X, y = co-ordinates of the centre of pressure.
Then T — wAh
If = TT / y^ . dx . dy over area.
=1 --r X moment of inertia of area about Gx.
Ah
i sz-^Jxy dx dy over area.
= -=- X product of inertia of area about Gx, Gy,
Ah
If the surface and the axis Gy be inclined at an angle ^ to tlje
vertical, T and x are unaltered, but the value found for y should
be mnltiplied by cos 9,
Pbessube on any Closed Suefacb.
The resultant preasure on the whole Immersed surface
of a body Is equal to the weight of the water displaced by
the body and acts vertically upwards through tlie centre of
gravity of the displaced volume. The upward force is termed
the displacement, and the point through which it acts the
centre of buoyancy.
90 DISPLACEMENT.
DISPLACSMENT, Etc.
Computation op a Ship's Displacement.
This consists of computing the Tolume of the body of the
vessel below the water-plane, up to which it is required to
know her displacement, by one of the rules used for findings
the volume of solids bounded on one side by a curved surface
(see pp. 54, 56).
Two processes are generally made use of in computings
a vessel's displacement, as the calculations in each process
are requined to determine the position of the centre of gravity
of displacement, or centre of buoyancy, and also because tho
two results iare a check on the oorreotness of the calculations.
One t>rooeeB conflists in dividing the length of the ship on
the load water-line by a number of et^uidistant' vertical
sections, computing their several areas by one of Simpson's
rules, and then treating them as if they were the ordlnates
of a new curve, the base of which is the load water-line.
The other process consists in dividing the depfch of the
vessel below the load water-line by a number of equidistant
horizontal planes parallel to the load water-line ; the
areas of their several planes are then computed by one of
Simpson's rules, >nd those areas are treated as if tiaey were
the ordinates of a new curve, the base of which is the vertical
distance between the load water-line and lowest horizontal
plane.
As the vessel generally consists of two symmetrical halves,
the volume of only half the vessel, below the load water-line,
is calculated, the ordinates all being measured from a longi-
tudinal vertical plane at the middle of t^e ship.
Usually the portion below the lowest water-line is treated,
as are also the stern, rudder, bilges, keeU, eto., as an
appendage, its volume being calculated by means of equi-
distant vertical sections. The water-lines that are * snubbed '
or cut short abaft the fore perpendicular or before the after
perpendicular are conceived to extend to these perpendiculars,
the extra volumes thus introduced being regarded as negative
appendages.
The displacemen't of a ship can also be obtained by dividing
the length into sections, spaced as required by Tchebycheff's
rule ; the integration in a longitudinal direction is effected
by simple summation. The wafer-lines are equidistantly spaced
and integrated by Simpson's rules as before. This method is^
generally speaking, more exj>editioud than is the <me pre-
viously described, since fewer ordinates can be employed, and
half the multiplication is dispensed with.
Both methods are Illustrated in the displacement Mieet^
given on pp. 94 ff.
DMPTACEMENT.
91
Determination op a Ship's Centre op Buoyancy fob
THE Upright Position.
The centre of bnojanej is also termed the centre of gravity
of displacement, as it occupies the same point as the centre of
gravity of the volnme of water displaced by the vessel, and its
position is determined by the rules used for finding the centre
of gravity of solids, bounded on one side by a curved surface
(see rules, pp, 66 and 67), with the exception that its position need
only be determined for its vertical distance from a horizontal
plane, and its horizontal distance from a vertical plane ; for the
ship consisting of two symmetrical halves, it must necessarily
lay in the longitudinal vertical plane in the middle of the ship.
Calculation of the centre of buoyancy is generally performed
on the displacement sheet (see pp. 94 ff.).
Curve of Areas of Sections.
This curve (see fig. 115) is of use in des^ning and in
estimating the resistance of a ship, for it fixes the distribution
ol displaoemeat mlong the length.
Fio. 115.
CURVE OF SECTIONAL AREAS.
Method of Conttruction.—Ctmpnte the area of eaeh
traasvene section up to the l.w.l.; and set it of! to scale
on a base of length. A curve drawn through tlie tops of the
ordinates will form the curve required.
Curve of Areas op Midship Section.
This curve (see fig. 116) is used to determine the area of the
immersed part of the midship section of a vessel at any given
draught of water.
Method of C^mtruction.^ Compute the areas of the midship
section from the keel up to the several longitudinal water-planes
Via. 116. which are used foi calculating
^ . the displacement ; set these
III ^ ^ — • areas off along a base line as
ordinates, in their consecutive
jj- -—^ order, the absciss® of which re-
j ^ ^^^ present to seale the respective
^^ *^^ distances between the longi-
tudinal water-planes : a curve
bent through the extremities
•»■ «■ of these ordinates will fonr;
the required curve.
92
CURVJS OF DISPLACEMENT.
OuBVB OF Displacement.
This curve is used to determine the displacement a vessel
has at anj draught of water parallel to the load water-line
(see fig 117).
Method of CoTutruetion. — This curve is constructed in a similar
manner to the foregoii^ curve, with the exception that the oidi-
Fio. 117
7M
nates represent the several volumes of displacemetit (in tons of
35 cubic feet for salt water, and 36 cubic feet for fresh water)
up to their respective longitudinal water-planes.
Curve of Tons per Inch of Immersion.
This curve (see fig. 11 8) is used to determine the number of
tons required to immerse a vessel one inch at any draught of
water parallel to the load water-plane.
To find the displacement per inch in cubic feet at any water-
plane, divide the area of that plane by 12 ; and if the displace-
Fig. 118.
Scale of Tons
meat per inch is required in tons, divide by 35 or 36, as the
case may be.
A = area of longitudinal water-plane in square feet.
T = tons per inch of immersion at that water-plane.
Tc for salt water ; T = ,- — — for fresh water.
12x35 12x36
COEFFICIENTS OF FINENESS.
93
Method of Conatruetion. — ^This cnrre is also oonstmcted
in a lumilar manner to the two foregoing carves^ with the
exception that the ordinates represent to scale the tons per
inch of immersion at the respective water-planes.
The coefficients of fineness of a vessel consist of the block
coefficients (3), the prismatic coefficient (7), and the midship
section coefficient {p). They are determined from the following
equations : —
y ~ volume of displacement in eabio feet.
L = length of vessel at load water-line in feet (or length
between perpendiculars, according to convention).
B = extreme immersed breadth in feet. (Occasionally this is
taken as the breadth at L.w.L. in cases where this is
less than the extreme breadth.)
D = mean draught of water in feet. (Take to top of keel if
bar keel.)
3 = Area of midship section up to L.w.L. in square feet.
V V _ 2
^ "" B.D.
3 =
V
^ = 7.M.
L.B.D.
Another coefficient sometimes used is that of water-line area
(a) given by X = — -- where A is the area of the L.w.L. Usually
LB
this is expressed as a coefficient , '—^. — r, this latter being
420 • ^^^ P®' ^^"^
equal to -r-
/^
Values of these four coefficients for typical ships are given
in the table below.
Table of Coefficients of Fineness.
Class of Ship
Block Co-
efficient
^ I.BD
Prigmatio
Coefflo't
>-3
Kid. Sec.
Coefflo't
BD
Waterline
Coefflo't
I.B
Tons
per Inch
Battleship (modem) .
Battleship (older)
First-class Cruiser
Modem Light Cruiser .
Torpedo Boat Destroyer
Steam Yacht
Fast Passenger Steamer
Large Cargo Vessels .
Sailing Tacht
Tug ... .
•60
•65
•66
•58
•65
•62
•59
•73
•2
•68
•62
•68
•62
•63
•67
•665
•62
•77
•5
•61
•965
•95
•90
•92
•82
•92
•95
•95
•4
•95
•73
•81
•68
•76
•76
•69
•70
•83
•75
•78
675
520
620
660
650
610
600
610
560
660
Note.—The ' length ' in warships is the length between perpendiculars.
Table showing Mbthod of Ck>MPUTiNe a. Shzp'sJ
LengUibetiraenveiveBdiealuB. 886 fort: luoeiidCli, tt f«et ; dkinglitatparpBDdicabn.
1
c
O
•s
1
~2
~8
1
"6
"6
~7
1
9
10
1
1
1
Ajppendaee below lowest Water-Bne
Watbb-lines
"S
i
h
•9
4-1
3-67
10-48
10
~9
1
"7
~6
1
1
"3
~2
1
"o
1
~2
"3
"1-
IijI
3»^
6s
-3
-45
-48
-61
7W.L.
• W.L.
«W.Ir.
4W.L.
• 8W.L.
BOCraOV'S MaLTXPUXBS
k
2
1
2
1
2
^—
—
1-6
6*0
9-2
12-8
8-46
20-6
11-96
262
13-6
26-92
12-6
21-2
8-3
U-8
3-7
4-4.
1-0
1-62
•88
—
•8
•8
8-6
2-6
4-66
4-66
6-M
6-94
9-4
9-4
11-7
11-7
13-9
13-9
16-86
15^86
1784
17-34
•4
6-2
4^66
18-88
9-4
234
13-9
31-72
17-34
36-52
18-7
87-48
18-06
34-2
15-4
26-4
10-8
13-8
3-8
8-0
•88
•8
l-«
80
60
•4
60
•6
•5
•2S
6-2
6-7
17 •«
—
—
—
1-76 8-62
3-52
8-1
81
1
•9
7-2
•27
1-6
•8
8-6
7-0
6-6
11-0
8-6
110
6-8
10-6
7-9
16-8
10-6
21^0
12-86
26-72
15-04
30-06
16-9
38-8
18-4
36-8
19-8
38-6
19-7
39-4
19-64
39-28
19-9
38*4
18-46
36*92
170
340
16-1
30-2
6-8
15-8
10-6
26-72
16-04
33-8
18-4
67
6-7
2
206
28-7
1-84
1-76
5-34
3^87
80
1-5
4-6
2-3
6-4
3-2
8-46
4-23
8-6«
8*54
1
867
22-02
7-64 7-64
15-28
11-8
11-2
11-2
2
6-24
52-4
9-9
19-8
19-8
18-6627-Sd
13-66 1
1
718
7-18
28-72
-54
12041204
24-08!
16-8
15-8
15-8
2
9-88
19-36
12-48
28-6
15-65
59-28
-67
-6
•63
-66
-66
-65
11 •2S
7-6
10-8
6-15
11-96
6-98
141
28-2
16-6
31-2
28-2
15-6
i7^e
17-6
35-2
1
12-48
24-96
190
190
19-0
2
14-8
16-66
28-6
180
131
6-56
16-64:33-08
33-06
18-26
18-26
18-7
18-7
18-74
18-74
18-06
18-06
171
17-1
16-4
15-4
13-2
13-2
10-8
10-3.
69
69
88
3-8
1-6
16
•76
•76
88-6
19-7
39-29
19*^
96-92
17-0
1^-9
19-9
39-8
^^
11
\'
251-88
10^3£
207
922
13-6
6-8
13-46
6-73
12-6
6-26
17-1
34-2
170
34-0
16-2
32-4
14-8
29-6
12-8
25-6
10-4
20-8
7-8
14-6
484
8^68
21
•8
1-6
•76
1^52
17-1
34-0
16-2
29-6
12-8
20-8
7-3
8-68
2-1
1-6
-88
80-96
aO-26
20-28
12
13
14
15
16
17
18
19
20
21
2
U-7
31-4
14-2
22-7
81
11-44
3-9
6-26
•27
31-4
80-84
20-24
40-4ii
19-9
8K-6
18-1
82-8
1
14-2
28-4
19-9
199
2
tl-S6
68-1
•67
-69
-71
14^8
10-6
6-3
19-8
19*8
1
81
4
~5
1
1
~8
~9
10
82-4
5-66
8-14
8-8
4-15
6-9
2-95
181
18-1
2
6*72
67-2
30-2
l6-4
16-4
1
8-9
23-4
-73
-75
-3
2-W
3-7
1-85
12-64
25-28
9-6
19-0
12-64
190
14-84
14-84
14-34
2
T
8-63
-64
86-82
8-94
2-2
1-1
U-6
11-6
2-16
•081
1-0
-6
6-0
tt-0
80
60
•76
1-66
6-0
6-0
8-4
8-4
60
6-0
8-4
2
I
—
—
—
•76
•88
•76
•38
10 0
—
•88. 1-6
1 1-6
1-5
9000 279-88 125-47
251-88
S8-00
19754
98-71
6
[ 284-94 83794 87688 4
r + 669-88 + 837-94 + 761-76 + 4
16 4 3
04-38
04dB
i
N.B.'^Th* dwk figoTttg are tlie oHiiMtM ; thi
liroduetB of the ordinates by their respective
592-6S
»licht
Simi>sc
1 -f 2849-40 +1861-76 +2256-28 + 8
flcorei muter jOwan and alao tp Vfffx xUikd an
m'8 mtatlDliers. which are itlaeedat the head
08-76
B the
T>IBFXiACS3fZNT, XTC., USING BlllPSON'S FiBST RUIjE.
18 feet ford, 14 feet aft. 18 ft. 6 in. xoaui ; wmtorlineB apttrt, 2 feet : ordisAtes H«rt. 19 ft 8 in.
r '~ •" ' ' ' • ^'
»W.L.
1W.L.
-2
•4
8-16
6-82
6-04
1206
:7-88
1-ee 11*66 U*9iU-94
s-as
It* 14 28-28
I8-28
16S$
IB-62
180
BO
16*26
36*0
1»*8
110
lB-7
»'4
18*7
ff-4
Rr-2
14-4
6-82
604
k9-8
IB6
ID14
i0-28j
li*4B20*46|80-5
iO-92 110 -25
41-0 98*6
10-25
I0-94 20-24
10-48
Yebticai. Ssctionb
o
I
a
8-8
1-66
6*84 6*84
817
6-9T
14*48 a8-9S|
7-28
16-H
8-27
18-8
91
19-88 19-9(
9-68
80-8
10* 1
80*8
10- IS
89-9
18*7
34-4
18-4615-46
»-92
L8-14 26-28
i6*26
L8-f6
n-12
71
14-2
10*56
14-2
3 M 1-62
C-4S
19*9
9-d5
19*04
9*£
17-78
8-88t
16*94
8*12
14-8 28-6
7-16
19*1
606
9*0 180
4*6
6-6
16-&
L28-88
36-4
145-71
40-4
20-6
41-0
20-8
39-8
19-04
S6-5S
16-24
12-1
816 2-681
9-681
8*8
23-19
44-01
66-29
88-47
109-69
218-18 6
291-42
L56-6
166-77
170-53
170-94
166-64
167-57
144 17
126-8S 253-66
106-41
66-07
83-18
14-74
o
I
1*66 10
46-881 _9
8
4401
132*58 7
88-47 6
128-88
158-6
388-64
170-53
340-48
165-64
315-14
144-17
105-41
80-71 161*42
66*07
66-86
7-37
o
s
si
1096-9
8
3
6
8
9
10
§
METACKXTBE8
:§
I
5^
a
16*6
417-4S 8-8
S52-« 6
928-M
530- ffi 11
515-62
874-96
317-2
338-64 20
5881-3
340-4^ 90
381-98 20
945-42
576-6€
1268-3
632-4616
7 1129*94 14
448*66
697-24
73-7
9
14
16
18
19
ao
19
19
17
12
9
34
26
94
46
3098 604f 28
54
4525
2 6029
86
8
9
04
76
24
8
0
16
TmUVCfMl
Loagitodinal
15
36
255
793
1702
7256
8242
8616
8615
8865
7881
6902
5600
4283
2924
1772
729
138
I
7S 6
25S
158(
1701 11
452S
1205£ 36
725( 19
16484 40
8611 »
17230 41
8366 20
1676S99
690S 19
I
6
18
16
11200 35
4281 16
ITK 19
146e 18
eg|
6
34
52
94
92
54
36
3
8
04
52
24
6
0
58
{
10 —
121
5oP4
10
59-4
50-72
129-64
71-64
144-60
66l6li
8
109-2
88-72
40-4
0710-48
1
41-0
40-6
119-4
76-16
177-6
97-44
7200-2
8 96-8
9162*0
10 26*8
9 534*4
8 405*'
6
9
lOi
^1.
907-
429-<
72S*C
264-C
327-6
77-4
40-4
41-0
81-2
358-2
304-6
8880
584-6
J 1401-4
8
774-4
1468-0
2680
424*48 488*7
+ 848*96 + 219-86
1
848*96 » 8706-78
3281-0
6344-06
6381-80
131488438-7 1087-00 9660-2
710-48
962-76 826-62
(Gontinoed on next page.)
Btegxfttodbr«lift propw mnltiplien, and the soma of these m-odncts added together, the two sumi
riUacreeif the caknlationa aie correct. In thig case Che anm thua obtained by two methods is sasi o
96
DISPLACEMENT.
8231-0 8
Displacement - main solid = —^ x-xi9.26x2 = 3159 tons.
8706*78 8
Moment below l.w.l. -main solid = ^^ x - xi9.25X4=17,026ft.-ton8.
: s
Moment abaft }£ - main soUd - 962-76 x -^ x (19-25)2 x 2 = 18,121 ft.-ton8«
Displacement - lower appendage = gg- x — x 19.26 = 147 tons.
125*47
O.G. below 7 "W.ii. — lower appendage = ^qq" ~ '68 feet,
28 4
Moment abaft 5C ~ lower appendage = 35 ^ 3^ x (19 -25)^ = 895 ft.-tons.
Below
li.W.L.
Abaft
X
Item.
Tons.
Distance.
Moment.
Distance.
Moment.
Main solid
3169
17026
18121
liower appendage
147
12*63
1857
896
Aft
5*7
1*8
7
197-8
1197
Pore „
•6
60
4
-198-6
-116
Rudder
2-2
8-7
19
194-8
427
Bilge keels
•6
9*6
6
—
Shafting
8-2
6-8
20
158*8
492
Shaft brackets
1*2
6-5
8
168*5
902
Propellers
-9
6*7
6
1781
166
0well
•6
5*8
8
121-6
78
Recess
-2*8
6*9
-16
146*6
-410
Negative appendage
aft
-7*0
11-2
-78
1760
-1232
Total
8811*2
6-70
18862
6-8
19235
Displacement 8811 tons ; c.B. 5*70' below i^.w.L., 5*8' abaft ^.
4
Area of L.W.I4. (main portion) = 488*7x-xl9.25 = 11,260 sq. ft.
Moment abaft 3C = 326*62 x | x (19*25)2 = 161,800 ft.8
Moment of inertia about 3C = 9860-24 x | x (19*25)» « 93,700,000 ft.<
Item.
Area.
c.P. abaft 5C
Moment
about ^
Moment of Inertia
about ^
Main portion
Appendage aft
11,260
100
198
161,800
19,800
93.700.000
8,900,000
Total
11,860
15-95
181,100
97.600,000
11,860 X a5'95)8 » 2,900,000
Moment of inertia about c.7. = 94.700.000 ft.^
. , 11,360 _^
Tom per uxeh =» "^gd" ~ ^'^*
DISPLACEMENT SHEET.
C.F. abaft ^« 15-95';
97
Longiladinal bm
94.700,000
=s818'. BG (with o in ii.w.li.) »6' approz.
8811 X85
.*. Longitudinal gm » 818 - 6 = 812.
3811 ^ 812
Moment to change trim 1 inch «= to x a^ ^ ^®^ ft.-tonB.
ini 40Q. 11 ^^
Tranaverse bm = ^'^S ** 3 3 ^^^ = 978 It.
Area of midship section = 170-5 x g x 2 = 454 sq. ft.
* A.' W.L. (S ft. abore L.W.L.)
aw.L.
8 W.L. 1
Mo.
8.M
1
SB m
CI oJS
1
on
1
1
k
1
J
u
1
i
1
1 1
0
§ 0
0
0
1 ^
1
s
1 ^
_
_
-2
,,
-5
2
a
8-45
6-90
41
82
3-16
81
62
8-1
80
60
S
1
6*66
6-65
294
294
6-04
220
220
5-7
186
185
4
2
9-66
19-10
871
1742
8-94
717
1634
8-54
628
1246
5
1
12-2
12-20
1816
1816
11-66
1581
1581
11-2
1405
1405
6
2
14-65
28-80
3144
6aB8
14-14
2KHH
ODDD
18-66
2549
6096
7
1
16-7
16-70
4657
4657
16-26
4291
4291
15-8
3944
3944
8
2
18-25
86-50
6078
12156
18-0
5882
11664
17-6
5452
10904
9
1
19-8
19-80
7189
7189
19-8
7188
7189
190
6^59
6869
10
2
2a-06
40-10
8060
16120
20-14
8181
16862
19-9
7881
15762
11
1
20-4
20-40
8490
8490
20-46
8552
8552
20-26
8316
8316
12
2
20-4
40-80
8490
16980
20-5
8615
17230
20-24
8292
16584
18
1
20-2
20-20
8242
8242
20-24
8800
8800
19-9
7881
7881
14
2
19-65
99-70
7821
16642
19-7
7645
15290
19-8
7188
14378
15
1
191
19-10
6968
6068
18-7
6539
6689
18-1
5980
5980
16
2
180
9600
5882
11664
17-2
50HK
10176
16-4
4411
8822
17
1
16-66
16-66
4616
4616
16*46
3690
8600
14-84
2960
2960
18
2
14-95
29-90
8841
6682
18-14
2274
4548
11-6
1561
8122
19
1
1805
18-06
22S2
2222
10-56
1174
1174
8-4
593
593
20
2
10-25
20-60
1077
2154
7-1
858
716
5-0
125
260
21
_
1
6-6
8*80
287
W
8-94
84
17
1-6
8
2
446-86
134148
194701
114291
A.' W.Ij. ■* .
FwMiiim Mult. ProdiAet. Mult. Moment,
of area.
L.W.L.
■j.W.Ii.
I W.L.
446-4
488-7
424-5
5
8
-1
7
6
-1
8125
2682
5757
424
5333
Displacement of layer
(lj.W.Ii. to A.W.L.) =
651 tons.
Moment about L.w.ii. —
5888 X I X 19-25 x ^ >< ^ =
652 ft.-tons.
Up to L.W.i>.
iayer l.w.Ij. to a.w.l.
Tont.
8911
651
S9^
Momefit
below ii.w.ii.
Displacement to A.W.L. - 3962 tons.
c.B. below L.w.L. =«= 'i;;;75r=4-60'
18862
-652
18210
H
3952
4 19-25
Tons per in. =446- 4 X ^^~^ «27-3
134148 4 19-25
Transverse BM=
8-23'.
8962x85^3*^
3
98 DISPLACEMENT SHEET.
2 W.li. -
Functimi Mult. Product. Mult. Moment. Displaoementof liiyer
of area. (ii.w.ii. to 2 w.Ij.) =
6186x|xl9a5x^x^ =
694 tons.
Moment about ii.w.x«« =
6157x|xlfl-26x^x A =
680 ft.-tons.
_, -mw X DiBplaoementto2w.ii. = 9677ton8.
Tons, Moment 18232
ahotU li.w.ii. c.B. below luy^.Ti. =~^gSS" = 6'82'.
4 19*25
Uptoii.W.Ti. 8311 18863 Tons perin.=4a4-6x~x ^^=26-9.
124701 4 19*25
Layer ii.w.ij. to 2 w.L. 684 630 Transverse BM= 5==— ^ x - x
li.'W.Ii
438-7
6
2194
8
1316
2 w.li.
424-6
8
8996
10
4245
8w.ii.
404*4
-1
5S90
4 4
5186
-1
5561
404
5157
8 W.I*.
2677x86 8 8
2677 182512 H'*'*
Funeticn Mult. Product. Mult. Moment, Displacement of layer
of area. (Ij.w.Ij. to 8 w.ii.) =»
4 ]d*95 X *2
L.W.L. 438-7 1 438-7 0 — ^^^^g^^a^ =1342tons.
2w.ii. 424-6 4 16980 1 1698-0 Moment about I1.-W.L. =>
8W.I*. 404-4 1 404-4 2 808-8 2606-8 x i xl^^^^= 2452 tons.
2511-1 a506-8 ^
_ ■», ± Di8placementto8w.li. = 2089 tons.
Tons. Moment ' 16410
ahout li.w.ii. c.B. below li.W.ii. — "goM **7'98'.
4 19-25
Up to L.W.li. 3311 18862 Tons per ln.=404-4 x - x -^ =24-7.
114991 4 19*25
Layer ii.w-ii. to 8 w.L, 1242 2452 Transverse ^^^^aoWx^^ 8 ^ "s"" ~
2069 16410 18-62'.
Explanation of Displackmbnt Sheet (see pp. 94, 95>.
The lensrth of the ship between perpendiculars is divided into
twenty equal intervals, and the immersed depth by seven equally spaced
water-planes, the lowest being 2 feet above the keel amidships, fielow
7 w.L. is treated as an appendage, it being preferable in all oase«
not to take the lowest w.i.. down to or very near to the keel. The
ordinates or half-breadths at the intersections of the vertksal eross
sections with the horizontal sections are measured o£E la feet, and
set down in dark figures (usually in red) in rows opposite their
ordinate number and under their w.L. number. Water-lines that B>rs
snubbed or cut away at the ends should be produced to the perpen-
diculars by eye for the purpose of these measurements ; the volumes
thus added are afterwards deducted as negative appendages.
The Simpson's multipliers (halved in order to reduce the labour of
multiplication) are placed against the ordinate and water-line
numbers ; each Ordinate is multiplied by the multiplier appropriate to
its ordinate number, the result being placed on the right ; it is also
multiplied by the multiplier appropriate to its water>line, and the
result is placed underneath.
Adding the former products in columns gives the functions of the
water-planes ; these are multiplied by the appropriate water-line
multipliers, and the products then added, giving a number (3231*0 in
the text) which is a function of the displacement. The displacement
DISPLACEMENT SHEET. 99
ot the raatn tolil is obtained by mnltiplytn; tbis f onction | x J^ x
cpacinsT of w.l-Ib x spacing of ordonates ; the factor ^ is derived from
Simpson's first rale applied twicfo in saooession, the 8 is 2 for both
sides of ship x 4 for tike half multipliers used twice instead of the
whole ones ; the ^ converts cubic feet into tons (for sea-water).
The fonctions of tilie water-planes are again maltiplied bj the
nnmber of interrals from the L.W.L. ; the sum of the products
(8706'78) being a function of the moment about the uw.t. The
multiplier — 8 x j^ x (spacing of w.Ti.*B> > x spacing of ordinates —
gives this moment in foot-tons.
Again, the products of tSie ordinates with the water-line maltipUcrs
are added in rows, the sums being functions of the transverse areas ;
these are maltiplied by the appropriate ordinate multipliers, and the
products added, giving the same function of the displacement
(3231*0) as before. These products (headed 'multiples of areas*)
are fnrtiier multiplied by the namber of ordinate spacings from amid>
ships (station 11) ; the products are added for each end of the ship,
and the difference betw^n the sums gives a function (962*76) of the
moment of the main solid about amidships. Chi using the multiplier
S X j^ X spacing of w.l.'s x (spaicing of ordinates) >, the actual
moment is obtained in foot-tons.
The lower appendage is dealt with by calculating the half -area of
each transverse section below the lowest w.l., and the vertical
position of its e.g. They arc tabulated on the left, as shown. Each
semi-area is multiplied by its Simpson's multiplier, and the result by
the number of intervals from amidships ; the functions of areas are
also multiplied by the distances of their e.g. below 7 w.i.. The three
results are added in columns, allowance being made for the opposite
signs of the two ends of the longitadinal moments, the sums are
c<m verted by the correct maltipllers as shown. The remaining
appendages are calculated by the ordinary rules for volumes and
moments of solids, rough approximations being alone required. The
'recess ' is that due to the emergtspce of the shafts. A table is set
forth containing the displacements and moments of each item ; the
total displacement and the position of the centre of buoyancy are then
found by simple summation and division.
To obtain the position of the longitadinal metacentre (see p. 188).
each ordinate of the load water-plane is twice multiplied by the
number of intervals from amidships. The difference between the sums
of the first products for each ends (326*52) is a function of the
moment; the multiplier-^! x (longitudinal interval)* — gives the
moment of the main portion about amidships. The sum of the second
products multiplied by | x (longitudinal interval) > gives the moment
of inertia of the main portion about amidships. Area of the main
portion is obtained from the function of area (438'7) multiplied by the
multiplier | x longitudinal interval. In this case the appendage aft,
being fairly large, has an appreciable effect ; its area, moment, and
moment of inertia are calculated (the last being equal to area x
(distance of e.g. from amidships)*) and inserted in a small table as
shown. Thus the total area, the position of the centre of flotation
(or e.g. of water-plane), and the moment of inertia about 11 are
obtained. The correction necessary to find the inertia about the c.f.
(see p. 70) is then Introdnoed, and the longitadinal bm-" moment of
inertia Skbont c.y. -t- volume of displacement. By assuming an approxi*
mate vertical position for the e.g. of the ship, the moment to change
trim 1 inch or ^^ ^'^ (lony.). j^ obtained. This* together with the
■ xa I4
position of the c.f., does not vary greatly with moderate changes of
draft ; and they are generally assumed constant.
13ie remaining particulars evaluated are the. tons per inch (equal
to area of water-plane -^ 420), the area of midship section (equal to
function of area for 11 x $ x water-line interval), and the transverse
BM. To obtain the last, the cubes of the ordinates of the water-plane
are multiplied by Simpson's multipliers, and the products added. The
BISPIACEMSI^T SHBBT. 101
sttm mulliplied by | x lonffitadidal i]i,ter7al 1« equal to th^ mointj'.k of
inertia of the water-plane aboat amidshipi ; on diriditts this bj the
volume of displacement, thie transTerse bm is obtaiaed (see p. 111).
Frequently the displaoemeiifc, tons per inoh, traasTerse ix, ftad
vertical position of the c.a. are repaired for other water-lines. Here
they are worked oat for 'a' w.i- <2 feet above L.W.L.), 2w.l., and
Sw.L. The process consists of finding the volumes and moments of
the layers between the respective wju and the I..W.L.; and adding to,
or subtracting from, tlie displacement and moment for the l.w.l.
The multipliers used at a.w.l. are 6, 8, —1 for volumes, and 7, fi, --l for
moments about middle ordinate (see pp. 46 and 68). For fiw.i*. the
multipliers are 5, 8,~1 for volumes, and 3, 10, ~1 for moments about
end ordinate. For ftw.ii. the ordinary Simpson's multipliers are employed.
When the after appendage is large, it is desirable to use the tons per Ineli
and 0jr*. both eorrectedfor after atneiidage, instead of the " fuootione of
area *' taken on p. 97. The transverse bm's are obtained by cubing the
ordinate^ as for the i:i.w.i<.
EXFLANATION OV BlSPLACXXKNT ShX^T USING TcnnTCHBTV'S RULB.
The horisontal water-lines are spaced eqnidistantly as witb the
preceding displaeement sheet ; an appendage is left below 7 w.L. The
vertical transverse sections are spaoed so as to meet the requirements
of Tchebjcheff's rule (see p. 46), using Ave ordinates for eaoli half
of the length, i.e. ten ordinates in all : the positions of tiie seottone
are indicated at the top of the sheet.* The ordinates are measured
from the half-breadth ^laa ; thfty are numbered I, II, III, IV, V for
the fore end, commencing from amidships, and I a, II a. III a, IV a,
Va for the after end, oommeneiag from amldshipe. The half-breadths
are measured off in feet and inserted in the table against the number
of the corresponding ordinate, and under the corresponding water-
line, in dark figures (usually in red). Under each water-line iM set
the correot Simpson's multiplier, halved for oonvenicnoe ; no multi*
plier is required opposite the ordinates.
The ordinates are first added in columns, the sums being functions
of areas of the water-planes. These are mult4Dlied by the corre-
sponding Simpson's mulciplaers and their sum (1($76'82) is a function
of the displaeemeat ; the multiplier required to obtain the displace-
ment in tons is 2 (for both sides) x § (Simpson's rule for half
multipliers) x ^ (salt water) x water-line fpaeing ■ t^ « The pro-
ducts of the functions of the water-line areas are also multiplied by
the number of intervals from the l.w.l. ; the results are functions of
vertical moments. The sum Of these multiplied by | x j^ x (waters
line spacing) s ^"^^n^ ^^ ^^® moment of the main solid about the
XhW.L.
Each ordinate is afterwards multiplied by its appropriate water-
line multiplier, and the products sidded in rows : the sums are
functions of the areas of the transverse sections. The sum of these
gives a function of the displacement, which should be the same as
that, previously, obtained (1076'82). The differences between the
functions of the areas for the fore and after ends, taken in pairs, are
written down and multiplied by the levers equivalent to the Tcheby-
cheff spacings expressed in terms of half the length ; in the example
the fonotions for the after body are greater in .each case than the
correspcmding ones for the fore body, bub, if this is not the oase,
allowance should be made for sign. The products are functions of
moments about amidships ; their sum muUipliod by | x j^ x water-
line spacing x^^ *- is the momei^t of the main solid about amid-
ships expressed in foot-tcms.
Unless a special body has been constructed with Tchebychefi
sections, the calculation of the lower appendage is the same as that
in the' ordinary displacement sheet, equidistant sections being used.
• Alternatively the ordinary * Simpson * sections numbered 2, 6, 7, 10,
12. 15, 17 and 20. may be taken instead of the exact sections required for
Tchebychefl'B rule with four ordinates repeated.
102
WEIGHT AND CENTRE OV GRAVITY.
Thd*rptb^aqi^ a]|$Dehdat?es are calQoIated as before; the filial table
has not been inserSed in tfiis csae. '
The caloulatlon for the transrerse metacentre is the wme as
before except that the cubes of the ordinates are added direct, and
their sum multiplied by tt x ^"^~* to obtain the transverse moment
of inertia.
For the centre of flotation, the differences of the ordinates in pairs
are written doirn, allowinfir for sign if necessary, and multiplied by
the Tchebycheff lerers. The sum of the products is a function of
moments ; and the distance of the c.v. abaft amidships is equal to
function of moments „ length, »,. _ * ii. _ i.' i • i
— ; — ; X — -f— The area of the water-plane is equal
function of areas 3 r -^
to function of area x fi x ^^ * To obtain the longitudinal moment
of inertia, the ordinates are added in pairs, and maltiplied bj the
squares of the Tchebycheff levers (*007, '0)8, '25. -472, *840). The sum
of the products multiplied by a twentieth of the cube of the length
ia the moment of inertia about amidships. The correotions for the
after appendage, and for the position of the c.v., aad the ealct^tion
of BK are simikur to those on the ordinarf sheet.
The remaining calculations are made as before.
WEIGHT AND CENTRE OF QRAYITT OF SHIPS.
In the early sta^^ of deaign, an approximation xnuat be
made to the weight of a ship in order to determine whether
it is equal to the displacement assumed ; the position of the
centre of ,gra?itjr is also required in order to determine the
stability and trim.
The weight of a ship is conveniently divided into six
items : hull, equipment, machinery, fuel, armour, and arma-
ment. In a merchant vessel the two last named are replaced
by the load to be carried. The proportions vary greatly in
differeiit ships ; those in the table are illustrative of certain
types :—
Type ot Ship.
Percentage Weight.
i
84
88
43}
84
61
55
-4
S
.s
10
17
20
34
19
27
o a
^1
H
7
12
25
10
16
•
1
Armament or
Load.
•
Battleship . .
First-clasB Cruiser
Light Cruiser. .
T. B. Destroyer
(deep condition)
Steam Yacht . .
Atlantic Liner .
3
4
6
4
10
included
in hull.
31i
20
121
18
14
6
3
2 (Passengers
and Stores) |
WEIGHT OF HULL.
108
Hull.
First Mstimaie,
The weight of hull is determined to a first approximation *
in a variety of ways. In ships of very similar type it may
be assumed to be the same percentage of the displacement,
c.^. 34<Vo in battleships, etc. Or it may be compared with
the product length X (breadth -\- depth) amidships, the co-
effi<nent being d^rmined from a similar ship, making allow-
ance for any great alteration of scantlings. Mr. J.Johnson
(in Trans. Inst. Nav. Archs., 1897) published a useful method
for approximating to the hull weight of a yessol built to
the highest class at Lloyd's or Veritas.
II N =» a modification of Lloyd's old longitudinal number
SB Length from after part of stem to fore part of stem post
on upper deck beams x {i greatest moulded breadth
+ depth from top' keel to top upper deck beams +
i midships girth to upper deck stringer}.
In spar- and awning-deck vessels the girths and depths are
measured to the spar or awning decks ; they are taken to the
main deck in one-, two-, and well-decked vessels,
w =3 finished weight in tons of the steel hull.
or logioW='xlogio (^j[^) - ^•
Where x and K or X are determined from the table below : —
Type of Vessel.
Three deck ....
Spar deck
Awning deck .. '. . .
One-, two-, and well-deck
Sailing
K
1-40
1-35
1-30
1-30
1-40
0-492
0-676
0-665
0-856
0-410
0-308
0-240
0-177
0-068
0-387
The distance of o abaft, the middle of length and above
the keel can be estimated from information available for
other ships, taking these distances proportional respectively
to the length and total depth.
Detailed Eitimate,
In the later stages, when scantlings are fixed, the weight
and centre of gravity of hnU are found as follows: —
The hull may be divided into two groups : (1) calculable
items forming About 60<Vi> (in a warship) of the whole, which
inelode the greats •peat of the structure, and .(2) ' judgment '
items, mcludang portions .of complicated structure and fittings.
The latter can only l>e assessed by comparison with known
weights in a similar ship (recorded weights if possible) ;
the centre of gravity of each item can usually be determined
* See also under " Desiflm ".
A
104 WEIGHT OP HULL.
with fair accuracy from its position. The former are directly
calculated from the scantlings ; the manner of so doing is
indicated in a few instances below. To eaoh item 3<Vo should
be added for fastenings {or such an addition should be made
at the end of the calculation).
If the stresses on the ship are also reqaired, it is Qoa->
venient to divide every item into portions wholly before and
wholly abaft the midship section. Moments are taken about
two fixed planes, one being generally the midship seetioi]
and the other the l.w.l. or the. keel.
Outer Bottom Fluting, — ^Assume all. of uniform thickness,
repeating as necessary for the portions whore the thickness
is in excess or in defect of that assumed. Divide the leqgtb
into sections spaced equidistantlv, and measure the half-girthj*
at eaoh section. Apply the metnod of Rule IX, par. 24, p. 58,
obtaining the ' modifying factor ' at eaoh section. (This is
the ratio of the slant length of the mean water-line intercepted
between the sections to their perpendicular spaeing.) Find
the height of the centre of gravity of each section by diyiding
it into four equal parts, and proceeding as in par. 16, p. 62.
Then arrange the calculations as in the following table for
the forward portion of a warship below the armour deck, the
percentage for laps, butts, and liners being taken as calculated
for an average-sized plate (say 120' x 4').
Transverse Framing. — There are usually several varieties,
such as web and ordinary frames, or bracket, lightened plate,
and watertight frames. To avoid calculating each one
separately, calculate the weight and height of e.g. for a
specimen frame at intervals of about A- length. Plot these as
curves on a base of length of ship, drawing separate carves
for eaoh type of frame. The weight and e.g. position can
then be read o£E for each frame, or a mean can be taken for
a group of similar frames coming together.
Longitudinal Framing. — Usually uniform in section, but
the height of e.g. must be taken at equidistant intervals^
and the mean taken.
Bulkheads. "^JfAiJi bulkheads are usually thicker towards
the bottom, and the e.g. is below the centre of area. Take
the minimum thickness of plating and calculate its weight
and e.g. without any allowance. Then add the additional
thickness, the stiffeners, and allowances for laps, butts, and
fastenings ; all but the first item have the e.g. at centre of
area. Then find thtf ratio of the initial to final weights, and
of the e.g. below oeAtre of area to the height of bulkhead.
If this be done for one or two typical bulkheads, the rest may
be determined from the simpler first calculation by allowing
the same ratios. The ratio of weight for the main btalkheads
of a warship is 1 : 1*9 ; for ordinary below- water buikheadji
it 19 1 : 1*66.
WBIGHT OF HULL.
i
i
1'
1
i
1
S
i
a
i
a
I-
siiiiiiiiii
i
1 i
1 '
i i
lli
i |i
? f s .
i 5 s T
i.iiii
rii ^
rill
■8 J
fl
■'ZZZZttlllZ
1
1
5isil1|iisil
I
H
SI
10-8
60-0
31-7
70-6
39-6
83-8
42-6
87-4
439
88-0
32-0
i
1
8
i
1
1!
— -'•-'™
1
mmum
i
§?S?S3
1
31-0
27-2
808
34'6
88-8
41-5
43-6
43-7
43-9
44-0
44-0
Si
r-.(Nao-*io«t-<»»g«
1
06 WEIGHT OF HULL.
For smaller balkheada between dacka, measure total leaelJi
nd multiply by mean distance between decka. From this
rea the weight of plating is at once fonnd ; 12i'>ii) ia a naual
■ercentage to add for laps. The length of ^e boundary
are is readily determined ; that of the stiffeners is found
y dividing the area by their spacing, adding a peroentago
a necessary for brackets at heada and heels.
Secki. — Take in sectioos, each seetioa having nniform
hicknesB. Find area and longitudinal e.g. of each portion.
Veight of plating ia foand aa with bulkheads ; that of beams
i eqnal to area X weight of beoni per foot ran ~ beam
paciT^, adding a- small allowance for beam knees or brsoketa.
i'or planking, find volume of wood and multiply by ita
density, allowing abont 6<Vi> for fasteningi, e.g. a 3 in. deck
reighs 14 lb. per square foot i( of taak, I^ lb. if of fir,
ndading fastenings.
Wben all the itema for the hull have been evaluated, tiiey
je tabulated in the manner shown in the succeeding pages,
rhe calculations therein given are thoae for a paddle tog
45' X 28' X 11' 4" (4 teet freebosjd) x 750 tons displacement.
Uaui bottom placinn. .
Slem . I ! *. ; ; ;
Stempoit ......
Bodder, eto
iDtercoatBl U.L. kcalaon
Tnuuiverst bnlkbeada . .
Trsaivciae flaming (.ez
Cant fiamv .....
Taep boaioa and v«b
Inmu
Side k«e1aona and bilgs
Top al reaarva ^ad tank
Upter daok
Loirer deck torvaid. .
Lower d«k aft ...
Bridge deck ....
Paddle.boiei ' * ' '. '.
Coal-banker bnlkheada .
niYiaiooi in fresh -w^ter
and ballast tanks . .
Main engine bsarars . .
Aiuillary onaiDc bearers
Boiler beann . . .
I Carried foneard ,
WEIGHT OP HULL.
107
Item.
Tons,
Brought forward.
Houses and fittings on
and above bridge
Sponson houses
Bubber . . • • •
Companions . . .
Skylights ....
Engine and boiler casings
Steering gear . •
Ventilation • • •
Pumping. ....
Pillars ....
Oathead ....
Mooring pipes and chock
Horn .-
Towing hooks and stiffen
ing
Towing beams and sop
ports
Bollards and fairleads
Samson post . . .
Boats davits . • •
Awning stanchions, etc
Ladders • • . . .
Fittings in galley
W.O. fittings . .
Fittings in other sponson
houses ....
Miscellaneoas upper deck
fittings ....
Shovelling flat . .
Side scuttles forward and
aft
Side scuttles to sponson
houses .....
Coaling scuttles . .
Engineers store - room
bulkhead ....
Cabin bulkheads forward
Cabin bulkheads aft
Corticene on lower deck
aft
Silicate lagging. .
Chain locker fittings
Fittings in-
Fore crew space
After crow space '
Cabins "forward .
Cabins and mess aft
Cabin and saloon
Fore holds • •
After holds . . •
Engineers' Stores
Topside fittings
Cement . • • •
Faint
2501
36
4-6
100
2-5
20
90
•1-6
1-2
1-6
•6
•4
32
•4
11
•9
120
•5
2-2
•8
1-0
1-2
'6
•3
60
1-2
1-0
•8
•6
347-4
Before 6.
CO. Mmt.
16*6
2-6
8
73
64
2-6
16
20
1-6
83
56
66
38
33
45
668-4
64'2
112
120
24-4
^2
830
Abaft 6.
CO. Mmt.
2
2
•6
11-8
220
60
?0
1-2
4-8
20
1-9
16-6
27-6
650
760
19-2
67-5
10880
260
57 0
76
2r
ii"o
3-6
18-8
18
48
60
367
44
48
48
Above base.
CO. Mmt.
1060*4
200
60
1-0
106-2
330
60
T-5
27-6
61-3
900
48*4
i6-6
21-0
219
10*8
43-2
200
110
f6 0
870
720
240
8*14 1831-3
10 8 0
746-3
270
18-6
160
18-7
21-5
190
166
ISO
100
18 0
21-0
190
220
19-0
20-6
166
180
220
23-6
160
170
17-0
17-0
16-6
1-6
140
200
16-3
2807-3
946
832
150'0
46-8
430
1710
248
21*6
16-0
60
8-4
60-8
88
20*9
18-6
1980
90
48-4
180
16-0
20-4
8-6
6-1
99-0
1*8
140
6-0
92
8-6
60
10-8
34
27
1-6
120
240
7-2
18 0
180
8-3
8-3
36
27-0
8-8
40 0
11 0 3881*0
CO. abaft 6, 2-14'. Above base, 11'.
Take 360 tons. 2-1' abaft 6. 11' above base.
108 WEIGHT OF EQUIPMENT, ETC.
EQUTPMBirr.
This is convenientlj divided (in warships) as follows :
Presh water (for 10 days allow abont 8*7 tons for 100 men) ;
provisions and spirits^ including tare (for 4 weeks allow
about 5'7 tons per 100 men) ; officers' stores and slojos ;
officers, men, and effects (allow 8 to the ton) ; masts, rigging.
Bails, etc. ; cables (500 fathoms) ; anchors ; boats ; warrant
officers' stores (4 months); torpedo net defence.
In passenger ships undergoing long voyages allow 1 ion
per 5 persons for passengers' gear, including baggage,
bedding, etc., also *025 ton per day pdr pdrsOn (average)
for water and provisions.
Machineby and Fuel.
In the preliminary estimate the weight of machinery is
based on the total power. Coefficients for various types of
inachinery are given on pp. 389, 390. Information obtained
from actual ships should also be utilized where possible.
The weight of coal assumed is frequently an arbitrary
amount less than the full bunker capacity. The full coal
storage can be, however, determined from the Volume of
the bunkers calculated by the rules on p. 54, the areas of
the sections being taken to underside of beams only. About
10 or 15 per cent (varying with type and shape of bunker)
is then deducted for broken stowage ; the net volume in cubic
feet, on being divided by 43 (North Country coal), 40 (Welsh
coal), 86 (patent fuel symmetrically stowed) or 45 (patent
fuel shot into bunkers), gives the stowage in tons.
The weight of liquid fuel is equal to the whole volume
In eubio feet divided by 88'5.
In all cases the centre of gravity of thd fuel is the o.g.
of the volume (see p. 66)*
Abmoub.
The weight and position of o.g. of armour in warships are
determined by a process similar to that adopted for the hull.
If the armour is not specified by its weight per square foot
of plate, this can be determined from its thicknes^^ sini^e it
weighs 495 lb. per cubic foot. Add l^o/o for bolts. Backing
is dealt with similarly to the planking of a teak deck.
Abmamenx OB LOAP.
The weight of guns, mountings, charges, and projeciileB
are known (see pp. 380-385), and the position of the e.g.
of each item can generally be spotted without difficulty.
Allow 30 to 40<Vo tare for cartridge cases.
SUMMARY OF WEIGHTS
109
The load in a cargo ship is generally determined before-
hand. Its e.g. is usually found by assuming the whole spao«>
available to be filled with a homogeneous cargo — the assump-
tion the most unfavourable to the stabilify ; the o.g. if
then that of the volume of the hold.
For passengers without baggage allow 16 to the ton with
men^ women, and children ; 14 to the ton with men only. In
pleasure steamers where the stability can be affected, assume
the e.g. of passengers seated to be 6 inches above the teat ;
for those standing^, 2 feet above the deck is generally a tafcf
assumption.
The final weight and position of oentre of gravity are
found by adding together the weights and moments of the
several portions as shown in the table below : —
SxJMMABY OP Battleship Wbiohts (580' x 90* x 27* 6" -
-44'deep).
Item.
Weight.
Moment
from
amittshipi.
Moment
Above
ZI.W.L.
Moment
below
Xi.W.Ii.
Weight before
amid-
Tons.
Ft.-ton8.
Ft.-toni.
Ft.-toni.
ehips —
Armament .
• • *
2076
236110
43020
1870
Armour .
1 • ■
3270
353300
42077
1901
Hull . .
• • •
3770
355340
25473
28796-
Machinery
Coal . .
» • •
820
670
29500
23000
1360
8200
5840
Equipment
» • ■
amid-
400
2763
55000
7420
46280
766
2190
Weight abaft
ships —
Armament . .
1052250
348020
Annour . ,
*
• •
3585
481250
28848
3563
Hull . . .
* • •
4933
597170
24849
38897
Machinery .
Coal . . .
• •
• •
1930
330
198800
21400
640
^9300
3060
Equipment .
•
335
48200
3564
52
Tot
C.G
24882
1694840
1052250
223531
114435
114435
642590
109096
al weight of ship :>
. abaft 3e=||
B 24882 ton
«90 ^ 25'
382
s.
7 feet.
C.G
. above L.W.i..=^ =
4-38 feet
•
110
STABILITY.
Fio. 120.
STABILITY.
If a ship be slightly disturbed from a position of equi-
librium, and if the forces then in operation tend to restore the
original positioOy the equilibrium is termed $table ; if the
forces tend to move it further from ihe original position,
the equilibrium is termed unstable ; if it shows no tendency
to move away from or return to the original position, the
equilibrium is termed neutral.
The equilibrium of a ship is always stable as regards
▼ertioal deflections causing an alteration of displacement ;
the only disturbances that need examination consist of inclina-
tions about iKMrizontal axes with the displacement unaltered.
Ot these the principal are : (a) inclination in a transverse
plane about a longitudinal axis, and (b) inclination in •
longitudinal plane about a transverse axis. The stability in these
directions is termed transverse and longitudinal respeotivelj.
TbANS VERSE StABILITT.
^ig. 119 is a transverse section of a ship heeled over
through a certain angle 0. w'l' is the water-line for the
Inclined position, and wr« is the water-line for the upright
position. Theee ^o
planes intersect each
other in a longitudinal
direction, and bound two
Z wedges l'sl and wsw
equal in volume to each
other, provided the dis-
placement remains the
same. The wedges are
called respectively the
wedges of immertion
and emeraicn, or the in
and out wedges. G is
the centre of gravity of
the ship and b' her centre
of gravitv of displace-
ment, or centre of buoyancy. The w>elght of the ship then
acts vertically, downwards through 0, and the resultant
pressure of the water acts vertically upwlardfl through B',
these two forces forming a righting couple, the arm of which
is oz — that is« the perpendicular distance between the lines
of action of the two forces. The moment of thb couple — that
is, the weight of the ship, or its displacement, multiplied by
the length of the arm 02S — ^iii the moment of tttrtictd stabUitff
of the ship at the given angle of inclination 0. This moment
18 generally expressed in foot-tons — that is, the weight of the
ship in tons multiplied' by the lengpth of the arm Oz in feet.
B is the centre of buoyancy of the ship when upright ; s is
the point of intersection of the two wnter-lines, I the point
where the vertical b*!! cuts the plane of flotation ; ff and ^
l^io. 119
STABILITY.
Ill
are the centres €$ gravity of /lihe emerg^ed and immersed wedges
respectiyely, ffh and ff'h' being perpendiculars dropped to
ff fOkd ^' from tiiie piano of flotation w'l'. The point M, where
the Tertical line Bsc,. drawn through the centre of buoyancy
B when the shipj is in fin upright ppoLtuHi, cuts the yertioal
line B'M, drawn through the centre x>f buoyancy b' for tiie
inclined position, is termed the trwa^wer^e meta^fentre when the
ship is inclined through an indefinitely small angle, and also
when the point of intersection is jthe same for all angles of
heel.
If the centre of gravity o is below the metacentre u, the
equilibrium is stable ; if 0 is above if, the vessel is unstable,
and will capsize or at least heel to a large angle ; if o
coincides with M^ the equilibrium Is neutral.
The intersection of tiie new vertical through b' is found
usually to pass very near the metacentre M for all angles of
heel up to 10^ or 15°. Within these limits the stability lever
oz is equal to GJtf • sin ft ; or the moment of statical stability
is w . OM sin 9.
For moderate angles the stability depends wholly on the
value of OM, which is termed the metacentric height. The
position of G is calculated by the rules given on pp. 102-9 ;
that of M is obtained by the process indicated below.
To obtain the height of the trantverM metacentre*
Assume the angle of heel 0 to
be small, let
y = half breadth WS or SL at
any station«
t? = volume of either wedge
wsw' or i»si«'.
g, j7i ^ o.g.s. of the wedges.
;^^Vii=^eet of perpendiculars
from g, pi on wsi*'.
T = volume of displacement.
I =s momentof inertia of water-
plane about longitudinal
axis through s.
^ S3 an dement of lengtib of
ship.
Then BB = - . hhi.
Also V . hhi :=: moment of transference of wedge8«
~- . -^ . <Zx approximately.
= |e J ^dz?= ».i.
/. BR. = B . i/v.
Also BR = ^ . BM approximately.
I
Whence BM=-
Fia. 121.
112 METACENTRIC DIAGRAM.
The height of the tmnsverw ntstaeentre above the centre of
huoyanof is equal to the moment of ineriia of the water-plane
area about the nxU of inclination (in this eaee the centre line)
divided bff the volume of displacement.
The Mtoal calculations for the transtene Bic at several
dratights of a ship are given in the displacement sheets on
pp. 94, 100. The moment of inertia I is there expressed \yj the
integral i^y^ »dx\ the cuhes of the ordinates are first integrated,
and the result muliplied by the factor ).
In a ship whose sections are circular b\ the neighbourhood
of the water-line, such as a submarine, the metacentre is
coiucident with the centre of the circular arcs.
Definition , — ^The surface stability of a ship is that obtained
when the centre of gravity coincides with the upright centre
of buoyancy.
ICetacentbio Diagbam.
The stability of a ship in various conditions is conveniently
exhibited by means of a metacentric diagram. In fig. 122,
which shows the diagram for tlie ship taken in the displace-
ment sheets (pp. 95, 97), and inclining experiment (pp.. 135,
138), the vertical scale of draught is intersected by a straight
line drawn at an angle of 45^. From the intersection of
'a.'w.l., L.W.L., 2 W.L., and 3 w.l., with this line are set up
(or down) the vertical positions of tiiie oentres of buoy^tncy,
and of the metacentres ; these being obtained from the dis-
placement sheets. Through the spots thus obtained are drawn
the curve of metacentres and curve of buof/aney^ giving the
positions of M and b at intermediate water-lines.
The heights of the e.g. are calculated for a number of condi-
tione of the ship ; they are here shown for the inclining
condition (see inclining experiment, p. 135), the legend (or
normal) condition, deep load condition, and light condition.
A cargo or passenger ship is frequently worked out for a
large number of conditions as regards stowage of oargo^ coal,
and water ballast. From the curves the position of the
metacentre at any water-line is obtained, and the vortical
position of G marked on ; the metacentric height is thus
determined and recorded.
Value of cm in Typical Ships.
A vessel having a low metacentric height is termed crank ;
one provided with a large GM is termed stiff, A crank vessel
usually rolls less and moves more easily among waves than
a stiff vessel ; for this reason the value of the OM adopted for
a ship is made as small as possible, consistent with safety
and other considerations. Typical values are given in the
following table : —
METACBNTEIC DIAGRAM.
113
F». ua.
BfETAOENTRIC DIAGRAM OF SMALIi GBUISER.
V
MEAN
MAUeN
TONS
,ois-
^PLACE'
MCNT.
TONS
PER
INCM
A
"5
•>
b
Cnrrvc
3iCctiiceiitx
••
eo
•
16-oK
4.. 140
*4
00
•
/
IS- 6
3.964
27-65
z
Z
s
1
14-6
3.623
2714
B
.
<
/
larve <
loyAno
13-6
3.303
26-65
/
c
bn
12-7
3031
C
Y
/
2-3
11-6
2.947
2,677
2615
D
/
P-6
2,0«7
24-81
^
*
If
A. I>eep condition. Coal 725, oil 142, reserve feed 92 tons.
B. Normal condition. Coal 450 tons.
C. Light condition. No coal ; no consumable stores*
D. Condition as inclined.
2^ot€, — ^The cnrye of buoyancy is generally nearly straight ;
!the tangent of its inclination to the horizontal is equal to
12 X depth of O.B. below W.L. x ToMPefi''°h
' Tons displacement
114
STABILITY.
Type of Ship.
Minixnam OM in feet.
First-claas Battleships — ^Modern .
Do. Older types and Cruisers
Torpedo Boat Destroyers . . .
Torpedo Boats
Steamboats
Large Mall and Passenger Steamers
5-0
3-5
2-0
1-6
1-0
1-0 to 2-0*
(maintained by water
ballast)
20
1-5
Very large-
1*5 to 6 (depending on
sail area)
Cargo-carrying Steamers . . .
Tugs
Shallow Draught Vessels ....
Sailing Ships
* In some very large modern liners the OM is greater ; e.g. in AQuitania
OM is 4 feet.
Approximate Formula for Height of Metacentre.
Depth of Centre of Buoyancy (Normand's Formula).
Depth of O.B. below water-line = } mean draught +
volume of displacement ^ _ . , , displacement in tons
i z — *T i ; or * mean drausnt + ^a^ . -. — r
8 X area of water-plane ' » ® 36 x tons per mch
I^ote. — ^If a bar keel is fitted, the mean draught should be
taken to the top of keel.
Alternatively, this depth can be expressed as a percentage
of the mean draught, which is about 42 for fine ships, 44 for
ordinary battleships, and 46 for many merchant vessels.
Distance (bm) between Centre of Buoyancy and Meta-
eentre.
(greatest beam)'* , . . . , , « • v x^i
BM = , y— ; where K is approximately 13 in battle-
ships, 12 in light cruisers, destroyers, cargo, and passenger
steamers, and 11 in steam yachts. In new designs it is
advisable to take the value of K found in a similar ship. .
Stability at Laroe Angles of Heel.
If the . heel be so large that the vertical through B'
(fig. 119) no longer intersects the middle line at a fixed point,
the metacentric method is no more applicable.
During the inclination of the ship the centre of buoyancy
moves from B to b', and b' lies in a plane parallel to a line
joining g and ^'. The distance bb' can be found from the
following expression : — v X aa*
BB' = — ^
where v = volume of displacement and v = volume of either
of the wedges ;
STABILITY. 116
BB = • — - — , where BR is.perpenaicular to B M ;
, , ^ V X hh' . _
and oz = BR - BG . sin 6 = BO . sin 6,
whence Atwood's formula for expresBUng the moment of
statical stability at any angle 9 is
M = w|^-^^^ — - - (bg . sin e)|
The moment of statical surface stability at any angle 9 is
BB X w^ being the righting moment obtained on the
assumption that the eg. of the ship coincides with B. The
angle of heel in fig. 119 is bmb' = lsl', and its sine is equal to
BR _GZ
bm~"gm
Dynamical stability is defined to be th0 amount of
mechanical work necessary to cause a body to deviate from its
upright position, or position of equilibrium.
Bynaihieai stability is expressed as a moment by malti-
plyii^ the som of the vertical distances throuffh which the
centre of gtayity of the ship ascends and ine centre of
buoyancy descends (i.e. the vertical separation of a and b),
in moving ^from the upright to the inclined position, by the
displaisement.
In -Qg, 119 during the inclination of the ship through the
angle 0, the centre of gravity has been moved through a*
vertical height OH — GO^ and the oentre of buoyancy has been
lowered through a vertical distance B'l-^Bff, and the whole
work to do this, or her moment of dynamical stability for
the given angle 0^ is
= w{(gh - go) + (r'i - Bh)}
= w(b'2 - bg) = w(b'r - BG . vers B)
/wv (gh+g'h') ^,
=w(^ — ^w" — ~ *^ • ''^^^ ^ *
whence Moseley's formula for the moment of dynamical
stability at any angle $ is
= wv(gh + g'h') - (w X BG . vers «),
where w is the density of water.
l^e dynamical stability of a ship at any angle 6 is the
integral of its statical stability -at the given angle — that is,
if M = the statical stability and u the dynamical stability, then
where d0 is vl very small angle of heel.
The moment of dynamii^l surface stability is expressed by
multiplying the weight of the ship, or displacement, by the
depression of the centre ef buoyancy during the inclination
•^that is, for the angle $
u = w(b'i - bh).
The Curve of Statioal Stability i» > caTve used to record
the Tftlae of the stability lever (07.) of a veaael at snj giTen
angle of heel.
Fio. 113.
Method of Conttruetlon. — Calculate the length of tbe uid
of the rightiiw conple, or oz (aee fig. 119), for several niocea-
Bive angles of heel taken betweea the upright positioD Knd
Ulat at which the Iwigtli of the arm beoomea zero ; set the
Isngtlu of these Drms o9 as oidinatea (see fig. 123] from
a base lias the ahacisas of which repFeaent to soale tbe
respective angles of heel : a oorve bent through tbe extremi-
ties of these ordioatea will form a curve of statioal stabUity.
Note.—Tha length of the perpandicnlar at 5T'3° (one radian)
intercepted between the tangent at the imtial portion of the
curve ^id the base line is equal to the metacentrio height.
The Curve of SpmmiBol Stability is constructed in a
similar manner to that of the onrre of statioal stability, with
the exoeptlon that the various lengths of the arm (b'z — bg)
= (B'n-so vers 9) (see fig. 119) are taken as ordinate*
instead of QZ. Or preferably the oarve is obtained by in-
tegrating the statical curve. The area up to eaoh ordinate of
the statical ourre expressed in d^reea X feet is divided by
G7'3° and set np as an ordinate of the dynamical ourve.
Xete.—Th.t angle at whloh the itatleal lever vanishes (and
at which the dynamical lever ig a maximum) is termed tha
range of stability.
STABILITY.
117
FlO. 136.
Cross curves of Stability.
TOO
Gboss CtmvBs OF STABiLrrr.
These curves may be termed 'vertical curves of stability * ;
tbey consist of curves of righting levers at venous
draughts or displacements for certain fixed angles of heel.
They hold a somewhat similar relation to the ordinary
curves of stability as the body plan of a ship does to its water
plane.
For cross curves (see fig.
125) the righting levers are cal-
culated at certain fixed degrees
of heel at various displace-
ments, and the levers are set up
as ordinates from an axis the
abscisssB of which represent the
displacement at which the lovers
for the fixed degree of heel are
found.
A number of such curves are
\- — y*-~» constructed for various inclina^
tions, and set off as in fig. 125.
Fig. 126.
/M Curve of Stabiutv atooo TNs.Dispuuait. i:
IS 30 45 GO 75
Scale of anqle of heel in degrees.
For finding such curves at various draughts and angles of
Fig. 127. heel, eay at 15® (see
!A / fig. 127), divide the
body plan by a
number of parallel
planes representing
various draughts of
water or displace-
^ ments,'
Drop a perpen-
%^ dicular through tho
point where the
hi£;hest water-line cuts
the middle line of the
ship, and then calcu-
late (by the methods
indicated hereafter) the horizontal distances d^, d^y d^, etc., of
118
STABILITY.
Fio. 128.
700'
OB.
SOALC or FCCT FOR LEVERS.
the centre of buoyancy up to each *incliBed water-plane from
the vertical ab.
By assumine the centre of gravity to be at S, and fixed
there rfor all draughts, the dis-
tances dif d^, d^f etc., would be
the righting levers at the dis-
placements, up to the respective
water-planes w^, W2, Wg, W4. _
These lengths are then set off g ®^^
as brdinates along an axis having ^
the several displacements up ^00
the water-planes as abscissae. q ^
The actual righting levers can »
then be determined, when the r
correct positions of the centres g
of gravity corresponding to the »
various displacements are fixed, z 300
by multiplying the respective dis- 5
tances A^, A2> ^3» ^^'i of the h
actual centres of gravity g^, ^2> g 2°°
^3, Qi below 8 by the sine of the *
angle of heel, and adding this
length to the arms already found
(see fig. 128).
The actual righting lever for the displacement ap to W«, w«
would be equal to (^4 + ?i4 sin 15° = ^4 + 84.
Up to Ws, Ws it would be equal to d^ + hz sin 15°=(?8 + Ss, etc.
Should any of the centres of gravity be above the point 8,
a deduction would have to be made equal to the distance h of
the centre of gravity above S multiplied by the sine of the
angle of heel.
To Calculate the Statical and Dynamical Stabilities op
A Vessel at Successive Angles op Heel.
Among the various methods that are used for calculating
the statical and dynamical levers, three are here desoribed— «
(a) Barnes' method, (fi) the direct method, (jo) the integrator
method ; the last named is by far the quickest and most
convenient. Either equidistant sections may be employed
using Simpson's rule or specially spaced sections with Tcheby-
cheff's rule (see .displacement sheet, p. 100). The former may
obviate the preparation of a special body plan, but the
latter rule is generally more expeditious on the whole. It is
generally assumed that all weights are fixed, all openines in
the sides and decks closed and made watertight, all appoaaagev
c^n be neglected, and that no change of. trim iakea place
during inclination.
Barnes* Method.
1. Body Plan. — ^Prepare a body plan (fig. 130) in which aU
the sections are taken perpendicular to the load water-line,
STABILITY. 1»
and *t vqoBl diaUnoes apart (if TohebjroliBS'a method ba
employed the sectlona ara ipaoad as shown in the dU-
plaoement Bhodt, p. 100). In ooostructingi it the sectiona
■hoolii be made fair continuous curvca, anj iri^nlaritlee
nhich might be caused bj embrasur '
(as ehowl
1 full li
. fig. 129, where the dotted
\ lines Bhow the actual
"■"* section of vessel), they
being' treated Eeparatel?
arte r wards as appendagci.
there are appen-
_^ t is a!B3 neccBsary
J have correct sheer and
half-breadth dranghte, in
order to calculate thcit
vottune, etc.
. Angular Interval. — The body plan has now to be crossed
j^o. isa. 1>J * numter of lines,
radiating from the
middle paint of the
' ' ' r-plano, and
at
Bqm
tervals from 6' to 10°,
r arranging if possible
that one passes
throi^h the edge of
the upper continnous
de^k amidships.
The equiangular
interval it determined
o( foUovtl : — Divide the angle which the rai^nting line, passing
through the ed^e of the upjier deck, makee with the load
water-line into such a number of equiangular intervals that
the line passing through the edge of the opper deck beoomea
a itop-point in the int^Fatjan to which these radiating lines
will be afteFWarda treated. If Simpson's first role is ased
the nomber of intervals must be even ; if 1>Sa tecond rule,
a multiple of three must be nsed, and so on.
3. Meatttring the Ordinatei. -'The ordinatea of tba
immersed and emecged lidea of the various inclined longi-
iudinal vster-plaoea are measured off right fore and aft for
rach aneceaaive angle of heel from the middle line of the
ship, and entered upon a set of tables, styled preliminary
(ablet, under tiieir proper heading. One of these tablet is
neceaary for eaoh separate angle of heel.
4. Pr^lirtUtiary Table, — Three operations am performed
(«e p. 122) upon the ordinate! entered in these tables. Firstly,
fhej ue affected hy a set of SimpBoa's multipliers, in order
120 STABILITY.
to find a function for the area of the immersed and emerged
sides of the respective radial planes. Secondly, the squares
of the ordinates are affected by the same set of multipliers in
order to find a function for the moment of the immersed and
emerged sides of the respective radial planes. Thirdly, the
cubes of the ordinates are affected by the same set of multi-
pliers in order to find a function for the moment of inertia ot the
immersed and emerged sides of the various radial planes about
the middle line of ship.
6. Combination Tables (see p. 123).— The results obtained
in the preliminary tables are made use of in these tables to
determine —
(1st) The area of the various inclined water-planes, together
with their centres of gravity.
(2nd) The volumes of the assumed wedges of immersion and
emersion.
(3rd) The position of the true water-planes at the different
angles of heel.
(4th) The moments of the corrected wedges of immer-
sion and emersion.
6. Areas oftlie Inclined Water-planes, — The area of an inclined
water-plane is easily found for any angle of heel by adding
together the sums of the functions of the ordinates for the
immersed and emerged sides of the respective water-planes,
and multiplying the result by ^ the longitudinal interval if
Simpson's first rule is used.*
7. Centre of Gravity of the Inclined Water-planes. — To find
the distance of the centre of gravity of any inclined water-plane
relatively to the middle line of the ship, proceed as follows :
- — Take the difference between the sums of the functions of the
squares of the ordinates for the immersed and emerged sides of
the water-plane ; divide the result by 2 and multiply the
quotient by ^ the longitudinal distance between the ordinates,
if Simpson's first rule is used. That product divided by the
area of the water-plane will give the distance of its centre of
gravity from the middle line.
8. Volumes of Assumed Wedges. — Take the sums of the func-
tions of the squares of the ordinates for both sides of each of
the radial planes contained in the wedges of immersion and
emersion, and enter them in their proper column in the com-
bination table, and affect them by a proper set of multipliers ;
add their results together, subtract the lesser sum from the
greater, and divide the result by 2. The quotient multiplied
by ^ the longitudinal distance between the ordinates, if Simp-
son's first rule is used (this division by 3 is generally done in the
preliminary tables) : this final product multiplied by i of the equi-
angular interval in circular measure, if Simpson's first rule is again
* iVote.— T]ie diTision by 8 is generaUy done in the preliminAEy tables.
STABILITY. Ifil
used, will give the difference between the volumes of the assumed
wedges of immersion and emersion. If there are any appendages
the necessary additions or deductions are made here.
9. Correcting Layer, — If the volume of the assumed wedge of
immersion exceeds that of the wedge of emersion, it shows that
the displacement up to the radial plane is too great, and that to
find the true water-plane a parallel layer must be taken away
from the assumed wedges ; but if the wedge of emersion
exceeds that of immersion, a parallel layer must be added to th^
wedges.
The titiekness of this layer is found by dividing the dif-
ference between the volumes of the two assumed wedges by the
area of the proper radial water-plane, having made any addi-
tions or deductions in the case of appendages.
10. Moments of Wedges for Statical Stability, — The sums of
the functions of the cubes of the ordinates for both the im-
mersed and emerged wedges are placed in the proper column in
the combination table, and are affected by the same set of
multipliers as were determined for the sums of the functions
of the squares ; the products are multiplied by the various
cosines of the angles of inclination made by the radial planes
with the load water-line ; the products are then added together
and the sum divided by 3 ; the quotient is then multiplied by 4
the angular interval, and that product by ^ the longitudinal
interval, between the ordinates, if Simpson's first rule has been
used (this division by 3 is generally done in the preliminary
tables) : the final result will be the moment of the wedges about
a line perpendicular to the radial plane, and passing through
the middle point of the load water-plane. The corrections for
the moments of the appendages must now be added or .sub-
tracted, as the case may be, also the correction for the layer, if
any, must be done here, its moment being found by multi-
plying its volume by the distance of the centre of gravity of its
water plane from the middle point of the load water-plane. If
the centre of gravity of the layer lies towards that side for
which the assumed wedge is the greater, the correction must be
deducted ; if it lies towards the opposite side, it must be added.
This final result, being divided by the total volume of displace*
ment, will give the length of the sirm bb (see fig. 119). Multiply
the height of the centre of gravity above the centre of buoyancy
by the sine of the angle of heel, and subtract the product from
BK; the remainder will be the length of the arm of the righting
couple oz ; OZ multiplied by the displacement in tons will give
the righting moment, or statical stability, of the ship for the
given angle of heel.
11. Moments of the Wedges for Dynamical Stability. — This result
is determined in a manner somewhat similar to that pursued
for the statical stability, the only difference being that the
L22
PRELIMINARY TABJLB FOR STABILITY.
Prelimh^ary Table por Stability at 30° Angle of Heel.
8
E
0)
1 1
Squares
Ordi-
nates
1
1^1
of
Ordi-
d
S
1 6
uates
E
t
I
Fiwctions '
of
Squares
Cubes
of
Ordi-
uates
e
s
-s.
FuQctions
of
Cubes
IMMERSED Wedge.
1
U
2'
S
4
5
6
7
8
9
91
10
lOi
11
•8
i
•4
•6
*
di
8-1
2
16-2
65-6
2
131-2
14-2
1
14-2
201-6
1
201-6
17-8
2
35-6
316-8
2
633-6
20-6
u
30-7
420-2
H
630-3
20-4
4
81-6
416-2
4
1664-8
20-2
2
40-4
4080
2
816-0
20-2
4
80-8
408-0
4
1632<
20-2
2
40-4
408-0
2
816-(
20-2
4
80-8
408-0
4
1632-(
20-2
H
30-3
408-0
U
612-0
20-3
2
40-6
4120
2
824-0
18-8
1
18-6
353-4
1
353-4
15-8
2
31-6
249^
2
499-2
10-6
^
6-3
112-4
•
66-2
3)547-3
3)10502-6
182-4
3500-9
•5
h
531-4
2
2863-3
1
5639-7
2
8616-1
14
8489-7
4
8242-2
2
8242-2
4
824^-2
2
8242-2
4
8242-2
H
8363-6
2
6644-7
I
3944-3
2
1191-0
i
•3
1062-8
2863-3
11279-4
12922-7
33968-8
16484-4
32969-6
16484-4
32969-6
12363-6
16727*2
6644-7
7888-6
595-6
3)204972-9
Immersed
Emerged
Both wedges 126914*7
68324-3
58590-4
Emerged Wedge.
1
H
2
3
4
5
6
7
8
9
H
10
10}
11
11
4
6-5
2
10-9
1
14-1
2
16-9
H
20-0
4
21-2
2
21-5
4
21-2
2
20-1
4
17-6
n
16-4
2
12-5
1
8-9 2 1
3-5
i
•5
13-0
10-9
28-2
25-3
80-0
42-4
86-0
42-4
80-4
26-2
30-8
12-5
17-8
17
3)508-1
169'3|
1-2
h
42-2
2
118-8
1
198-8
2
286-6
1*
400-0
4
449-4
2
462-2
4
449-4
2
4040
4
306-2
H
2371
2
156-2
1
79-2
2
12-2
.4
•6
84-4
118-8
397-6
428-4
1600r0
898-8
1848-8
898-8
1616-0
469-3
474-2
156-2
158-4
6a
3)9146-4
3048-8
1-3
274-6
1296-0
2803-2
4826-8
8000*8
9628-1
9938-4
9628-1
8120-6
6369-4
3652-3
19531
705-0
42-8
h
2
1
2
1*
2
4
2
4
H
2
1
2
i
649-2
12960
6606-4
7240-2
32003-2
. 19066*2
89753-6
19056*2
82482-4
8039*1
7304-6
1963*1
1410-0
21*4
3)176771*31
68590-41
124
STABIIJTY.
Bums of the functions of the cubes are multiplied by the sines
of the various angles of inclination instead of the cosines ; the
sum of the products so obtained being divided and multiplied
by the same nimibers as were used for the statical stability, in
order to find the moment of the wedges uncorrected relatively
to the respective radial planes. The corrections for the
appendages are then made, that for the correcting layer
being subtracted in all cases. The moment for the correcting
layer is found by multiplying its volume by half its thickness,
that being about the vertical height of its centre of gravity
from its radial plane. This final result divided by the totsd
volume of displacement will grive the length of the arm b' b,
from which if bg . vers 0 be deducted, the remainder will equal
the length of the arm for the dynamical stability, or the vertical
height through which the centre of gravity of the ship has been
lifted and the centre of buoyancy depressed.
12. Geometrical Mode qf Calculating Dgnandcal StaHlUy. —
The dynamical stability of a vessel at any given angle of heel
is the sum of the moments of the statical stability taken at
indefinitely small equiangular intervals up to the given angle
of heel, and is therefore equal to the area of the curve of sta-
tical stability included between the origin of the curve and the
angle in question. It must be noticed that the abscissae of a
curve of statical stability is given in angles, and therefore the
longitudinal interval is taken in circular measure.
But, as the lengths of the arms for statical stability are
generally used to construct a curve instead of the moments of
stability, the area, as above found by the rule from such a curve,
will necessarily give the length of the arm for dynamical
stability and not the moment.
Example (see fig. 123). — To find the length of the arm
for dynamical stability at an angle of 30^ inclination.
Angles of Heel
Lengths of Statical
LeA'ers az
Simpson's
Multipliers
Products
0 degrees
10 „
15
20 „
25 „
30 „
•0
•2
•42
•68
•97
1^30
1-66
1
4
2
4
2
4
1
•0 •
•8
•84
2-72
1-94
5-20
1-66
1316
J of angular interval in circular measure = •0291
1316
11844
2632
Dynamical lever for 30°- -382956
8TABIUTT.
126
Fro. 181.
13. Curre of StabiUtp for Light Draught.— The lengths of
the arms for this CDrire can readily be approximated from the
results obtained for the curve in the load condition.
In fig. 131 WL b the
load water«line, and wl
the light water-line, for
J^4|i the upright position of
' ,l' the Teasel. If the vessel
is inclined through an
^ angle 9, and wV is the
true position of the in-
clined wator-plane for
the load condition, then
the true position of the
water-plane for the light
condition will run
parallel to it, as w'V, To determine its perpendicular dis-
tance from yf'i/, divide the volume of the layer contained
between the light and load water-planes by the area of the
assumed inclined water-plane hh^, which was found for the
inclined load condition. Let B be the centre of buoyancy for
the upright load condition, b' for the inclined load condition^
and b' for the inclined light condition, br is perpendicular
to the vertical b'm, and br' is perpendicular to the vortical
b«m'.
Let V equal volume of light displacement.
n
n
V s=s
volume of displacement contained between
the light and load water-planes.
distance of centre of gravity of assumed in-
clined water-plane from the vertical through
A, assumed positive on the emerged side.
„ Q2 and G'z' = the lengths of the arms of the righting
couples for the load and light condition
respectively.
Then o'z' « GZ - oa' sm $ + -^ — '-
Example. —In the ship illustrated in the tables (pp. 122,
123) find the lever of statical stability at SO"" when light,
assuming the displacement diminished by 200 tons, and the
eg. raised by 1*5 ft. B is 6*5 ft. below original upright w.L.
Here v = 200 X 35 « 7000 ; V =» 86767 - 7000 = 79800 approx,
OZ = 1-65 ; og' = 1-5 ; sin « = J ; BR = 4-78 ; ba = 6-5 ;
c = - 116.
f , fl« «e L 7000(4-78 - 3-25 - 1-16) „„,
GZ' = 165 - 75 + 79800 ^ == 93'.
Direct Method.
Lay a piece of tracing paper over the body plan, and on
it draw a trial water-line at the correct inclination. Trace the
126 STABILITY.
wedge sections, replacing the curved portions by one or two
straight lines approximating ,as closely as possible to the
curves. Find graphically the areas of the triangles and
quadrilaterals, and thence determine the volume of each wedge.
If thesis are not nearly equal raise or lower the water-line, and
proceed as before until there is practical equality in volume.
B*ind the e.g. of each triangle or quadrilateral (see p. 59)
and calculate the moment of its area about any line peppen*
dioular to the inclined water-line. Thence find the momentp
of the volumes and add; The total moment divided by the
volume of displacement is equal to bb (fig. 119), whence GZ
is at once determined.
The direct method is, perhaps, the most convenient one
when an integrator is not available..
Amsler-LAffon's Mechanical Integrator.
By means of this instrument, the area, moment, and moment
of inertia about any axis, can be obtained for any curvilinear
area by tracing its outline with a pointer.
Its principal use is that Of obtaining the stability of a vessel
at various angles of heeil and at various drafts. It is usual,
when using this instrument, to first calculate the righting lever
for a number of displacements dXme inclination, say 15**. Then
the same for 30®, 45^, and soon ; tlie cross curves being constructed
YiQ, laa before the ordinary curves.
Let fig. 132 be a body
plan drawn for both sides of
a ship ; let WL be its upright
waterline intersecting the
middle line at s. Through
S draw inclined waterlines
at the required Inclinations,
and let w'l' be any one of
them, say at 15®. The first
step is to find the displace-
menjb at w' L' as it is gene-
rally different from that at
WL. The pointer is passed (i) round the two end sections,
(ii) round the dividing sections, and (lii) round th^ intermediate
sections* ; the pointer in each case passing along the waterline
and round the section, as w'l'aw'. Readings are taken at the
start and after passing round (i), (ii), and (iii), so that after
subtracting, the readings due to each of the three series of
sections are known. Reading (ii) is multiplied by 2, and (iii) by
4, and the two products added to reading (i). The total is then
multiplied by the common interval and the constant of the
instrument and divided by 3 times the square of the scale used.
Tne result is the volume of displacement, which is then reduced
to tons.
* See Simpeon's Rules.
STABtLlTT.
127
If in the same way st, the line through s, perpendicular to
w'l' is made the axis for moments, and the readings for
moments are treated in the same way as those for areas, it is
evident that the final result will be the moment of the under-
water portion about st as axis (obviously, the total must now
be divided by 3 times the (nibe of the scale instead of the square).
This divided by the volume of displacement will give the per-
pendicular distance of the inclined centre of buoyancy from
ST ; that is sz, when b'z is parallel to ST.
The righting lever, or GZ, is equal to sz + SG sin B when G is
below s as at G| ; and equal to SZ— SG sin 9 when G is above s.
The righting lever GZ is set off at its proper displacement on
the cross curve for 15**. This is done at different waterlines and
the cross curve thus completed.
The following is the actual form of the calculation for sz.
Sections 10'6 apart. Scale of body J" to 1 foot.
I Areas .
40
Moments - .
1000
Anglb of Hesl 1$*^
Areas
Moments
Seetions
1
1 Slmpson^s
1 Multipliers
1
n
i
a
5
1
s
1
1
I .
.5 o
Initial
4029
_
982
_
End ordinates, .
4111
82
1
82
986
4
1
4
Dividing ordinat.
10502
6391
2
12782
1398
412
2
824
Intermediate „ ^ j
17309
6807
4
27228
1819
421
4
1684
40092
S512
•
1
Displacement in tons
* The 4 maltiplier is the reciprocal of the scale of the drawing.
123 STABILITY.
Tohebycheff's rule (see p. 43) can be Tery usefully cm-
ployed instead of Simpson's rule in the above ; the saving of
time due to its adoption is, for a complete set of cross carves,
more than sufficient to compensate for the time of preparing
the special body plan, which need only be drawn in fairly
roughly. In this labour may be avoided, by using the sections
numbered 2, 5, 7, lOfj. 12, 15, 17, and 20 from an ordinary
body plan whose equidistant sections are numbered 1 to 21.
It will be found that these coincide nearly in position with
those required with Tchebycheff's rule for 4 ordinates^ repeated.
This was pointed out at Inst. K.A., 1915, by Mr. W. J. Luke.
Example. — ^Length of ship, 210 feet ; number of sections, 8 ;
scale of body, il' to 1 foot. Machine constants as before.
20 210 1
Displacement in tons = -^rrv x 16 x --- x -- x area reading.
4 -~ the scale
Moment reading ^ 2 = ratio of
Area reading machine con-
stants.
Note, — ^The above or ' all-round ' method is fhe simplest,
since it gives directly the stability lever desired. A more
accurate and expeditious method, however, is that known as
the 'figure-eight'. The pointer is passed around the outline
of the wedge sections, only, taking them in the opposite
directions on the two sides of the ship ; e.g. commencing at s
(fig. 132) the pointer reaches the poin£ l', l, 8, w, w', 8
in the order named. The result is to give the difference of
the wedge volumes (by the area reading) and the sum of
their moments (by the moment reading). If v bo the original
volume of displacement, v the increased volume register^ by
the machine, and M the moment registered (the last two being
found from the readings as in the 'all-round' method),
and BS the distance of the upright C.B. below s,
M - V . BS sin e
FORMULiE FOR STABILITY LeVERS IN SPECIAL CasES.
1. Ship with concentric circular sections, cylinder. — ^The
metacentric method is here applicable to all angles of heel
and statical lever GZ = qm sin 0, dynamical lever = gm vers $.
2. Wholly immersed vessel. — ^The metacentre and centre of
buoyancy are coincident, and the above formulas apply if B
be substituted for Af .
8. Wall-sided vessel, parabolic cylinder. — Statical lever
OZ » Bind (gh + } BM tan^0).
Dynamical lever = GM (1 — cos d) + J BM (sec 9 + cos 9 — 2).
For BM, its value when ship is upright is intended.
STABILITY. 129
Change of Metaoentbio Height dub to Shall Chang ne
IN Dimensions.
Let the beam of a ship be increased by — of itself, all
transverse ordinates being augmented in the same proportion.
Similarly let the draught be increased by — of itself. If
these changes are moderate, and the iheight of the e.g. above
the keel be assumed to vary as the draught, the increase of
metacentric height is given by —
m "" ni V m^ n% \ in^ ' ' ' ^ '
where m is the original GM, 9m the increase of GM^ and a
is BG.
If the beam only be increased, — = 0, and
= -^( 1 +—}... (2)
If the d^ught be diminished S6 as to n^intain the same
displacement as before, — = , and
^wi 1 / - - 4a\
U 1(8+1")... (8)
m
In the preceding case if the total depth be unaltered (the
freeboard being increased to compensate for the diminution
of draught), and if /x represent the height of the e.g.
above the keel, originaUy,
If in the preceding case it be assumed alternatively that
the freeboard^ is nnaltered (the height pf e.g. above keel
varying as the total depth as before), and if 8 represent thet
original ratio — ftreeboard -i- draught,
In the general case, determine the effect on GM of increasing
the beam by one foot, assuming that BM « Lb'/A. The increase
of GM roughly varies as that of beam.
Example 1. — ^In making a preliminary estimate of the
dimensions of a new desigti, the beam is assumed 86 feet, the
distance bCf Is 8 feet and gm is 1 foot. It is desired to double
the metacentric height, while maintaining the draught un*
altered. Find the beam required.
Using formula (2), 8w = 1, w = 1, a « 8.
2 11
Whence —(1 + 8) = 1, or — =r^.
ii\ n xo
K
180 STABILITY,
Therefore the beam required is 36 (1 + ^) or 38 feet.
Note that if it is desired not to alter the displacement, the
ength must be diminished by ^ of itself.
Example 2. — ^In a battleship having^ beam 89 feet, mean
draught 27 feet, GM 5 feet, G abo7e water-line 6} f6et> and
BQ 18 feet, find the effect on the metacentric height of in-
creasing the beam by 1 foot, assuming that owing to a change
in the distribution of weights the e.g. is 0*35 feet higher
above the water-line in the new design. The displacement
and length are assumed unaltered.
Using formula (3), w = 6, nj = ^, a = 18.
™— t'-sC*?)-'"-"-"-'-
But this assumes that the height of G above water-line
becomes 6-6 (1 + ^) or 6'5(l - ^) or6»43feet; itiBactoally
Hence the metacentric height is 5 + '98 — (6*85 ~ 6*43) »
5 -56 feet.
Alteration of Stability Cubvb due to Small Ghahoes in
Dimensions.
Assume the beam increased by — of itself, and the draught
1 n-i
by — of itself as above. This process is applicable to any
two ships of fairly similar form, even if they depart somewhat
from exact proportionality. Given the curve of statical
stability (oz) for the first ship, it is required to construct
the corresponding curve for the desired vessel without con-
structing the body plan or performing the usual calculations.
KuLE. — 1. Construct the curve of dynamical stability of
the first ship by taking areas of the oz carve (see p. 116).
Two or three spots are safficient, as great aooaraoy is not
required.
2. O)rre8ponding to the angle 9 at which the stability lever
is required in the new ship, determine an angle ^ from the
formula- (i + 1) ten* = (l + i) tan».
3. Determine GZ, the statical lever, and z the dynamical
stability lever for the original ship at the angle ^.
4. Determine Sm, the increase of metacentric height, by
the methods of the previous page.
5. Then the stability lever G^z' of the new ship at the
angle B is given by —
O'z' - GZ = «m sin (> + J (— H — ) (gz - w sin ^)
Kl 1\ ^ *** '
) (gz cos 2 ^+2sin2 ^ (a+z)-(3m+4a) sin ^).
STABILITY.
181
By calculating g'z' for about 3 values of 0, the stability
curve can be described by the aid of the tangent at the origin
as given by the OM.
LOKQITUDINAL STABILITY.
Definitions, — 1. The cintre of flotation is the centre of
gravity of the water-plane ; it is denoted by F in fig. 133.
For longitudinal inclinations without change of displacement
the water-planes intersect on a transverse axis passing through
the centre of flotation. ' '
Fio. liSS.
1
2. The difference between the draught forward and that
aft is termed the trim. If the former is greater the trim
is by tiie bow, and vice-versa. When not stated the draughts
are supposed taken at the perpendiculars ; they are actually
measured at the draught marks which are frequently placed
at the extremities of the straight keel.
3. Change of trim is the sum of the changes of draught
forward and aft if one is increased and the other diminished ;
otherwise it is the difference between the changes of draught.
To determine the draughtB and trim at, the draught marks
given those at the perpendicular, and the converse.
Let Ij « length of ship between perpendiculars,
a, 5 = distance ol forward and after draught marks from
amidships.
Di, Da = draughts at F.p. and A.p.
Ds, l>4 =' draughts at forward and after draught marks.
Ds = — 7i 1- — (Di - Da).
D4 =
2
Di + Dj
L
+ -(Dl^DiJ.
d
132 STABILITY.
a + b . .
D4 - i>8 = —£- (Da - Di).
(fl + 6)Di = D8(2 + ^)—^*(|- ^)
(a + 6)D2 = D4(|+a)-D8(^- &)
To determine the displacement of a vessel floating Cut of
her designed trim.
Let D bo mean draught amidships, w the corresponding
displacement as obtained irom. the displacement sheet, T the
tons per inch, d the number of inches excess of trim by the
stern, L the length in feet between perpendiculars, and 0
the distance of the centre of flotation abaft amidships in feet.
cd"
Then virtual mean draught is D -{ — — •
Ij
T cd
Hence the displacement is W + — — tons.
Ij
Ex, — ^In a ship where L = 400, c = 15, t ^ 80, the dis-
placement deduced from the mean draught is 14,000 tons
where the ship has a trim of 2 feet from the bow. If the
normal trim be( 1 (foot by the stern, find the true displacement.
Here <i = — S6", and increase of displacement is
80 X 15 X 36 i/voi^
"" 400 ~ tons*
Hence displacement is 14,000 - 108 = 13.892 tons.
Note. — 1. The distance c expressed as a fraction of the ship's
length has the following average values : — ^Battleship ^^ light
cruiser 3^, T.B. destroyer ^, steam yacht ^, channel steamer, ^^
cargo steamer j^.
a. The centres of buoyancy and gravity lie abaft the midship
seotion at a distance, which, expressed m a fraction of the
ship's length, has the following average values : — Battleship ^^
light cruiser -^^ T.B. destroyer ijfj^, steam yacht ^, channel
steamer ^, cargo steamer 0.
8* JFor a change of trim t without change of displacement, the
draught forward is altered by-j7-+ •— and that aft by;:
' To find the changes of draught and trim in passing from
salt to fresh water, and vice versa.
Let the «ymbols w, x, and t) above refer to salt water. Let
8 inches be the sinkage in fresh water, and D' the final mean
draught.
• 9 w W
Then D' = D + 8/12 ; i^ — ^^^^
It is assumed above that the fresh water occupies 35*9 cubic
STABILITY. 188
feet to the ton. If the water is t^Kiokish, and occupies
35 -f* <P cttbie leet to the ton, the latter formula becomes
35t
The change of trim is nsnalljr very small. If 0^ be the
distance of the centre of flotation abaft the centre of buoyancy,
and M the moment to change trim in salt water, the change
of trim by the bow on passing from salt to brackish water is, in
inches, -rrr; ; or ,- , - for fresh water where aj is '9.
£ap^ — Find the changes of draaght and trim in a light
cruiser on passing into fresh water if w «» 3000, T =& 25,
M = 650, o' =11.
8000
Increase of mean draught = qq q^ok = 3-1 inches.
Change of trim by the bow = ^ ^ q= 1-3 inches.
To defermina the positions at whieh a weight must he
added or removed so aa to leave the draught at one end
constant,
BuiiE. — To maintain constant draught at a distance y abaft
{or before) the centre of flotation, place the weight at a distance
X before {or abaft) the centre of flotation, where x » — . If
constant draught is desired at either perpendicular, the two points
for the weight are situated at a distance —- very nearly from the
C.F. TU8 dUtanea U about ^, or about ^ in many ships.
To determine the vertical height of the longitudinal meta*
centre above the centre of buoyancy.
Divide the moment of inertia of the water-plane relatively
to a transverse axis passing through the centre of flotation by
the volume of displacement (for example, see displacemciO:
sheet and ^cplanation on pp. 94, 99).
* Note, — ^This height i^ frequently greater thaa the ship's
length, so that bo is negligible in comparison ; then QU. =«
BM approximately.
Moment to alter trim of a vessel, — ^In fig. 133 let the
weight p be moved forward throogii a longibadiual distance
d, changing the. water-line from wl to w'l'.
_, , w.crar wxGMXtrimin feet ,
Then Pa=W.GGi .= — -2 — = • — ; hence trim
. . . 12pdl. ^
d
134 STABILITY.
The product vd is the moment caasing.trim; i^;^^'^^^^
weights are moved, their moments are added* allowing for
sign. Note that the moment to change trim one inch is
equal to the expression -^^ — . This is fairly constant
within moderate changes of draught, and practically unaffected
by vertical shifts of weight.
Approximate formula* «
1. (J. Ar Normand). Long. BM = 18,000 "z^, L « lengUi on
BV
li.w.L in feet, B = beam in feet, V = volume of displacement in
cubic feet, T » tons per inch.
2. (Derived from the preceding) . ^b
Moment to change trim 1" = 30 :--
3. Moment to change trim 1" = l^b/10,000 in ships of
ordinary form, TL -f 18'6.
Effect op Adding Weigiits op Modeeate Amount.
The weights added are supposed insufficient to affect
appreciably the transverse stability, or to cause relatively
large heel, trim, or immersion.
BuLE. — ^Find the distance of each weight from the middle
luie plane and from amidships. Calculate the moments,
positive and negative (weights removed are considered nega-
tive), and add
Mean sinkage =^ weight added -f tons per inch..
„ , . _ ^„ „ transverse moment
Heel m degrees = 57-3 x duplaeemfeat x am
Trim in inches =»
longitudinal moment about centre of flotation
moment to change trim 1 inch,
longitudinal moment about ^jh (w x c.F. abaft v)
moment to change trim 1 inch
using + sign when the net weight (w) added or subtracted is
before amidships.
' Effect op Adding Weights of Considebablb Amount.
Bulk. — ^Add the weights and their moments as above, in-
cluding in addition the vertical moments required to find the
rise or fall of the e.g.
The new mean draught id found from the enrve of dis-
placement, or more accurately from the curve of tone per inch,
by estimating the mean tons per inch between the twQ water-
lines. If necessary make the correction due to the position
of the centre of flotation as described on p. 132.
To obtain the heel find first the vertical position of a ; from
the metacentric diagram the new QM is obtained. The lateral
STABILITY.
185
moyement (go') of 0 is found by diyidlng the transTene moment
by the new displacement. A moderate angle (9) of heel is given
by the formula tan 9 ^ qo'Igu, It 9 i& very large, construct
a carve of stability for the new condition, using the cross curves,
and find by trial the angle 0 at which the relation oo' » GZ sec $
holds.
For the trim the method given on the preceding ^Age is
usually sufficiently accurate. If, however, the sinkage is very
great, construct a curve of moment to change trim 1 inch on
a base of draught, also one giving the longitudinal position
of the centre of buoyancy. l%en at the original displacement
if the trim be by ihe stem, the distance of 0 abaft b is equal
to the trim in inches X moment to change trim 1 inch at
that draught (found from the curve) -f displacement.
Knowing the longitudinal position of B from the curve, that
of 0 is obtained. The change in this due to the added weights
is then determined ; and the above process, reversed and
using tho final moment, positions of b and o, and displaoe-
ment, gives the final trim.
Examples of the above methods are given in the inclining
experiment described below.
To Determine thb Vektical Position op a Ship's Centre
OF Gravity bt Experi¢.
In fig. 134 let MZ^ be
the upright axis of a ship ;
her centre of gravity then
lies somewhere in that axis.
M is the metacentre, and
GM its vertical height above
the centre of gravity G.
If a weight P be moved
transversely through a dis-
tance PP' = d, it will heel
the vessel over through an
angle 0, and her centre of
gravity will then shift in
a direction OQ^ parallel to
that in which the centre of gravity of the weight has been
shifted. Let mt be parallel to go' and tq' parallel to QM ; let
p = weight shifted in tons, and w = displacement of ship in
tons : then
HT s GG' = — — , and GM = gg' cotan 0 = — =— cotan 0. -
w w
In practice the ballast is usually in the form of pig iron
arranged in two paraUel rows on the port and starboard sides
of the upper deck.
The method of performing the experiment is Illustrated by
tho calculations below, which correspond to a light cruiser,
whose metacentric diagram is given in fig. 121.
Density of water, 85-1 cubic feet per ton. Length of ship, 885 feet.
Shift, 28 ft. 6 in. Pendulums, 16 feet long, two in number.
(li tons baUast, 8. to P. . For'd 4 A" Aft 4 ~
ft
15
Readings i Ballast replaced
1 7i tons baUast, P. to S.
•*•* ft t» ft • ft
Draught at marks j For'd
(4^' abaft F.P. ; 50' before A.P.) 1 Aft
«*
•I
. Port 10' ir Starboard 10'
. .. 13' 7^ ., 18'
MeaBurexnents ma
from initial zero
11"
Deep Condition ; Weiqhts to go on Boabd.
About L.w.r..
Item.
Fresh water
Fresh water in filters .
Boats
Officers, crew ....
Water in gravity tank .
Provisions .....
Officers* stores and slops
Wireless
Paint
Sails and awnings . . .
Rigging and blocks . .
Powder and cases . . .
Saluting ammunition .
Shot and shell ....
Practice shot ....
Small arms & ammunition
Torpedo heads ....
Toii)edo bodies ....
Maxim ammunition . .
Reserve feed
Coal to fill upper bunkers
Coal to fill lower bunkers
Oil fuel
Water in boilers . . .
Water in condensers, pipes
pumps,- etc.
Feed tanks (half full) .
Engineers' stores ...
Water in evaporators and
distillers
Mess tables
General fittings . . . ,
Water in sanitary tank . ,
Total
Weiijlit
of
Item.
27
1
14
•8
88
•0
1
•0
28
•0
25
•0
8
•5
11
•0
8
2
4
-25
11
94
•1
20
76
1
•16
1
45
1
•1
3
2
45
92
0
577
8
147-
>7
142
2
41
2
16
5
5-
5
7
1
1
5
8
0
1
0
4
0
About 11 Ordinates.
I
82
1235
15
106
24
128
42-5
460
1280
840
128
•3
1-7
14-9
26-3
§
2214
420
2625
386-56
12-8
882-3
53-36
185-6
92-4
67-6
27-6
961-41
2200-73
1083-56
12
13
27
21
83
4
40
13>8
S'O
2-0
18-5
64-6
6<
118
77
8
118-6
45-1
S
27
810*8
8740
40
140
42*66
84
6-1
19-7
6-0
21-5
94
23
64
640
2680*7
1064*26
819*0
837*8
115*6
855*5
180*4
i
6*5
6*2
4
10
17*3
6*1
391*66
190*0
21-6
329*0
70*4
372*0
20-8
3679-26
12
10
7*6
6
8
8
7
7*
10
6*
10
6-
11'
5
5
3-7
11*4
4-2
8*35
6*0
4*0
3*1
91 — 11134-92
6999-61 — 4871*82 —
Ballast . .
Lumber
Men . . .
Oiljuct ._ .
Total
To i
some out.
30-0
—
80*8
909*0
10'8|
834
—
6
—
7*8
48*8
8-4
60*4
—
5
82-0
1600
14*0
70*4
— —
1-9
—
1-0
1-9
—
10-2
4i
42-9
— 1114
7 — 444*4
Engineers' stores .
Boatswain's stores
Machinery
To be shifted.
'm
161
8
Total
53C — 161 —
8
To go on .
To come off
To shift .
123;j
-42
Summary,
11134-92
■f-1114-7
536-0
To add, net
.1193-0 4
72' 12785-62
7160-51
5C25-11
8999-51
161*0
7160*51
4871*82
+ 19-39
8*0
4894-20
4679171
214-49
— 4218-
— +444-
~ 16-
-- 4CT9*
STABILITY. 187
G. flotation abaft ^g = 16 ft. Trim assumed in metaeentrie
diagram = 1 foot.
Assume moment to change trim 1 inch as 5S6 ft. -tons at all
draughts.
Mean reading for 15-ton shift »
J{2(4A + 4A + 4 + 4) + (8i + 8A + 8A + 8A)} = 8'31 in.
Draughts at perpendiculars (see p. 131) are 10 ft. 6i in. for'd
and 14 ft. 0} in. a^t ; giving 12 ft. 3^ in. mean dranght, and
42 in. trim between perpendiculars, i.e. 30 in. excess trim.
30
Correction for C.F. (seep. 132) is r^ x 16 x 26-4 = 33 tons.
Hence displacement when inclined »
2914 (from displacement curve) + 88 = 2947 tons.
_ Td ^^ 15x28-5 15X12 „ ,^,^
Hence GM = ^ cot 0 = ^^^^ x -g^^T" ^ ^'^^^^'
u above l.w.l. as calculated (see note 5 below) = 4-28 ft.
.*. G above l.w.l. = 1-1 ft.
For deep condition add 1193 tons. G 4*72 ft. abaft, ^having
a moment of 214 ft. -tons above L.w.i«.
Hence in deep condition displacement »= 1193 + 2947 = 4140 tons.
, 214 -f- (2947 X 1.1) „^,,
G above L.W.L. = -~iVTK = -78 ft.
4I4U
H above L.W.L. = 3*61 ft. (from metacentric dia^am),
.•. GM in deep condition = 3-61 - -75 = 2-83 ft.
G of weight added before centre of flotation»16— 4*7 »= 11-8 ft.
nu * * • v I. 1193 X 11. 8 ._ .
Change of trim by bow = rrr = 23 m.
.*. Final trim by stem = 42 - 23 = 19 in. Mean draught
(from diagram) == 16 ft. OJ in.
Draught F.P. = 16 ft. OJ in. - 6 - 7 x ^^ ^Jg^^ ^ = 15 ft- 2J in.
Draught A.P. = 16 ft. OJ in. + 6 -f 7 x ^^^ = 16 ft. 9f in.
The normal and light conditions are dealt with similarly?
Note. — 1. The experiment should be performed in calm
weather^ ship bein^ moored head and stern, or allowed to
drift, so as to eliminate as far as possible the effect of all
external influences on the result.
2. An acco.unt of all weights on board should be previously
' made. No moveable weights such as loose water or oil should
be allowed ; men on board should remain in deilnite positions
when the readings are taken.
3. The readings are taken along horizontal battens, so &?
to be directly proportional to tan 9, The penduluidi bobs
can be allowed to hang in water, if necessary, to render them,
steady. If there be diificalty in obtaining a really steiidy
reading, wait until the oscillations are diminishing fairly
138 STABILITY.
regultkTljy and then note the reaxling at the ends of three
consecutive oscillations. The mean can then be taken as
the earn of one-quarter of tiie first and third readings
and one-half the second. About 3 degrees is usually
a suitable angle of heel. The quantity of ballast required'
should be estimated beforehand from the probable QM ; usually
from ^ (S&rge e^psf) to 1 per cent (small ships) of the
displaoement is sufficient.
4. The readings taken in the middle of the experiment arf^
for checking purposes only, and, if small, should bo ignored ;
all readings being taken from the original zero. If the check
readings are large the cause of the discrepancy should be
investigated and removed, and the first readings repeated.
5. If the vessel be greatly out of trim when inclined,
greater accuracy is obtained if the positions of B and M be
re-calculated in lieu of taking their positions as given in the
metacentric diagram. The distance BM is readily found from
the ordinates of the inclined water-plane ; if the trim be by
the stern bm is generally increased. The height of B can be
found exactly by taking a wedge o$ buoyancy between the
water>line and one of the original water-lines, treating this
as an appendage ; but, for an approximation, the rise of b
above its position at the same disp-laoement on the metacentric
1 T^
diagram is equal to —BMl ^ where BMj. is the longitudinal
a li
BH and T the trim in feet.
6. The draughts should be read as accurately as possible
before the experiment, and checked afterwards. If there is
a slight " lop " in the water, a glass tube 3 feet or more in
length and ^ inch or more in diameter, if held against the
side of the vessel, will give a water-level whose height can be
accurately measured.
7. The method of allowing for the added weights is
described on p. 134. If great accuracy be desired in the
estimation of the trim (supposed large), it is preferable to
adopt the second method, and calculate the longitudinal posi-
tion of G, using curves of moment to chang*^ trim 1 inch and
of longitudinal position of b. Note thati a lies abaft B by
BQ sin 0, where 0 is the angle of trim.
Buoyancy and Stability as affected by admitting Water
INTO Watertight Compartments.
The compartments are supposed to be empty, unless other-
wise stated. The volume of frames, plating, etc., is neglected.
let COM,
Water admitted into one or several compartments bounded
by a flat so that they are entirely filled.
Treat as- added weight, using the methods of pp. 134 ff.
If the amount admitted is large and so placed as to immerse
partly the upper deck or in any way to change greatly the
STABILITY.
189
stability, a process of trial and error must be nsedl This
should be continued until the line joining B and 0 is vertioal,
i.e. perpendicular to the assumed inclined water-plane.
2nd case. (Fig. 135.)
Water admitted into a central watertight oompartment,
which is not entirely filled.
Jjet H (fig. 135) s= metaoentre with free water on board.
$ s= angle of inclination,
n and b' = centres of buoyancy when upright and
inclined respectively.
b and 6' s= centres of gravity of free water when
upright and inclined reepectiv^y.
G ss centre of gravity of ship and free water
when upright.
8SS intersection of upright and inclined
free-water surfaces.
= weight of ship and free water in tons.
=3 volume of displacement in cubic feet*
s=s volume of free water only in cubic feet,
es moment of inertia of free water-sur*
face about fore and aft axis through s«
Fio. 135.
S
W
V .
The ship, as she inclines through the angle 9, has the centre
of buoyancy B carried to b', and the centre of gravity of the
free water carried from b to b\ It Ci and Cg be the centres
of gravity of the wedges of emersion and immersion respec-
tively of the free water, and Vq be the volume of either wedge,
then
CiCa X V0 ^ bb' X Y,
It is evident that CiCa X Vo==ix sin 0.
Then 66' X V = ♦ X sin 6, or bm =-. — ==— , where m is the
sin tf V *
intersection of the verticals through 6 and 6'.
Then for any small angle of inclination, the water in the
ship will shift round until its centre of gravity is in a vertical
line with m, so that for heeling purposes its centre of gravity
may bo considered to be at f» instead of 6.
/
140 STABILITY.
•
This will raise G, the centre of gravitj of the shif and wat^,
to a', so that oa' x w =^ bm X ^5 x v,
_, , bm X 35v bm X '^ i v i
Then gg' = 6= ,= -^ X — = —
W V ■ V Y Y
So that the loss of metacentric height, due to the mobility of
the water, is equal to the moment of inertia of its free
surface, about the middle line^ divided by tli0 total yolume
of displacement.
The moment of stability for a eonall angle 9.
=wxG'MXsin^=wx(GM-GG')x8in^=wx(GM-:^) sintf.
Note. — 1. It is immaterial whether the leivel inside be or bo
not the same as that outsidei; If the free water-surfaoe be^
divided by longitudinal bulkheads which entirely prevent com-
munication between adjacent compartments, find the moment
of inertia of each portion about a longitudinal axis througU
the centre of gravity of the free surface of that portion, an^
add them to obtain 'i'. If, however, any compartment be
in communication with the sea, its moment of inertia should
be taken about the centre line of ship. At larger angles of
heel, the shift of b can be estimated by methods, e.g. with
the integrator, similar to that used for the shift of B in
estimating the intact stability. Oil or ballast tanks are
generally assumed half full.
2. Allowance should be made also for the weight of water
added, regarded as solid. Since the loss of stability due to
mobility depends only on its free surface, and the gain of
stability depends on the weight and position of the water
added, there is usually a net loss of initial stability for small
depths of water and a net gain for large depths.
3. If oil or other fluid whose density is o- times that of the
water be used, the virtual rise of G is ~^« For oil fuel as
compared with salt water v = *91.
4. If the compartments are not entirely empty, both the
weight of water admitted and the loss of qm (if^ arei
reduced in a certain proportion. In bunkers or spaces filled
with coal, three-eighths of the space is void. For example,
if 800 tons of water were admitted with a consequent loss of
0*8 feet of gm with empty bunkers^^ these figures would veapec-
tively become 300 tons and 0*3 feet if bunkers were fully
stowed.
Zrd case*
Water admitted into a non-central compartment which is
not entirely filled.
If the damage is very large, use a process of trial an4
error as in the first case ; allowance must here be made for
the adjustment of the water-plane as the vessel trims and
STABILITY. 141
I heels. For moderate damag^e, where metacentric methods arc
available, proceed as follows :—
1. Find the mean sinkage by dividing the volume of water
admitted up to the original water-level by the area of the
intact water-plane.
2. Assuming first that the water-Iin6 rises parallel to itself
by the above amount, find the new displacement and height of
0, iiudoding tlie lirater thai added.
3. Find the position of the Intact centre of flotation by
deducting the moments off the areas bilged.
4. Find the intact moments of inertia about lonfi^itudinal
and transyerse axes through the intact c.F. ; thence determine
the new transverse and longitudinal stabilities in conjunction
with the new Q and b.
5. Using these new data, calculate the heel and trim duo
to the water assumed added in (2) above.
JSxample.—A rectangular pontoon 300' X 12' X 4' draught
is divided by a longitiidinal bulkhead at the middle line, and
by four transverse bulkheads equidistantly spaced. An end
compartment is bilged on dne side. Find the heel and
draughts, if the metacentric height is originally 2 feet.
14 y 10
Original BX » B«/12d = 12 x 4 ®'-
B above ke^ = 2'. Hence G above keel = 3 + 2-2 = 3'.
Volume of water added to original waterline = 60 X 6 X 4 =
1440 cubic feet (a 2' above keel).
,* • w 1440 4'
Mean sinkage.^^^^^ ^ ^g) - (60 x Q^T
Tetal volome of displacement =^14400 +1440 = 15840 cubic ft.
VT *. . u* .^ u 11 (H400X 3) + (1440x2) ^ ^,,
New height of G above keel =- 15840 ^=2-91'.
l^eW height of b above keel => 2 • 22' (approx.). Hence bg = • 69' .
Calculation for c.F. and M.i.
(a) transrerse.
Item. Area. ^enSe. ^™** ^•^- ftl>out centre.
Original water-plane 3600 — — Ax300xi25 = 43200
Portion lost * • . 360 3 1080 JxeOxB'' = 4320
Intact water-plane . 3240 i 1080 38880
3240 X ( J)2 = 360
M.I. abotit asis through new c.F. = 38520
(b) longitudinal.
Orijrifi&lvmter-plane: 8600 — — ^fj x (300)' x 12* 27000000
Portion lost . . . 360 120 43200 360x{120)2 == 6190000
(appro:£iinately)
Intact i*hter-plane . 3240 13| 43200 21810000
8240 X (13 J)« = 576000
Mt.l. about axis through new C.F. - 21234000
/
142 STABILITY.
88520
Transverse BM = , ^ = 2 '43'. Hence GM = 1'74',
T .. J. , 21234000 ,^,^, „ ,«^^,
Longitudinal BM = = 1340'. Hence GM = 1840'.
Moment caaslng heel » 1440 X 3 ft.^
1440 X 3
Hence tana = 15340 x 174 * '^^^ ; and heel = nearly 0®.
Moment causing trim =[1440 X 120<
„ ■ . 1440 X 120 X 800 . . .,
Hence trim = — ,^- .^ ,^ .. — = 2*44 .
15840 X 1340 * ** •
The heel and trim tak6 place about axis through the new C.F.^
which is situated }' transversely and 13jt' longitudinally from the
centre. The central immersion is increased by —
i X angle of heel + 13} x angle of trim »
Jx .157 + 13Jx^=a6'.
Hence the draughts at the middle line are —
for'd 4-0 + -44 + -16 + 1-22 = 5-82' = 5' 10"
aft 4-0 +-44 + -16 -1-22 = 3-38' = 3'4J"
Note. — ^In the above' method the water added up to the
new water-line obtained by adding the mean, sinkage that
would be obtained with a central compartment to the ordinal
water-line is regarded as added weight. That entering the
ship afterwards due to heel and trim is not included as
weight ; if required it is found from the additional immersion
of the original C.F. due to the heel and trim about the new C.F.
In the above example the total amount of water entering
the ship = ('44 4* *16) X intact area of water-plane.
= -6 X 300 X12 = 2160 cubic feet.
This is half as much again as that assumed up to the deep
parallel water-line.
Note. — Alternatively, if desired, -the whole of the space bilged
may be regarded as lost buoyancy instead of added weigh^. The
displacement and G are then unchanged, the position of B alone
is altered. The same results are obtained for heel and trim, bat
the GM is greater than that found above, the product w x OM
being the same.
Stability of Ship Aqround.
The displacement is less than the weight ; part of the
support is then provided by the bottom of the ship, 'and it
supposed concentrated at one point.
To find the stability, subtract the displacement from the
weight of the ship, getting the pressure due to the ground.
Find the position of uie resultant support due to the pressure
and to the displacement (acting at the metacentre). The
WAVES.
143
height of this point above the o.g. of the ship is the virtaal
metacentric height.
This is of importance when doeking a ship having con-
siderable trim by the stem, as there is a tendency to
instability whea the keel is on the point of tonehlng right fore
aad aft.
Let w = weight of ship.
A = displacement when just aground fore and aft.
a = distance of groand support abaft centre of
flotation.
A = height of metaoentre above keel at displacement
A.
X = hei^t of eg. above keel,
p = ground support in tons.
Bi =^ moment to change trim 1 inch.
8 = original trim in inches.
Then Pa = mS, giving p
A =3 w — P, giving A, and hence h.
Virtual GM « -— - - K.
w
This must be positive if the ship remains stable.
WAVES..
Sea Waves*
In the ordinary sea wave, or wave of oscillation, the form
alone has a translatory motion, as the particles composing it
revolve at a uniform rate in droalar orbits, the radius of
^==B^-
these orbits varying with the undisturbed depth, but rejiaining
constant for particles in any subsurface or subsurface of equal
pressure horizontal when undisturbed ; the form of wave-
surface thus formed being trochoidal (see fig. 136), as also the
form of any subsurface (see fig. 137), the only difference
being that while the diameter of the rolling circle of the
subsurface remains the same as for the wave-surface, the
lengtii of its tracing arm diminishes in geometrical progression
In going downwards^
/
144
WAVES*
V, V* are columns of water which are vertical in still water.
8, 8' are subsurfacOB of equal pressure horizontal in still
water.
FOBMULiE.
T = periodic time of wave in seconds.
L ss length of wave in feet (from crest to crest or trough
to trough).
V sss velocity of advance of wave in feet per second.
Vi= velocity of advance of wave in knots.
B =B radius of rolling circle in feet.
r ft= radius of tracing arm for wave-surface in feet.
g S3S accelerating force ot gravity =s 32*2 nearly.
V BB linear velocity of wave-surfaee particle in its orbit.
8 = sine of steepest slope of wave-surface.
A =: height of wave in feet.
- 1 r. «. h wh
h =2r L = 2irR s = ;— = —
2b
304
V
^ g 5-123 6023
V = 5-123 T = \/5-123L = \/^=l/t
Vi= 3- 03t = \/1-8l
L = =F^r^ = ^='^^ = o =6123T-
g 5 • 123 1 • 8 2t
. ..-A^=16.iA=9mA
2l *l V V|
«=^' = ''\/E='11
Pbopebties of the Subsurfaces and Wave Inieriob.
1. To find the ratio in which the or hits and veloeitiee of th^
particlee are diminished at a given depth below the ^ave-
smface*
Using the above symbols, in addition let
D »: depth in feet of centre of orbit of subsurface particles
below centre of orbit of surface partioleS'.
WAVES. 146
r' =s radius in feet of tracing arm of subsurface.
s= half the height of subsurface wave.
Then r' ^re * or loge--? = tt <>' iogio -7 - — z—
r a r jj
Note, — Approximately the orbits and velocities of the
particles of water are Aminished by one^half for each addi-
tional depth below the surface equal to one^mnth of a toave"
length.
Examfile /^P*^ ^ fractions of a wave-length 0 ^{f ^, etc.
^ \Proportionate velocities and diameters 1 11 J A* ©to-
For table of exponential functions see pp, 708-10.
2. To find how high th0 eentre^ of the orbit of a given
particle is above the level of that partiele in etill water*
Multiply the square of the height by --- (*7854) and divido
by the leng^ of the wave. Symbolically ^he distance a*
Note. — This gives a method of calculatiiig the area beneath
a trochoid, which is the same as that up to a straight line
rV^B below the line of centres.
S. To find the pressure at any point within a trochoidal
wave*
The pressure to which any partiele is subjected in the wave
is the same as that in its corresponding position in stiU
water.
If 41? is the density of the water, the pressure =
«(»-^)=«'('>-2^(^-''""^)
4. To find the mechanical energy of a maee of water of a
given horizontal area and of unlimited depth agitated by
waves.
Multiply the area by one-eighth part of the square of the
height of the waves and by the density of the nuid (641b.
per cubic foot for sea water).
Note. — ^The exact expression for a trochoidal wave Is
w X area x h^ /. i^h\ . , i _a * _. j^ n
energy = ~ (1 - r-g j ; the last term is usually
negligible, and vanishes entirely in an irrotational wave.
One-half of this energy is potential and one-half kinetic.
146
WAVES.
5. To find the momentum of a wave.
A trochoidal wave has no momentum ; but an irrotationaJ
wave, and probably an actual sea wave, has forward
momentum equal to (w x area x /i^)/8v, or kinetic energy -r ~
Table
OP THE Periods and
Lengths of Sea
Waves.
Velocity in
Knots.
Velocity in
Feet per
Seoond.
Velocity in
Statnte Uilm
per Hour.
Period in
Seconds.
BadluBB.
Length in
Feet.
1
1-688
1-15
'33
•09
-56
2
3-376
2-80
•66
-36
2-25
3
6-064
346
•98
-80
606
4
6 762
4-60
1-31
1-43
9 00
5
8-44
6-75
1-64
224
1405
6
1013
6-91
1*97
3-22
20-2
7
11-82
8-06
2-30
4-38
27-5
8
13-50
9-21
2-63
5*72
360
9
16-19
10-36
2*96
7-24
45-5
10
16-88
• 11*61
3-29
8-94
662
11
18-67
1266
3-32
10-8
680
12
20*26
1381
3-65
12-9
80-9
13
21-94
14-96
4-27
15-1
96-0
14
23-63
1611
4-60
17-6
1101
15
25-32
17-26
4-93
20-1
126-4
16
2701
18-42
6-26
22-9
14S8
17
28-70
19*57
6-59
25-8
162-3
18
30*38
20-72
6*92
29-0
1820
19
32-07
21-87
6-26
32-3
202-8
20
33-76
2302
6-58
35-8
224-7
21
36*46
24-17
6-91
394
247-8
22
37-14
26-32
7-24
433
2720
23
. 38-82
26-47
7-57
47-3
297-3
24
40-51
27-62
7-90
61-5
323-6
25
42-20
28-77
8-23
65-9
351-2
26
4389
29-93
8-56
604
379-8
27
46-68
31-08
.8-89
652
409-6
28
47-26
3223
9-21
701
440-5
29
48-96
3338
9-64
752
472-5
30
6064
34-53
9-87
8()'6
605-7
30-35
61-23
35-0
10
81-6
6123
33-38
66-36
38-5
11
98-8
619-9
36-42
61-48
420
12
1173
737-8
39-45
666
45-6
13
137-9
865-8
42-49
71-7
49*0
14
1600
1004
46-62
768
62*6
16
183-8
1163
48-66
820
560
16
209
1312
WA\'E8.
147
Shallow-water Waves.
In shallow water of uniform depth the orbit of each
particle ia an oval, approximately an ellipsOy the orbits
becoming more flattened the nearer the particles are to the
bottom.
As an approximation water may be taken as shallow when
the depth is between ^ and ^ of a wave-length.
Using the same symbols as with deep-water waves, in
addition let H = depth of water measured from the centre
of orbit of the surface particles,
then
v2 = S;
tanh
2»H
2irL ^, 2irH
coth
g L
Note. — Tanh x =
sinh X e* - e~*
coth X =
tanh X
cosh X e' + e^
The values of these hyperbolic functions are given in the
tables on pp. 708-10 ; for certain depths the velocity, etc., can
be found from the table below.
H 5 2tH
If — is more than about t^, tanh -— - = 1 approximately,
and the formulee become those above for deep-water waves.
IT 1. 2vH 2'WTT
If - is less than about r^r, tanh -— = — — approximately,
and then v^ = gB. and T^ = — . The period is then very large,
and the velocity is almost independent of the length.
Tablb of this Batios of Wavbb fob Shallow Water
TO THE COBBES PONDING QUANTITIES FOB DEEP WATEB.
I'
«D
a
S «8 «
=1*
r-l og
O o
1
36
3
36
4
3e
5
S3
RATIOS
Velocity
foora
given
Length
•417
•579
•693
•776
•838
Length
and velo-
city for a
given
Period
•174
•336
•481
•603
•703
Length
for a
given
Velocity
IP'*'
CO
fe CO
RATIOS
Velocity
for a
given
Length
5-76
2-98
2-08
1-66
1-42
36
H
?l
an
15
36
•884
•940
•969
•985
•995
Length
and Velo-
city for a
given
Period
•781
•884
•939
•970
•989
Length
for a
given
Velocity
1^28
M3
106
1-03
101
i
148
WAVES.
Btpples.
For waves of less than 4 inches in length the surface
tension has an appreciable effect. Using symbols as before.
let
T = surface tension of water = '00496 lb. per linear foot in
fresh water. Then v^ = ^ +
__2' . 2ir£r
= 612 L +
016
2v ' wIj
Minimum velocity is 9*1 inches per second or half a mile
an hour, the wave-length being then about '67 inches.
Sea Waves.
^y Lieutenant Paris.
Mean Wave
Period in
Lensth In
District.
Height lA
ftAMUlds.
Feet
Feet.
(dMlnoed).
Atlantic (region of trade
winds) . • • .
6*2
5-8
170
South Atlantic
140
9*5
460
Indian Ocean (south of)
17-4
76
800
Indian Ocean (region of
trade winds)
9*2
76
30O
Seas of China and Japan
10*5
6*9
240
West Paciflo ....
10-2
8*2
340
By J)r, Vaughan Cornish,^
Some large waves observed : (1) Western Meditenanean,
length 330' X height 22' in moderate gale ; (2) China Sea,
828' X 21' in violent north-east storm ; (3) North Atlantic, aboat
600' X 80' to 40' in hard gale, 850' X 80' (maximnm 48') in
strong gale; (4) North Pacific, generally smaller than the pre^
ceding ; (5) South Atlantic, 770' (mean) to 1300' (maximum) x
40', also 750' x 45' ; (6) South Pacific, maximum height 40'.
In addition to the above-described storm waves, Long swella,
whose heights are frequently about one-half those of the
corresponding storm waves, are produced ; during the storm
they are masked by the steeper storm waves, but they deg'rade
knore slowly, and can be observed afterwards. On the south
coast of England, such waves having periods of 19 to 22^
seconds (lengths 1,800' to 2,600') have been observed ; and
similarly off the north coast of Ireland with a period, of
17 seconds (length 1,400').
♦ Prom ** Waves of the Sea, and other water-waves " (T. Fisher Unwin)
and Cantor liccture, published in the Journal of the Royal Society of Arts.
1914. •
WAVES.
149
Dimensions op Waves finally peoduced in the Open Sea.
Velocity
Description of
Wind.
Beaufort's
Number
for Wind
Force.
ofWind(v)
in statute
m.p.h.==
Velocity
of Wave.
Period in
Beos.«=v-J-
8-498.
Length in
it.='V»-S-
a-883.
Greatest
average
Height in
ft.=vxo-7
Length -r
Helght=
vx.fioO.
Urong breeze
6
25
7-2
262
17-5
15 0
lioderategale
7
31
8-9
404
21-7
18-6
Fresh gale .
8
37
10-6
675
25-9
22-2
Strong gale .
9
U
12-6
813
30-8
26-4
^hole gale .
10
53
15-2
1180
371
31-8
Itorm. . .
11
64
18-3
1720
44-8
38-4
larricane * .
12
77
22. 0
2489
^~~
* Breakers of this length have not been observed.
5y Sir W. S, White.
Jjongeit waves : North Atlantic, 2,750'; South Atlantic,
1,920' ; Bay of Biscay, 1,300'. Longeat Atlantic storm waves^
50<r to 600'.
Batio of length to height : commonly 25. From French
observations its mean value is 17 for waves under lOO' long,
20 for waves 100' to 200' long, and from 23 to 27 for waves
200' io (»50' long. The minimum value (corresponding to
the steepest waves) observed was ; 5 for waves up to 100' longi
about 10 from 100' to 300% about 16 from 300' to 650'.
Appabent Wave Period when Vessel is in Motion.
Iiet t' = apparent period of waves, i.e. Uie time elapsing
between the impact of two successive waves at
the crest,
v' = apparent speed of waves, i.e. the distance along
a vessel's length through which the wave crest
passes aft in one second (if wave passes forward
v' is negative).
Vo = speed of ship, assumed positiye when ivith the
wave travel, negative when against it.
ct = angle made by course of ship with the direction
of travel of ware.
Ii, V, T = real length, speed, and period of wave.
Then V = (Vo- v') cos o; l = v't' cos o ; t = v't7(Vo- v').
If the vessel is meeting the waves, change (Vq- v') to (Vo+v')
in the first and third formules.
150 ROLLING,
ROLLING.
TjNRESrSTED ROLLING IN StILL WaTEB.
Ta= period of complete double oscillation, i.e. from star-
board to port and back to starboard, in seconds,
m s= metacentric height in feet.
K = polar radius of gyration about the e.g. of ship, in feet.
2irK —
ThenT = -7== l-llKlVm
Vgfn
Note. — 1. K^ is equal to polar moment of inertia of ship
about G 4- 'W (displacement). K can be calculated by a
laborious process ; but its value can generally be inferred with
sufficient accuracy from the known periods of ships of similar
type.
For many ships, including battleships, K — ^B approximatelj^
where B is the greatest beam ; in that case T = '37 B/^m.
Sir J. H. Biles gives the ratio e/b as '29 in Paris and New York,
and about 'i in lar^e Atlantic liners.
2. (a) The period is reduced when the metaoentric height
is increased^ e.g. if the GM is large in the ' deep loiuL '
condition, the period is appreciably diminished, although E
is then slightly increased.
(6) The period is increased when the radius of gyration
E is increased, i.e. when weights are winged or placed away
from the centre.
Example. — ^A ship of 12,000 tons displacement has a meta-
centric height of 2'5 feet and a period of roll of 15 seconds.
Find the period when additional weights are introduced
aggregating 1,500 tons, whose mean position is 40 feet from
the e.g. of the ship, the new gm being 2*7 feet.
Original E = 1^ = l^A^ = 21-6'
* 1-1 1-1
Original polar moment of inertia =WK*= 12000 x (21 • 6)^= 6700000
M.I. of weights added » 1500 x 40 X 40 = 2400000
Total moment of inertia = 9100000
New radius of gyration ^ A/9100000_g, ^
(neglective alteration due to shift of g) ^ 13500
, llE 1-1 X 25-9
New period = -~-r= = Tz^ — = 17'3 seconds.
Vr» . y2'7
ROLLING.
161
Table giyino the Periods of Boll of Ships.
Ship.
T
16
U
15
Ship.
T
13
9
7
Ship.
T
11
90
10
Older battleshipi .
Modem bfttfleahf ps
Irt okuB craisen .
itaidftndSrdelafli
cmisera
Modern light
cmisen
T.B. destroyen
EttMmjMhts
LaigwtAfUntio
linen
Cross-ohannel
steamers
Alteration of Period with Largb Akoles of Boll.
The period is coiutsnt (neglcobing resistance) when the QZ
curve is straight.
If this be coQcaTe, as in a ship with circular sections, the
period is increased at long angles ; the amount when GZ
is a sine curve may be seen from the table of pendulum
periods on p. 86. , i .
If the GZ curve be convex, the period is diminished at
large angles. In a ship with fairly small gm and large
freeboard, this may be considerable. With no initial sfcability,
T = n — 7= where a is the angle of roll in degrees.
31a VBM ** ^
Period of Dip.
t'= period of a complete dipping oscillation , i.e. including an
upward, followed by a downward, movement.
To = tons per inch, and w ~ displacement in tons*
Theni'
= 2irV^
w
12Togf
3^2 V^^ seconds.
For a battleship t' is about five seconds. For many ships it
H a^oiit one-third the rolling period.
Axis of Oscillation of Ship when Eolljng.
All ships when rolling rotate about an axis which passes
through or very near to the centre of gravity.
Assuming first that the resistance to rolling is small, the
axis passes through G if the distance of G above all the
inclined water-lines is constant: Generally, the rolling is
necessarily accompanied by dipping oscillations of period i T
whose amount varies according to the distance from G to th«
mean centre of curvature of the sections near the water-line,
and inversely as the amount t^-^4t'^. Generally speaking,
heavy dipping, and therefore ' uneasy ' rolling, results if t^
wedges of immiersion and emersion ace very unequal, and if
fhe period of toll approximates to twioe the period of dip..
152
ROLLING.
It was shown by Mr. A. W. Johns (Trans. Inst. Nav.
Archs., 1909) that the reststanees to rolling influmice the
motion of G, and oaufie also lateral moTemetits ; these have
been deterniined experimentally in the Elorn, where the axis
was found to be always slightly above 0.
BeSISTED BOLLDTd IN S^HL WaTEB.
Cv/rve of declining angles, — ^If a ship be held over to a certain
angle of heel and then released, it describes oscillations of
diminishing amplitnde. If ^e extreme angle B be noted at the
end of each half-roll and plotted as ordinates to a base of the
number n of single rolls (i.e. after intervals of } T) from the eom-
meseement, a curve (fig. 138) drawn through the tops of the
(Kdinates is termed a curve of declining angles.
Fig. 138.
CURVE OF
OECimiNG ANeUS.
Fio. 1S9.
CXTIlfCTlOH
L^--^
M.AXfMUM OF ANGLE (&.
CURVE OF EXTINCTION,
Curve of extinction. — ^By drawing a tangent to the curve of
declining angles at an angle 9, the reduction of 6 in a single roll,
i.e. - J- can be assessed. On setting up this quantity on a base
of 6 the curve of extinction (fig. 139) can be drawn. In this,
d&
supposing the ordinate — -j- were 2° for an abscissa value of 10°,
it would follow that a roll of IV to starboard would be followed
by one of 9° to port, very nearly ; the difference 11° — 9° being 2°,
the extinction.
Vahie of extinction. — The extinotion varies rather irregularly
with the angle 8. Within certain limits it can generally be
do
represented by the formula — ^ « a© + 6©^, where a and 6 are
two numerical coefficients. With a certain loss of aoouracy,
either coefficient can be neglected, and the w^hole resistAnce
assumed represented by the other. For instance, in tig. 139,
if b be neglected, as is frequently done in mstheniatioal
investigations, the resistanee is assumed represented by the
straight line, whloih id fairly accurate for -small angles of
roll. If the a term be neglieoted (as is done by French
invesjtigators), the curve in the figure wouid be repla^ee^
ROLLING.
153
by a parabola with axis vertical. The following values of
a and b have been found (all angles in degrees}:—
Ship.
Period T.
a
»
Sultan
17-75
•027
•0016
Devastation ....
13-5
•072
•0150
Inconstant • • . .
160
•035
•0051
Inflexible ....
10-7
•040
•008
Revenge, with bilge keels
(200' X 3' each side)
15-5
•065
•017
,y without bilge keels
15-2
•0123
•0025
Greyhound, with bilge keels
•
(100' X 3J' each side)
7^75
•0198
•0462
,, without bilge keels
775
•044
•0032
Elorn (tug) ....
4-5
—
•016
Modern battleship (small
angles) ....
135
•15
«
To find the angle of roll after n single rolls.
Let 9 » aagle of roll alter n single rolls.
a s an^e of roll at commencement.
a, & » coefficients of extinction, as above.
ajh
Then e =
(i+«y«_i'
or conversely n. = — lege — ■ —
1 +
ha
1 «
If 6 = 0, © = a«-«'» ; orn. = — ■ log^ —
If a = 0, e =
a o — 8
—. — -r- ; om = —r^
l + ahn ate
Tables of exponential functions and hyperbolic logarithms are
^ven on pages 708-15.
Work done in extinction.
Let w = displacement in tons.
m — metacentric height in feet.
6 = mean angle of roll in circular measure.
A9 =s decrement of roll in a single swing.
b' = value of b coefficient for circular measure.
= =67*36. (a is unaltered.)
w K angular velocity of ship in circular measure
per second.
The energy lost =» wm0 . A0 ft.-tons.
= wm0(oe + 6'02).
t ^agld 9l tPU WW tpo small %p d«tennliie b.
164 ROLLING.
If the resistanoe be supposed due in part to a couple yarying
directly as the angular velocity, and in part to one varying as its
square, i.e. resisting moment = Ki « + Eg 0^, then, approximately,
work done in single roll = t- Ki e^/T + -^— Ka e'/T*.
Since work done = energy lost, o * ^ , 1} = J , or
135 WW T*
The resistances represented by K^ or a are actually non-
existent, but the assumption accounts for the energy of the
waves propagated and for the virtual increase of polaf moment
of inertia of ship caused by the concomitant movement of
the water.
The resistances represented by K2 or b are those due to
bilge keel, deadwood, and frictional resistance.
Note. — ^In all cases the motion of a ship can be represented
1 11
by a model of ~ the linear dimensions, ^ the weight, and —7=-
the period.
Investigation with b eoefioient ad««fi^**-The motion of
a ship can be completely investigated when the resistances
can be assumed to follow the linear law, so that — :r" = <*®«
an
In that case, if 0 be the angle after time t^ 0 being the initial
The successive rolls diminish in geometrical progression,
and the period of oscillation is slightly increased — ^approxi-
mately by ^
Bilge Keels.
The effect of bilge keels on the Revenge and Greyhound
is shown in the table, p.. 153. At sea thev rediv^ed the
rolling in those ships by one-half. It was found that the
period was slightly increased.
The extinctive effect of a bilge keel cannot be directly
calculated, since its resistance is enhanced by the contrary
movement of the water round the bilge. Crenerally speaking,
it should be placed slightly above the turn of the bUge, and
made as deep as practicable. Since its efficacy varies «*
the square' or its distance from the axis of roll, it should
generally be not much longer than one-half the ship's length.
To reduce increase of resistance to motion ahead, it is placed
in a diagonal plane passing through the axis of oscillation.
When in.. motion ahead, the extinct|04 I9 in all oase^
ROLLING. 155
increased ; at 12 knots in Revenge by 40o/o, £naT.B. destroyer
by lOOo/o.
In practice the length of bilge keels varies from one-half
(in large warships) to one-third (in destroyers and merchant
vessels) of the ship's length. The usual depth is about ^ the
beam, slightly more in warships ; in lightships ^ beam (see
paper. Trans. I.N.A., 1912, by G. Idle and G. S. Baker).
"BZEX* DUB TO GUNFIBB.
w — displacement in tons,
m = metacentric height in feet.
w « weight of projectUe in pounds.
to* — weight of cluuge in pounds.
h = height of gun above water in feet.
T =^ period of double roll in seconds.
D ^ mean draught in feet.
V » muzzle velocity of projectile in ft. /sec.
a B angle of initial zoU in degrees*
^ (w + jio') 1? (ft + JD)
** aOOwTm
ROLLINa AMONG WAVES.
'JBffeoiive wave slope, — ^This is the slope perpendicular to
which a ship, moving among waves, tends to place herself.
Little is known of ite actual amionnt | but it wjas stated by
W. Froude that it was approximately the slope of the wave
subsurface through the centre of buoyancy of the ship. With
large ships the slope does*not generally exceed 3^ or 4*«
If T^ is the apparent period of a sinrple wave, and a the
maximum wave slope, then the wave slope at any instant is
a sin ^— + 7), where y is the initial slope. Ti is determined
by the methods of p. 144 ; it depends on thie speed and course
of the ship as well as on the dimravyon^y, etc., of the waves.
If the ship's length is perpendicular to the direction of wave
advance, a is equal to ith/l, ^heire H is the height and l the
lengi^ of the wave subsurface. If the ship is partly along the
waves, a is diminished ; approximately a varies as the cosine of
the angle of obliquity.
In a complex sea the wave structure is assumed divisible
into simple waves of the type described, and the effects of
each oompon-ent wave are superposed.
Rolling in a simple component wave. — It is assumed that
the resLrtance to rolling is due solely to an 'a ooefficient',
i.e. that it is directly proportional to the angular velocity.
By choosing a suitable value of a, the still water decremental
equation ~ Tf ^ <^® ^^^ ^ made to represent, within limits of
e, the actual decrement ; the error, when large, provides
a limit to the usefulness of the investigation, and to the
accuracy of the results.
156 ROLLING.
The wave slope being ei sin Tmtj'iu the inclination 9 of the
ship is found to consist of the sum of two terms —
1. Ke" r sin (- — 'V 1 — ^+ fih termed the 'free* roll,
which is exactly the same as' the natural roll of a ship in still
water. This osdllatiou is arbitrary in initial amplitude (a)
and phase (/3), but the formier diminishes in geometrical
progression for successive rolls.
3
--tan-i— ^J^V(l-~,) +^ ~,
termed the forced roll, whicih is in the period T^ of the wave.
The phase of thiis is such that among diort wtives the ship
rolls against the wlavie slope, but slightly in advance of the
direct opposite ; among long waves the ship rolls with the
wave slope, with a slight timed lag ; at or near synchronism^
when the ship's period T is nearly equal to the wave period
Tx, there is a phase difference hi about 90^, i.e. the shipt
reaches her greatest inclination when the Wave slope is zero
or very smaU. The variation of the amplitude of the forced
roll is sthown in the highest curve on fig*. 140, the value of
0 being *157. Among short waves, or equally when the ship's
period is relatively large, the rolling is very fimiall ; at syn-
ohronism it is very largpe ; among long waves, or when the
ship's period is relatively small, the rolling ifl reduced but
is always more than the effective wave slope. The dotted
curve in the figure is the corresponding forced roll when the
resistance is absent, showing the resistance has comparatively
little effect on the forced roll, except near synchronism.
Theoretically the free roll should die out almost im-
mediately, leaving only the forced roll ; actually, a ship roUd
generally in her own period, showing that it is continually
re-introauced by irregularities in the waves. Increase of
a, i.e. of the resistance, is therefore beneficial in restricting
the free roll component ; resistance also operates in other
waySj e.g. by increasing h and lengthening the period.
General Conclusions.
1. To diminish rolling among waves, a ship's natural
period should be as large as possible ; hence, her metacentric
height should be small, and her weights * winged ' or removed
as far as practicable from the centre.
2: In such conditions a ship will roll against the waves,
showing the necessity for a sufficient range of stability, since
her virtual angle of heel is greater than her inclination from
the vertical.
ROLLIKO. 167
3. For a ship to roll with the waves, her period shoald be
small, and her metacentric height lar^e, so that fehe will float
like a raft. The gain in safety ana eeaworthiness in such
conditions is, however, more apparent than real, except in
very small boats, as the vesel is likely to meet small
snppl«mentary waves which may synchronise and cause heavy
and dangerous rolling. •
4. Heavy rolling is caused by the periods of wave and
ship being approximately equal ; it may generally be obviated
by a change in speed or course of the Sup, wMch alters T^,
the apparent wave period.
5. Besistances, e.g. those provided by bilge keels, are
chiefly operative in preventing heavy rolling during approxi-
mate synchronism. They serve also in moderating the rolling
in all conditions ; but no considerable lunitations of the rolling
experienced among non-synohronous waves can be ensured
by any practicable resistance, however large.
6. Owing to the coexistence of forced and free rolls, the
combined roll generally consists of groups of rolls vrhich start
and end in comparative quiescence, and attain a maximum
in the middle. The period of the group is approximately
TTi
i .:: — zr ; tl^e maximum roll is X times the forced roll, where
T — Ti
A lies between 1 and 2, being equal to 1 + « ^aiolr^ where To is
the group period.
' Effect of Watea CRAHBEsa on Bolliko.
^ese were found by Sir Philip Watts to favour the
reduction of moderate rcdling, against which bilge keels are
comparatively inefficient.
Fig. 140 shows the effect of water chambers of closed
channel type, determined from theoretical considerations
(see paper by L. WooUard, Trans. Inst. Nav. Archs., 191S).
The Ship has an a coefficient of extinction of *157 ; the water
in the channel is also resisted, the corresponding a coefficienif
being ir times the values given for Kg/p. The curves represent
the ship's forced roll on a base of t^/t or wave period -f-
ship's period. The 'damping coefficients' marked on the
curves are equal to — times the corresponding a coefficients of
the free roll, which in this case vary, being augmented when
the water is throttied in the channel.
The conclusions derived are briefly as follows : —
1. The tank should be placed as high as possible in the
ship ; the horizontal portion in particular should be near, or
above, the ship's centre of gravity.
2. The period of the tank depends on its shape, particularly
the area of the constricted channel ; it is almost independent
MAKIUUM AN6LE4 « OF 'FOHCCO OSCILLATION OF SHI».
Curve I'. No t*nk, oo resistance to tolling.
Curve I. Ko tank, bMp resisted (Ei/ji ~ 05).
Curve II. With tank ol period -89 that ot ship, resistance
coefficient (Ki/p) - ■
Curve III. „ ,, ,,
Cnrve IV. „ ,, „ ,,
Curve V. ,, -71
Note.— The ordinate^ o( the currea represent value* ol O/a <
Maiimum angle ot ghip'a forced oacillation
Angle oF virtual slope ot wave
PITCHING. 159
of the amount of water contaiiied, and of the d^ree to which
the flow is resisted.
3. The rolling will be a Tninimnm when the period of the
tank is rather less than the apparent period of the waves
meeting the ship.
4. A tank having a period capable of adjustment would,
therefore, if practicable, prove advantageous ; failing this,
the tank period should be made about 70 per cent of the ship's
period in large ships.
5. A moderate resistance to the flow of water in the tank
is on the whole favourable, e.g. the flow can well be throttled
until the amplitude of movement is reduced by one-half after
each single oscillation.
6. The tenk is useless or diaadvantageoos among very short
waves^ and should then be put out of action.
Pitching.
The considerations governing the pitehing of ships are
largely the same as those affecting the rolling ; but there are
the following differences :-* ^
(a) The period of pitehing is relatively small, being in
most ships slightly more than half the rolling period. It is
governed by the same formula, being increased by placing
weights at the ends of the ship, and diminished by an increase
in longitudinal stebility. In applying the formula of p. 1^0
to find the period of pitching, teke K to be about }l.
(6) The vertical wave slope a is small, particularly when
the ships are long in comparison with the length of the waves.
It lA a maximum when the ship is head or stern to waves,
though she may be turned throufi^h a considerable angle in
each direction without greatly reducing a.
(c) Comparatively little is known of the resistence to
pitehing, except that it is augmented by full ends, by flare aft
the bows, and by flatness at the stern.
(<f) The period of pitehing is teo small te enable it to be
made large in comparison with the wave period^ as is done
for rolling ; it is also desirable that the vessel should piteh
with the waves, so that the bow and stern rise and fall with
the water. The ratio t/t^ is therefore made as small aa
possible ; and weights are brought longitudinally as near the
centre as convenient in order te reduce the pitching period.
(e) With these conditions satisfied,, a vessel at rest, or
running before the sea, generally follows closely the effective
wave slope. This is also the case with a ship head to sea up to
a certain speed, at which the apparent wave period is shortened
sufficiently to approach synchronism with the ship's pitching
period. The pitching then increases, and continues to do so
as the speed is raised, since the synchronising conditions
would rarely be reached and passed.
160 SPEED.
Obsbrvations of Boixino.
When observations are made by batten, or a constant
direction maintained by a gyroscope, no special errors arise.
Pendulums are convenient, but are liable to special errors as
follows :—
Let B be the true angle of ship to vertical.
6 + 4> be the actual angle recorded, ^ being the error in
excess.
T be the still water rolling period of ship in seconds.
Ti be the apparent wave period in seconds,
a be the virtual slope of wave.
I be length of pendulum in feet.
1. In still water any pendulum hung from the centre of
osoiUation (i.e. near the ship's centre of gravity) will indicate
correctly the angles of roll. It may be hung from a&y point
at the same height towards the side without large error. If
hung from a' point h feet above the e.g., ^ = ^ii^hBlgT^f or
4>/^ = l'23fe/T^ with a short pendulum, the error beang in
excess. With a longer pendulum ^/6 = 1'23^/(t*- 1-230 ;
the error is in defect with very long pendulums. The height
h should be made sufficiently small for this error to be
negligible.
2. Among waves a long period pendulum hung at ihe e.g.
indicates the correct vertical, and a short period pendulum!
the effective vertical, i.e. the perpendicular to the effective
wave slope. For a length I, the error is given by
4) = ± ag-Ti^ligri^ - 4ir«Z) or <Pla = ± Ti^Kl • 23Z - Tfl . To reduce
this error to reasonable limits a pendalum of very long period
must be employed — ^in general a delicately balanced compound
pendulum of equivalent period (2ir\^llg).
For example, among waves of 6 seconds period, 0 = a if
I is small or if Z = 59 feet, so that a pendulum 60 feet long
will cause the same amounit of error, though in the opposite
sense, as one of a few feet in length. To limit the error to
a/10, a pendulum of length 320 feet is necessary, which can
only be ensured by using a compound pendulum of equivalent
period, i,e. twenty seconds.
In the Mallock rolling indicator the period of the pendulum
is forty seconds.
3. The two errors of a short period pendulum incorrectly
placed may be made to neutralize in certain conditions ; where
such an instrument only is available, it should be placed about
mid-draught in the majority of ships, where it may serve for
rough indications of rolling.
SPEED AND H0R8£P0W£B*
General Bbmabks.
N&te. — ^Unless otherwise defined, v always represents speed
of ship in knot ; 1 knot = 6,080 feet per hour.
SPEED. 161
The horse-power required to drive a tAdp at a oertain speed
cannot be asseased by any mathematioal calculation alone. It
can be dedneed frcm obserrationa obtained from similar ships
or models ; in some cases it oan be estimated! with fair
accuracy from the performances of other ships having forms
resembling^, thongh not exactly similar to, the one desired.
Indicated horse-power (I.H.P.J is the power actually
exerted in the engines. In reciprocating engines it \%
calculated from the areas of the indicator diagrams. It
cannot be directly measured in turbines.
Shaft horae-^ower (S.H.P.) is the power transmitted by
the shafts. It is measured by means of a torsion-meter
whidi indicates the small twist in the shaft over 8 short
length. l?he shaft is calibrated previously, whence the torque
corresponding to the twist and the power are readily calculated.
This affords the only means of measuring the power eixerted
in tnrbine engines ; in reeiprocating engines, the S.H.P. is
usually about 85o/o of the I.H.P., the difference beine due to
frictional losses in engine, thrust blocks, and shaft bearings
forward of the torsion-meter.
Effective horse-power (E.H.P.) is that required to tow the
naked hull (i.e. without propellers, shaft supports, or rudder)
tiirough still water.
The difference between E.H.P. and I.H.P. or S.H.P. is
accounted for by (a) the losses at the propeller, (6) the
friction of engpbae and shaft (in part only if S.H.P. be con-
sidered), (c) the resistanoe of appendages, (dl) the inter-.
action between propeller and hull. The effect of the last is
usually small.
The resistanoe (b) is the force required to tow the hull,
as above. If thb is expressed in pounds, E.H.P. = rv/326.
PBOPULSIVE Ck>BFFICIENT.
E H P
The propulsive coefficient (m) is the value of the ratio ■^' '
for ships with reciprocating engines, or g'^'p* for turbine
driven vessels.
It is of the greatest importance that ii should be predicted
as aoonrately as possible when powering a ship, but it is very
difficult to estimate it directly. It is preferable in a new
design to adopt a value of ii based on that obtained from a ship
of fairly similar type having engines working under
somewhat similar conditions. Allowance should be made fof
any difference in the oircumstances in the two cases (e.g.
a change in the speed of revolution may affect the propeller
efficiency, etc.).
T3ie propulsive coefficient is equal to the product of
4 factors : —
1G2
SPEED.
1. Bnffine and ikaft ^fflcienop. This is from *82 to *S8 in
most screw ships, nearly 1 in tarbine-driven ships (based on
S.H.F.), and about *8 in paddle steamers. At low speeds these
ratios are smalls.
2. Propeller efficiency. This can only be .estimated by
designing the propeller ^ abont '70 is the greatest possibkk.
value (see p. 192).
3. Hull efficiency, depending on the interaction of httU
and screw (see below).
4. Ratio of resistance of naked hull to that with the
appendages. The appendages -iSorm a serious part of the
resistance in many ships; The effeet of shaft bossings is givei]
by Mr. Luke (Trans. I.N. A., 1910). As regards rubers,
Mr. T. G. Owens (Trans. I.N.A., 1914) gave the following
comparative results for a 25,000 tcm ship at 85 knots, having
balanced rudders ; —
Knmber
of
Rudders.
Arrangement.
Total Area^ Area
of Longitudinal
Imniersed Plane.
Pereentftge Increase
of Besiatanoe
caosed by ladder
over Naked Hull.
1
2
2
Tandem on middle
line
Twin, side by side
1
GO
1
46
1
83-5
1-3
2-3
7
These data are useful as a basis of comparison between two
vessels.
General remarks on propulsive coefficient, — Cet, par, fi is
usually highest in high speed ships, and in a given ship it
increases with the speed ; but there are many exceptional
cases, as, for instance, where the propeller efficiency is
diminished by excessive slip at certain speeds, leading to
a reduction in /t.
fx Is generally rather greater for. single screws than lor
twin screws (v. hull efficiency).
In turbine-driven vessels, the hig)i sp^ed of shaft lieeeSiBary
to obtain good turbine efficiency leads to a considerable veduo-
tion in the propeller efficiency. The propulsive coefficient is
then commonly equal to or even less than that obtained with
reciprocating engines, in spite of the gain in the first factor
of /i (engine and shaft efficiency) due to the non-inolusion of
most of the machinery loss in turbines. Geared tuvbineB
permit a variation between the revolutions of turbine and
propeller shaft and enable the efficiency of them both to be
greatly improved.
SPSED.
1(
The followinsr table gives the propulsive coefficients
iypieal vessels of a vairlety of classes. In general the
ooefiieients have been obtained ia favourable droumstanoes,
even after several modifications in the propellers. It
deairable, therefore, to adopt, when powering a new ship,
value of fi less than that given below bj from 2 J to 5 per ce
of the whole ^i.e. from 5 to 10 per oent of itself) ; this wi
provide a margin for eventaalities or nnfaTonrable condition
Tn>eof Ship
li
1
>
1
>
P
Wake
Factor w.
Thrust
Deduction
t
«
Remarks* 1
Battleship (turbine)
4
•M
•47
fl6
l-ao
•la
•16
1-01
1»01
Inne
Oute
,• (MolpnMsstiiMr)
a
•98
•47
•14
•17
•95
BaMle omiser (tarbine)
4
1-1
•fi3
/•13
1-19
•10
•la
1^02
1^05
Inne:
Oute
1st class cruiser Creoip.)
a
1-00
•58
•10
•10
•99
find class .,
a
1-oa
•48
•06
•10
•95
8rd ol^as ,. „
2
1*06
•48
•05
•08
•97
T.B. destroyer (recip.)
2
1-7
•61
-•01
•04
•97
,» (tarbine)
3
1-76
•6a
•04
•oa
•05
•06
•99
•97
Innei
Oute:
Mail steamer (turbine)
4
•03
•46
•80
•aa
•17
•ao
1-08
•95
Innei
Oute
Cargo vessel
a
•8
•48
•ao
•15
1'02
f* ft
1
•8
•58
•84
•17
111
Sloop
1
•98
•45
•21
•17
1-00
Steam Piimace
1
1-5
•50
•10
•15
•94
Submarine (in surface)
a
— * '
—
•16
•10
104
(diving)
a
"" '
•ao
•la
1^06
Wake and Thrust Deduction.
A moving ship is surrounded by a enrrent of water whic
is moving, on the whole, in the same direction as the ship
This current is termed the wake.
The speed of the wake varies from point to point ; thi
magnitude of its forward velocity is of particular miportanc
near the prcmellers. This velocity is assessed from mode
ezperimepts by comparing (a) the spaed v of the screvf
when moving with the ship in its correct position and rotating
so as to exert the proper thrust ; and (6) the sp^ v^ a
which the screw most be run in the open at the same numbc
of revolutions so as to exert the same thrust. Evidently th<
screw acts as if there were a forward current v-v^ ; this is
termed the wake Telocity. The fraction
V-Vl
Vi
is termed thi
the wake factor and denoted by w. Hence y » Ti (1 + t&).
* The nomenclature devised by Mr. Froude is here followed ; Mr. D, W
v-Ti ...1...... , , x« 10
Taylor defines wake factor as
, which is equal to
1 + w
164 SPEED.
The screw exercises a suction on the stern of the ship,
augmenting the resistance from B (without sorew) to T (with
screw). T is thus the actual thrust exerted by the screw when
propelling the ship. The ratio is denoted by t, and
termed the thrust deduction coefficient. Hence R=3T (1-0*
The effectiye power of the sorew, or the rate at which it
does useful work is Ty^i the power required to tow the ship
(i.e. E.H.P. ai^^ented to include effect of appendages) is
Ev. The ratio or (1 + to) (1-0 is termed the hull
efficiency.
The values of w and t are of importance in propeller
design ; the hull efficiency is a factor of the propulsive co-
efficient, and thus directly affects the determina^on of the
power. Average values for different types of ship are
included in the above table.
From model experiments made by Mr. W. J. Luke, the
following conclusions may be drawn (Trans. I.N.A.^ 1910
and 1914) :>-
With single screws decre^e in wake and hull efficiency
followed increase in diameter. With twin screws, outward
turning, the reverse was experienced.
Twin screws should be placed laterally as near the hull as
practicable.
Outward turning screws are much preferable to inward
turning when the effect of shaft bossilng is included. Such
bossinfi^ should be plaoedi as nearly norn^ to the surface as
possible.
The adoption of contrary turning screws of equal pitch on
a common axis may increase the propulsive coefficient by 15
or 20 per cent of itself with a full model (block coefficient
*65); the gain was much smaller with a fine model (block
coefficient '60).
Mr. D. W. Taylor has suggested the formulae :—
w
--— — a t = ^ .2-f- .65iS (hull efficiency 1) for twin screw ships, and
— — -sa - '05+ *5i3 (hull efficiency rather more than 1) for single
screw ships as an average ; fi is the block coefficient.
With quadruple screws Mr. Luke also gave the following
results, which show the relative advantages of inward and outward
turning: — ^8-bladed screws, 4" diameter, 1*0 pitch ratio, • 6 disc
area ratio. After screws 8" before A.P., 8^ from centre line;
for'd screws 18^ before a.p., 7' from centre line. .Immersion to
tips 3" ; clearance from hull i".
SPBBD.
165
Screw.
Wake w.
After screws (0)
(ff) Without for'd screws
(6) With for'd screws (O)
W « « (I)
•21
•16
•13
Thrust
deduction
t.
U
•13
•12
HuU
efficiency
1-04
101
•99
After screws (I)
(a) Without for'd screws
(ft) With for'd screws (0)
W « „ (I)
For*d screws (0)
(a) Without after screws
(6) With after screws (O)
(0 « „ (I)
•20
•15
•10
•11
•12
•10
107
101
•99
Tor*d screws (I)
(«) Without after screws
Ih) With after screws (0)
W „ „ (I)
•24
•22
•22
•24
•23
•23
•13
•13
•12
•10
•12
•10
1*08
106
107
112
108
110
O = Outward tanuag ; I = Inward turning.
Components of Eesistance.
The tow-rope resistance b of the naked model is divisible
into 4 parts : (1) skin friction ; (2) wave-making resistance ;
(3). air resistance ; (4) eddy resistance.
The air resistance is small and is usually neglected ; in the
extreme case of a large mail steamer with high freeboard it
is estimated to be 4o/o of the whole. Against a head wind
this proportion would be much larger ; the resistance would
then be further augmented to an indeterminate extent by
waves and rough water.
Eddy resistance is oauaed by discontinuities in a longi-
tudinal sense, particularly in fittings that have blunt endings
aft. In bilge keels, shaft brackets or bossings, rudders,
propeller bosses, and any other appendages it is frequently
present, but is then included in the difference between e.h.p.
and engine power. Eddy resistance in the naked hull (which
is all that is dealt with nere) only occurs rarely, and is then'
due to badly formed sterns or water-line endings. Any
abrupt or rather quick change in the direction <^:- the after
water-lines or buttocks may lead to the formation of eddies
(see p. 174).
166
SPEED.
It is usual to include ware-makii^ and eddj iwialiance Id
one group, which is. termed residuary resistance.
Skin Friction.
The resistance due to skin friction is estimated from data
obtained by experiments on planks hj W. Froade. Ho
deduced the formula — ft/=/SY"; where B/==frictional resistance,
v = speed of plaiik (or ship), s = wetted sur&ice, /, n— coefficients
depending on the nature and length of the surface and liquid
density and temperature.
The values of / and n can be deduced from the following
table.
Froude^s Resistances per square foot in lbs, of various Surfaces
at 600 feet per minute.
Nature of Surface
Length of surface or distance from cutwater in feet i
2 feet
8 feet 1
A
B
0
A
B
0
Varnish
2-00
•41
•890
1-85
'825
•964
Paraffin
1-95
•38
•370
1-94
•314
•260
Tinfoil .
2-16
•80
•298
1-99
•278
•268
Calico .
1-93
•87
•725
1-92
•626
•504
Fine sand
20
•81
•690
2-00
•583
450
Medium sand
2*0
•90
•730
200
•625
•488
Cear«M»^6and.
20
110
•880
200
•714
•520
Nature of Surface
Length
of 8urfac<
) or distance from
cutwate]
r in feet
20 feet
50 feet 1
A
B
0
A
B
0
Varnish
1-85
•278
•240
1-83
•260
•226
Paraffin
1-98
•271
•237
— .
—
Tinfoil .
1-90
•262
•244
1'83
•246
•282
CaHco .
1-89
•581
•447
1-87
•474
•428
Fine sand .
200
•480
■384
206
•405
•837
Medium sand
2-00
•584
•465
2-00
•488
•456
Coarse sand .
2-00
•588
•490
—
Columns A give the power of the speed to which the resistance is approxi-
mately proportional. Columns B give the mean resiatance per square foot of the
whole surface of a board of the leugtlis stated in the table. Columns O gire
the resistance in lbs. of a squaitB foot of surface at the distance steruward ^m
the cutwater stated in the heading.
Coefficients for Computing Effective Jlorse-power required
to overcome Skin Friction based on Mr. Froude's Constants,
(^Corresponds to varnish on models or clean bottom on ehip.
The constants in previous table show effect of fouting,\
If s is the wetted surfaoe in square feet then
EhH.P. = / 8, where / has the value given below.
SPEED.
167
SiMd
LeoBth ot 8hiv j
41 Feet.
in
Knots.
100
150
900
800
400
£00
eoo
800
1000
40
•9490
•9343
•9271
•9181
•9199
•9049
•8:89
•8890
•8792
39
*8836
•8698
•8331
•8547
•8183
-8424
•8356
•8276
'8186
38
•8209
•8082
•8020
•7942
•7830
•7828
•7776
•7691
•7606
37
•7611
•7494
•7436
•7£6i
•7306
•7253
•7209
•7180
•7051
36
•7047
•6938
•6885
•6818
•6765
•6720
'6676
•6603
•6530
35
•6508
•6407
•63r8
•6296
•6247
•6206
•6164
•6097
•6030
34
•6996
•5902
•6867
•6800
•6765
•6707
•6679
•5617
•6665
33
•6499
•6414
•6372
•5320
•5278
•5243
•6208
•5161
•5094
3Z
•6060
•4972
•4934
•4886
•4848
•4816
•4784
•4732
•4680
31
•4624
•4662
•4617
•4473
•4438
•4409
•4379
•4322
•4274
SO
•4210
•4146
•4113
•4073
•4041
•4014
•3988
•3944
•3900
29
•3826
•3767
•3738
•3702
•3673
•3649
•3624
•3585
•3646
28
'3466
•3412
'388S
•8«58
•3327
•S8:)6
•3283
•3247
•8211
27
•3126
•3078
•3094
•30«5
•ao&i
•2981
•2961
•2929
•2887
26
•2811
•2767
•2746
•2719
•2698
•2683
•2662
•2688
'2604
26
•2616
•2477
•2458
•2190
•2434
•2416
•2399
•2c83
•2357
•2831
24
•2242
•2207
•2169
•2162
•2138
•2124
•2101
•3078
23
•1988
•1967
•1942
•1923
•1908
•1835
•1882
•1831
•1^41
23
•1763
•1T26
•1713
•1696
*ie8s
•1672
•1661
•1643
•1625
21
•1687
•1614
•1502
1487
•1476
•1466
•1466
•1440
'1424
20
•1340
•1319
•1308
•1290
•1286
•1277
•1263
•1254
•1241
19
1169
•lUl
•1132
•1121
•1113
•1105
•1098
•1086
•1074
. 18. .
•0996
•0979
•0972
•C962
•0956
•0948
'0942
•0931
•0921
17
16
•0846
'0713
•0833
•0702
•0827
•0697
•0819
•069O
•C812
•0686
•C837
•0683
•C832
•0675
•0793
•0668
•0784
•0661
16
•0594
•0585
•058a
•0676
•0670
•0667
•0563
•0567
•0561
li
•0489
•0481
•0478
•0887
•0473
•0469
•0?81
•0466
•0463
•0458
•0463
13
•0397
•os9a
•orsi
H)378
•0375
•0371
•0367
12
•0316
•0312
•0309
•Oc07
•0304
•0302
•0300
•0296
•0393
11
•0246:
•0243
•0241
•0229.
•0237
•0236
•0233
•0231
•0228
10
•0188
•0183
•0184
•0183
•0181
•0180
•0179
•0176
•0174
Wetted Surface.
The wetted surface is determined by tke methods indicated
in pp. 67, 58. . That due to appendlages, e.g. bilge keels^ is
easily c^dcalated and added on.
Approximate rules are : —
1 . Suriace = 1 • 7 LD 4- Vq/d (Denny^ .
2. Surface « 13-6 V^WL = 2-64 VVoii (Taylor's).
3. Surface = 3 • 4 Vo* + i LVo*.
I, =■ length, D — mean draught, Vo = volume of displacement,
w = displacement in tons.
T^e first formula is generally the most accurate ; the other?
are only reliable for warship forms.
The surface can also be determined fi^om Blechynden'a
formula for mean girth (Trans. I.N. A., 1888).
4. Mean girth = •937M + 2 (1 - 7) ©.
7 = prismatic coefficient of fineness, M = midship wetted girth.
168 SPEED.
. Power absorbed in Shin Friction,
The power absorbed in skin friction is about three-qnarier9
the total E.H.P. in ^ips of moderate speed. At the highest
speeds of fast vessels it constitutes about 40 per cent or the
whole effective power. It is therefore imi>ortant, particularly
with ships of moderate speed, that tiie form should be
arranged eo as to avoid unduly increasing the wetted surface,
on which the friction depends.
The following broad principles affecting variation of weit-ed
surface with change of dimensions have been enunciated by
Mr. D. W. Taylor :—
(a) For a given displacement, surface varies nearly as
the square root of the length.
(6) For a given displacement and length, surface varies
only slightly within limits of beam And draught possible in
practice. The most favourable ratio of beam to draught is
a little below 3. [Extreme proportions of b/d, such as are
obtained with shallow draught vessels^ increase the surfaoe;,
e.g. 6:1 may increase it by 15<yo.]
(c) For a given displacement and dimensions the surface
is affected very little by min^r variations of shape, etc.
Extremely full sections are Somewhat, and extremely fine
sections are quite, prejudicial to small surface.
(d) Kext to length, the most powerful controllable factor
affecting wetted surface is probably that of deadwood.
Besiduaby Bebistakce.
This oonsists almost entirely of tiie resistance absorbed in
wave-making.
The waves accompanying a dhip in motion are divisible
into two classes — ^transverse and diverging. The former have
transverse crests, spaced at the longitudinal distancci appropriate
to their speed, which is the speed of the ship. The crests of the
latter form an angle of 30^ or 40o with the middle line of the
ship ; they advance perpendicularly to the crests with a
velocity equal to the component of the speed of the ship in
that direction.
Both the bow and the stem form a system of transverse
and diverging waves. The residuary resistance of the ship
corresponds to the energy expended in maintaining i^e
combined system of waves.
The diverging waves leave the ship immediately on
formation ; the resistance due to them is thus the sum of the
resistances due to the bow and stern systems tiJcen separately.
The bow transverse system spreads directly sternward,
and at the stern combines with the stem transverse system.
If the crests of one system coincide approximately in position
with the troughs of the other a small resultant system is
formed, and the resistance is comparatively low. On the
other hand, if two crests coincide, the resultant system is
lar^e, and excessive resistance is experiei^ced.
SPEED. 169
It follows that residuary resistance depends separately
upon three featnres :—
1. The size of ship.
2. The length of ship in relation to the speed.
3. The form or shape of ship.
A general account of the influence of these three features
is appended, prior to giving the methods used in practice for
powering ships.
Effect of Size.
Froude^s Law of CotnparUon,
If the linear dimensions of a vessel bo I times those of a model
(or another exactly similar vessel), and the resistance of the latter
at speeds Yi, Vg, V8» . . . be Bii Bt» Be . • . » then at the
corresponding speeds Vi V/, VftVZ, VsV2, . • • of the former the
resistances will be BiZ', B^Z^, Bs^. Or equally, if Hi, H^, Hs be
the powers expended on the model, those for the ship at the
corresponding speeds will be HiZ'/*, B^W^, HgZ''^, . . .
So that in order to compare the reaistanoe or power of
a model with that of a ship, the model must be run at the
' corresponding ' speed which varies as the square root of the
dimensions. If the model be constructed to a scale of i inch
to 1 foot, ^^isy ^^^ speed at which it must run in order to
compare with a ship at v knots is 'j-^
This law is exactly true for residuary resistance only.
It is approximately true for frictio!nau. resistance, and
consequently for total resistance, when the ships compared
are not greatly different in dimensions. When comparing
two ships of quite different dimensions, the frictions!
resbtance must be estimated separately and deducted from)
the total, or some equivalent method of correction should be
adopted.
Separate Variation of Displacement and Speed,
The E.H.P. (h) can be expressed in the f orm H = KW«V«
where W =^ displacement, Y ^ speed, K == a coefficient. The
indices m and n are not constant, but vary at different speeds ;
over a shprt range of speeds they are, however, practically
constant ; n is slightly less than 3 at low speeds ; at moderate
speeds it increases to 5 or 6, at very high speeds it again becomes
approximately 3. m and n are connected in similar ships by the
relation 6w + n = 7.
At low or very high speeds, when n == 3, tn == } ; hence the
power then varies as the square of the dimensions appjroximately.
At certain intermediate speeds, when » =£ 5, m «= i; hencei
the power varies^ directly as the linear dimensions.
It is above assumed that when the displacement varies
a corresponding variation takes plaoe in the length. The
results oo not in consec^uenoe ftpply to the same ship ivt
A
170
SPEED.
different^ draughts. It is fomid thatr m ii tliea weiy nthrlj
unity, so that the following rule is obtained : —
The change of power due to a modjerate Yariation of
draught in a ship without change of speed ean bet determined
approximately by aasuming the power to vary directly as
the ddsplaoement.
Effect of Change of Trim,
In ennooth water the effect of moderate change of trim on
the resistance of a well-designed ship is very small. At-
speeds that are very high in comparison with the dimensions,
e.g. in high speed motor-boats, considerable change ef trim by
the etem or ^ squatting' is apt to take place. This, apart
from its effect on the seawor&iness, increases the resistance.
It is prevented by the broad, flat stern usual in such cases ;
this, however, increases the resistanoe at low speeds owing to
the additional wetted surffuse.
Relation between Speed and Length.
This Was originally investigated by W. Froude (Trans.
Inst. N. A., 1877). By varying the length of parallel middle
body he obtained the results shown in fig. 141 for a scries
of ships having identical entrance and run. As would be
anticipated, the fricticmal resistance, shown below the base-
line, increases steadily as the length ^ets greater ; but tbe
wave-making reststance fluctuates considerably at tiie higher
speeds owing to interference between the bow and stern
transverse wave systems. At a particular speed, certain
SPEED.
171
lem^tlis, shown at Qy Q29 and Q3, are faTonrable, other lengths,
indicated at %, B^, B31 are unfavourable to propulsion. Inter-
mediate between these poan^ are others — P^^, p^, P3 — ^at
which an increase of length is beneficial, but a decrease of
length is detrimental. Froude found further that at different
Bp««ds the lengths should follow the law of comparison,
i.e. should vary as (speed)^. The fluctuations are of small
importance at small speeds or for great lengths.
These results have been confirmed' and extended by
Mr. G. S. Baker, who has deduced simple formulsB applicable
to all vessels of ordinary form (Trans. Inst. N.A., 1913
and 1914). On plotting a coefficient representing the resis-
tance of ships on a base of speed, 'humps' were obtained
at cisrtain speeds. These speeds correspond ezaetiy to the
mtdVay points p^, f^, ^8) shown above on the reiistaAce-leogtk
curves. At speeds somewhat greater than these, or eqoali^
with lengths somewhat lessened (at points Q^, Q^f ^3) ^®
resistances are comparatively low. At rather lower speeds
however, or for rather greater l^igths, the resistanee is oom-
Sarativeiy Urge. It was found that the portion of tho humps
cpended upon the eoeffi(»e&t p, determined by the fbrmulh
P = -746-/= where V is in knots, L in feet, and 7 is the
prismatic coefficient (see p. 93).
In vessels of usual type, the points Pi, P2, Ps* i.e. the
intermediate points in the resistance length curves (fig. 141),
or the humps on the resistanee speed curves, oceur whea
** =" ^' V2' VS' ^*^-
dence, for favourable propulsion, P is somewhat gteater,
so that vVrl' is approximately 2*39, l-OS, or 0»65.
This gives the following stiitiAble lengfths and prismatie
coefficient for various speeds : —
.3 .
II
Approximate favourable water-line lengths in feet for various
priamatic ooeffloi9nt9 (7).
7= -6
7 = -7
7 = -8 1
p=vi
«vl
= Vif
=vl
^V^
= vA
=vi
-V*
«VA
sa
30
27
24
21
18
15
12
9
760
625
510
400
30^
225
155
100
56
940
720
525
365
235
130
820
570
370
200
650
640
435
345
260
190
135
86
48
800
600
440
320
200
110
700
500
320
175
570
470
380
300
230
170
120
75
42
890
700
540
400
280
175
100
620
440
275
150
172 btHitiU.
Lengths midway between those given in the table will
generally give bad propulsive results. It should be noted that^
generally speaking, in high speed ships, increase of length is
beneficial to propulsion ; but when practicable [y should also
be varied so as to comply with the relation in the preceding
table and formulae.
With extremely high relative speeds, e.g. in destroyers and
steam pinnaces, another hump occurred in the resistance speed
curve when p = 1'5. For a prismatic coefficient of '6, thi«
gives the following lengths and speeds : —
Speed in knots . • 36 33 30 27 24 21 18 15
Length in feet . . 530 450 370 300 240 180 130 90
Lengths equal to, or rather greater tlian, the above are
nnfavourable ; but lengths smaller than these are fairly
favourable, oomparatiyely speaking, although the speeds- are
then very high compared with the size of vessel.
Length of Entrance.
In the paper above referred to, Mr. Baker also stated
that in some oases a hump occurred in the speed curve
depending^ on the length of entrance (lis) alone. This may
be of importance in full ships ; and unfavourable conditions
then result when V = 1*095 '^Lb, approximately.
Efpect op Form and Peopoetions.
No rules of ni^versal application can be formulated. The
following brief summary represents - conclusions that are in
general sound, but to which there are many ezoeptions.
1. Shape and proportions of transverse eeotions. — ^If the
area of tiie midship section is kept unaltered, change in the
ratio of beam to draught affects both skin and residuary
resistance to about the same extent. The deep and narrow
ship is usually easier to drive, but the dUQoreoioe ila hardly
appreoii^le when the beam-draught ratio is less than about
3*5* From some observations by Colonel Bota (Trans. Inst.
K.A., 1905) it appears that at ordinary speeds (up to v/ Vl « 1*4)
the totei resistance for ratio 7 is some 15% more than for the
most favourable ratio (about 8 or less).
Variation in shape of midship section, apart from variation
of area and proportions, has very slight influence on the
resistance.
At the ends the shape of transverse section is not of great
importance at low speeds, but at moderate speeds WVIj '8
to 1*2} u sections forward and v or hollow sections ait should
be adopted.
2. delation between length and displacement. — This is
represented by the ratio l/W* (Froude), or w/(xJir^)' {Taylor),
Unless the speed is very low the resistance per ton of
srjEO.'
17S
a ship having large beam and draught is much greater than
that of a ship having small lateral dunensions with the same
length, speed, and coefficients of fineness. The length-dia-,
placement ratio id thus one of the most important factors
affecting the resistance at high speeds. In practice, how-
ever, the length is determined by a variety of consideratifms.
In large walrships it is often virtually fixed from the lengths
of the internal compartments of the ship. In fast ships
due attention must he given to the relation between length
and speed as well as to that between lenff th and displacement ;
the standard results of Froude and Taylor (see p. 179) can be
used in such cases as a guide. It must be remembered that in
all cases additional length involves increased friotional resis-
tance ; BO that a compromise between conflicting considerations
has generally to be made*
3. Prismaiio coefficient and curve of areas of sections.—^
Mr. G. S. Baker (Trans. I.N.A., 1914) has recorded the
results of a systematic series of experiments made on models
of a ship 400' X 62i' X 23-2 of mercantUe form. There
were three groups of trials, designated "F, O, H, having lengtiis
of parallel body of 41-8', 120', and 200 resj^ecMvely. The
midship section coefficient was *98, there being no rise of
floor.
The entrance and run were varied for each group in the
following manner : —
No. of
experiment
Remarks
En-
trance
Bon
Whole
ship.
group F
Whole
ship,
groapG
Whole
ship,
gronpH
1
2
3
4
5
•72*
•672
•672
•672
•625
ATI
•638
•712
•691
•691
•691
•67
•775
•768
•768
•758
• 741
•85
•828
•828
•828
•816
Straight line bow
Medium „ ,,
Hollow „. „
6
7
8
9
10
AU
•672
•578
•638
•638
•638
•70
•664
•691
•691
•691
•719
r
•737
•768
•758
•768
•780
•812
•828
•828
•828
•843
Straight line stem
Medium „ ,,
Hollow „ ',,
♦ -764 for H.
The mean displacements were 9,400, 10,300, 11,260 tons
in the three groups. The results were plotted on a base of
V
p, orr~7T~7=f where y is the prismatic coefficient of the whole
ship. The conclusions as regards the total resistance were
briefly as follows : —
174 S?EED.
Fineness of entrance, — Set F. 5 better than 3 at all
9peeds« 3 better than 1 when P is more than *6. Highest
eoonomical value of p is '75.
Set G. 5 better than 3 when p exceeds '53. 3 better than
1 when p exceeds *57. Highest economical value ol P is *61.
Set H. 6 better than 3 when P exceeds '45. 3 better
Hktok 1 when P exceeds '375. Highest economical value of
P is '46.
Shape of entranee, — Set F. The medium bow (8) best
over the useful rang6 of speeds ; straight best and hollow
worst when P exceeded *75.
Set G. Hollow bad. Mediam best for P between *53
and *62. Straight best at higher speeds.
Set H. Straight best especHally when P is above *45;
Fineness of run, — Set F. 10 best when P exceeds '65.
9 and 6 about equal.
Set G. 10 always bad, 6 slightly better than 8 when
p exceeds *55.
Set H. 10 causes serious eddy-making, 6 superior to 8.
Shapie i>f run, — Set F. The straight (7) best up to p
sss *7* Tlk9 hollow slightly better at higher speeds.
Set G. Straight and medium about equal. Hollow
iaferior.
Set H. Straight best. Any hollow causes eddy-making.
FiBAlly, the offset of introdaoing various lengths of parallel
middle body was investigated. For the mean curves (Nos. 3
«r 8) it was found that when this length wfoa 42 feet (the
total length being 442 feet) the propulsion was more favoar-
able, reckoned on the basis of C (p. 177), than f6r greater
lengths, particularly at high i^^eedst, except when P waa les9
than '38, when total length of 520' wias superior. At all
speeds a 600' length (parallel body 200') was inferior.
In regard to eruiser and other forms suitable for higher
speeds than the above types, Mr. B. iE2. Frpude states that
except in extreme cases where serious eddy-making might
be introduced, the resistance of a form is dependent princi-
pally on — ^_
1. The extreme beam.
2. The curve of sectional areas.
8. The shape of the waterline forward.
1 and 3 have been considered. Two of the main features
of the curve of sectional areas whieh affeot resistaiioes
(a) The prismatio coefficient which is propartional to
the area of this curve, and equal to its nuean ordinate divided
by area of midship section ; and
(6) The distribution of this area fore and aft.
SPEED. 175
(a) Tho iDfluence of prismatio coefficient for ships of
good form is sliowii in Mr. D. W. Taylor's results.* For
speeds up toy/ Vl»1, this coefficient should be low. F«r
higher speeds, ih&te is a certain ooefficient which la assooi-
ated with the minimum residuary resistance per ton ; thus :—
v/Vl . . . . . I'l 1'25 1-60 and above.
Best prismatic coefficient about '57 •60 *65
Broad ships require rather higher^ and narrow ships rather
lower, coefficients than those given. At the higher speeds
a large variation can be made from the coefficient given
without appreciable loss.
(o) Having decided upon the prismatio coefficient suitable
to tlie particular speed, there are certain general principlea
to be followed in fixing the shapie of the curve of sectional'
areas.
Generally speaking, the fore-body should be finer than
the after-body. The section of maximum area and the centre
of gravity of the curve of areas should be dightly abaft the
middle of length.
The curve should be emooth, the variation in area of the
sections being as gentle and gradual as possible. Any abrupt
change in seetionu area tends to produce increased reeistanoe,
and this is particularly true pf the shoulders of the curve in
the fore-body. The buttocks of the after-body can, however,
be made relatively abrupt without detriment, ptovided the
curve is smooth.
Besistance is generally more sensitive to variation of the
fore- than the after-body, and perhaps particulariy to variflr
tions at the ends of the fore-body. , •
At values of -^ below '6 where a relatively small pro-
portion of the total resistance is due to wave resistance,
resistance is not sensitive to riSght variation of the curves,
bat it is almost as neoessary to obtain a curve of areas which
will give a teinimum reGistanee per ton of oarrying power of
the ship at these low speeds as at higher speeos.
y
For vttlue of --r- 7 to 12 the forward half of the curve of
areas in the fore-body should be decidedly convex to the base-
line, thus tending to become tangential to the base-line ; the
ending of the curve may, however, be made abrupt wltiMut
affecting the resistanoe adversely.
Mr. Froude (Trans. I.N.A., 1905) showed that for oruisers
y
within these limits of speed— ^'7 to 1*2, rather hollow -bow
lines were distinctly preferable to straight ones.
• The Sveed and Power of Ships, by ©. W. Taylor (Chapman & Hall).
176 SPEED.
For still higher values of -^ the fore end of t|i« curve of
the fore-body may be straightened with advantage aa the value
inoreaaea until it may even be conoave to the base-line*
From the above it will be seen that a carve of areas which
may Buit one set of speed conditions will not be at all suitable
for another. It is therefore very diesirable to adhere as
closely as possible to the form — ^particularly to the forward
portion — of a ship Or model that has been found successful.
If it is desired to increase the displaoement without seriously
affecting resistance it will generally be found that it can bie
added at the buttocks of the after-body curve of areas,
provided the general smoothness of the curve is still
maintained.
Methods of estimatino Hobse-poweb.
1. By Model ExperimenU.
This is the only reliable method of estimating the power
required in a new design. It is advantageous and economical
in two ways : (a) a smaller margin can be allowed on tiie
estimated power, since the only uncertainty lies in the corzeot
prediction of the propulsive coefficient ; (b) the effects of
varying the lines or proportions of the ship are readily in-
vestigated ; important propulsive economies are thus
frequently eifected at a trifling cost.
As stated on p. 169, the resistance of the ship is reduced
from that of the model, by means of the law of comparisonj
which is applied to the residuary resistance alone ; the fric-
tional resistance of both ship and model is estimated separately
by a formula. By multiplying the fresh-water tank resistance
by T^pr t allowance is made for the difference of density
between fresh and salt water.
The results are convenientihr recorded by means of the
' constant ' system of notation devised by Mr. B. E. Proude
(Trans. I.N. A., 1888 and 1892).
The method may be described as follows : — ^The propor-
tions, and to some extent the lines, of the hull are
characterised by numerical values and diagrams, representing
not absolute measurements of hull, but measurements stated
in terms of a unit-dimension proportional to the cube root of
displacement. The performance and proportions of the ship
are represented by 'constants', designated as follows: —
Let L, B, D » principal dimensions in feet,
w = displacemeiit in tons.
8 ^ wetted surface in square feet.
B = res stance in tons (salt water).
Y s= spee^ in knots.
Then ipeed constunt K = —r x -6894.
IkaiEtanoe conBiwit 0 = —j-j x 2B38.
-^ '""■'■
Iiength-gpeed eonslant I = -jj x 1 ■ 0S63.
X.
" VM.
1
Le'ngth cdogfamt ■■ a ^ x -8067.
Brm<mty>rdra[]^tconBtuitBorS= r~ X (b or d).
Skin ooQstuit i" i^x -OgsM.
IP aOOFT.LONa)
E
Via. 110.
'1
1
/^
;S^
K
■2
1
1
A, muo «^tet-line» viz. anrra of j^ &FflftB -^ mewi dEftosM. s, load water'
Bbt. c. waMr-llBs ni tt^ man dnii«lit. s, imrtMt ihUod.
!I%B shape and propoitions of &e hull are indicated in Qg. 143.
Tlie imdingB of the Imes and Sgures given denote, not abEolute
iineu dimenBioos, but the ratios of these dmensions to the
lioeac nnU w^, m ezenqiliSed m the above oonslants.
OwiBg-to tte-lcv-fni rtin friotidn reaiitaitoe b«og dSueCt
friHB tlut for reiidauf r^aiitanoe, the canattfnta. (which Ae
* Iterdpteaent TftTlot'al«wth-diBiilH*iii8D(oosIDeleDt(wJ(iigrJ'),tlien
Mlfl=j86TO or M X V« = BO-fiT.
-• Comotlan eatVBt for vbiIdiu lenEtb of ship. OcdinHtea donnwudi
ln>m aeo (eel line to be sdded to. uid npwudB to be dedooted from, the
C ludiiigB o[ the coDBtant cnire.
178 SPEED.
based on the law of compariaoB) will need correction as the
eize o£ ehlp varies.
To make the correction, let small letters denote model,
and large, ahip ; the ratio ol linear dimeiuions being ».
Let dashes refer to that part of resistance which is causMl by
friction only. Let r, f, be coefflciunta of iriotioa for ship
and model.
C'^E:
wiv"
=• constant x fSv
= constiuit >
SL'
■175 ^
-cere.
Similarly, e' = constant x 8L"'™ x fl
Hence c - C = c' - C, (for that part of c which la dne to
residuary resistance does not alter with dimensions), = 8L~
X (o - o) irhere the ' o ' ooeflicienta depend apon the friction and
length. Typical correction curves are shown in %. 142.
For a smooth surface, o can bo found from ihs following
table :—
TABLE OP VALUES OIT 0 FOB VARIOUS LENGTHS. 1
Js
So
fr?
%r,
r.-F
So
tl
-o
11
So
s=
^■s
5-a
-".
r=
51
g
20
■11470
eo
160
•08218
«0
■07!ia
10
«
^Sl
™
fl^i
18
SU
■0M64
-08M1
•^
■07412
Pig. 1J4 shows the variation of o for different values of m
nd also through a limited range of beam ; draught ratiot'
iuoh a diagram, when interpreted by approjyate scalet, is
SPEED. 179
in effect a diagram showing E.H.P. at a certain speed, for
all the torms reduced to a common displacement, plotted to
length of hull.
2. By Curves obtained from Methodical Experiments,
Among the methodical experiments that have been carried
out are the following : —
(a) By Mr. R. E. Froude (Trans. I.N.A., 1904) .-The results
are also given in a different form in article ' Shipbuilding ' in
Encyolopcedia Britannica, 10th edition, by Sir Philip Watts,
K.G.B. The parent ship is of cruiser form 350' X 67' X
22' ; displacement 6,100 tons ; block coeff. *4865 ; midship
section coeff. '8775 ; prism, coeff. '5385 fore-body, '570 after-
body, '555 mean. The cross sections were also varied, so
that for the same length the displacement varied from 2,500
to 10,600 tons ; or H varied from 7*884 to 4*886. Two ratios
of iieam to draught 57 : 22 and 66 : 19 were tried. The
speed was altered so that K varied from 2 to 4*8. Finally,
the parent form was modified so that on about the same
displacement the length was reduced in steps from 350' to
310', increasing the prismatic coefficient from *555 to '618.
(6) By Mr. B. W. Taylor (Speed and Power of Ships,
Chapman & Hall). — The prismatic coefficient varies from
'48 to *80, so that uie forms vary over a very wide range of
types. Two ratios of beam to draught 2*25 and 3*75 were
tried. The length-displacement coefficient (w/(7v^l)') varied
from 20 to 160, so that X varies from 11*3 to 5*6. The
midship section coefficient was *926. The ratio v/Vl varied
from -6 to 20.
ic) By Mr. G. S. Baier (Trans. I.N.A., 1913 and 1914).
—The 1914 set are described on p. 173. The 1913 set con-
sisted of five groups, each consisting of four or five models
having the ratio length entrance : length run varying from
*55 or more to nearly 1*7. The ratio breadth : draught was
2*25. The prismatic coefficient varied from *60 to '84, chiefly
by altering the proportion of middle body. The length was
about eight times the beam in all cases. K varied from *9 to
3*0. The variations of resistance due to change in the contour
of bow and stern were also investigated.
(<Q By Professor Sadler ^Trans. American I.N.A., etc.).—
These have reference mainly to the effect of varying the
length of parallel middle body at different speeds. Length
was eight tii^es the breadth ; but the breadth-draught ratio
varied from 3*0 to 2*143, and the prismatic coefficient from
• 734 to • 76. v/ Vl varied from • 2 to • 9. Other variations in form
were also tried.
(e) By Sir John Biles (Design and Construction of Ships,
vol. ii, Griffin & Co.). — Particulars of the resistances of
thirteen vessels of widely differing types are given. The
results are presented in a form very convenient for practical
application to new designs.
180 SPBBD.
Of these results, those by Mr. B. W. Taylor cover by
far the widest range, although Mr. Froude's series infdade
very many useful cruiser and battleship forms. The applica-
tion of tne results to the powerlog of ships is in each case
eefsy. The data will be found useful both for powering and
for determining suitable forms for such ships as oome within
the ranges dealt with.
For fishing-boats and fine-lined commercial niotor-vessels,
a table of approximate B.H.P. and speeds is given by
Mr. Linton Hope, Trans. I.N. A., 1910. : » '. >
8. By the AdmiraUy Coefficient.
wi v* "w" V*
The ratio , or for ships driven by turbines, is te];;med
I.H.P. S.H.P.
the Admiralty coefficient. It is connected with f'roude's resistance
" constant " C by the relation \C = 427-1 /t, or X = 427-l/i(C ;
where A. = Admiralty coefficient, /t = propulsive coefficient (t?. p. 181) .
This coefficient is not constant, even for the same snip at
different speeds. Its value depends on the efficiency of pro-
pulsion, the form, and the speed of the ship. For edmilar
ships at corresponding speeds it is constant provided thait the
means of propulsion are also mmilar, and that the aetaal
dimensions are not so greatly different as to necessitate an
appreciable frictional correction. For similar ships at very
low speeds it is almost constant.
It is to be noted that a high Admiralty coefficient indioates
comparatively low power, i.e. economical or favourable
propulsion.
The factors on whieh the Admiralty coefficient principally
depend are, in order of importance : —
(a) Type of ship, which generally includes mode of propulsion^
affecting m.
if)) Speed-length ratio - V/ -/l. j^
(c) Length-displacement ratio M or— -v X -305^ in Frotide'a
notation.
(d) Prismatic coefficient.
(e) Form including B, D, shape of sectional area curve, shape
of water-line forward.
(/) Absolute size. Small boats or ships have relatively low
coefficients owing partly to the relatively greater skin friction.
(e) and (/) are to some extent included in (a),
Tlie data in the following table are obtained from actual
trials of ships ; they can be used for predicting tiie Admiralty
coefficient, and thus the power of a new ship, provided it
resembles fairly closely one or more of the ships in the table,
can be estimated. The accuracy of the prediction and tiie
reliability of the method depend entirely on this approximate
similarity between ships and engines ; when this is not fairly
good the method is quite untrustworthy. If the propulsive
coefficients of the two ships compared are probably not the
same, their variation should be allowed for.
6PBBD. 18]
Ko attempt is madid in the table to cover all forms and
proportions of frequent praciioal occurrence ; a t9w ships of
each of the leading iyp«s are dealt with. The arrangement of
the data that is adopted in the table will generally be
lonnd ccmvenient for recording the results of the triius of
ships in a form that facilitates their utili^tion for approxi-
mately powering new designs . L = length b . p. ; w = displacement
in tons; y = speed in knots; b/d = ratio of breadth to mean
diaoght ; /B= block coefficient ; R, T= reciprocating or turbine.
The priamatic coefficient can usually be inferred from $
and the typo of ship ; in some cases the midship section
coefficient ia added.
Admiealty Coefficients
OF VARIOUS Vessels.
Ship.
I.
410
w
•V
•70
^
1
p
2-0
$
— 1 ;
H.P.
•
R-
T.
r
B
M No. of
Screws.
1-1
Cargo . . *
Passengetr boat
1180(
14-16
%30
•75
600b
29
4T0
13201
16-1
•74
120
2*5
•64
7900
R&T
4
29
47t)
1320(
14-6
•(J6
120
2-5
•64
610J
B
4
31
»» •
166
«6(
10-1
'86
230
1-7
•61
241
B
1
431
>t • •
170
106C
11-16
•86
216
21
•60
437
B
1
89
»» • •
247
3070
10-6
•68
20&
1^9
•70
818
B
1
301
Oil tank . .
470
17^00
14-2
•65
165
2-0
•81
6000
B
1
32i
Motor cargo boat
860
360
9m
9500
It
8
•68
•42
200
200
2-2
2-2
•78
•78
2160
850
B'
2
271
27(
Liner . « •
760
37080
25-6
•93
85
2*7
•60
760C0
4
24i
^ ^M A^^^ • w V ■
760
37080
23-7
•86
86
2-7
•60
51300
T
4
281
>t • • .
760
37060
20'4
•74
86
2*7
•60
29500
T
4
31(
ff • • .
760
37080
15*8
•67
86
2-7
•60
13400
T
4
321
Oil tank . . .
402
16700
12-fi
•62
240
2-0
•78
8300
B
1
3H
Salvage steamer
186
1000
12*3
•90
166
8-6
67
690
B
1
27(
Ore oarrler •
320
6-430
18-25
•74
166
2-3
•78
2365
B
1
30(
Carflfo ...
279
4780
ms
•Z4
218
2-0
•76
1577,
834
B
1
331
River • • »
195
685
10-7
'76
.92
9^:5
•81
B
2
1(H
Passenger . .
420
9600
17*8
•87
130
2-0
•64
6780
B
1
371
Training ship .
270
4870
10
•61
222
2-2
•64
932
B.
2
281
Steam collier .
280
4188
12
•72
191
2-2
-79
1880
B
1
2l<
Channel ateamer
260
2110
15
•93
120
2-9
•63
1640
Jl
2
331
Oable steamer .
243
2526
IB
•83
177
2-2
•70
■1790
B
2
22
Cargo and
passenger
180
1287
18-3
•92
212
2-1
•66
770
B
1
27'
Biver . . .
310
2460
16-4
•87
82
4^9
•64
3076
B
2
21J
Cargo and
passenger
300
4268
16
•87
158
2-3
•68
2811
B
1
311
466
15970
16-4
•76
160
2-1
•7^
7520
B
2
37!
Dredger . . .
105
299
6-5
•63
2;:8
4-5
•63
100
B
I
12!
Trawler . . .
10&
300
10-^16
1^10
275
2-2
•48
406
B
1
Ill
Liner . . .
650
21660
20^6
'88
130
2-0
•69
20000
T
3
34(
^6rry CiTeared
tarbine) .' .
2&0
«65
18
1*14
55
6-3
•63
2500
T
2
21(
250
865
16
1-01
55
6-3
•63
1520
T
2
24!
99
250
«66
13
•82
56
6^
•53
440
T
2
461
99
»
250
866
10
•G3
55
6-3
•53
200
T
2
45(
182
SPEED.
Admiealty Coefficients (Tugs, Small Ceaft, btCv).
Ship.
Tug:
Passenger boat
Yacht . . . .
Grab hopper
dredgrer . .
Hopper barge .
Harbour launch
Passenger pad-
dle steamer .
>>
»>
)i
it
i*
_ »»
Stern wheeler .
»
»
19
»»
Launch
Tunnel steamer
tt
TunneUifeboat
Tunnel steamer
>i
Tug . . ,
Paddle ferry
Tender . .
Vedette boat
140
120
110
140
100
90
80
Tin
70
65
60
45
40
135
130
176
160
140
140
100
125
147
100
165
205
12J
170
130
220
110
76
107
135
120
83
46
66
60
65
60
67
160
140
67
125
60
115
134
80
100
w
440
235
230
620
240
230
120
110
116
90
57
29
24
340
260
560
4 6
278
920
280
420
676
160
250
450
76
270
130
620
160
393
153
177
160
61
7
11
12
23
40
2S
200
152
37
123
12
400
2:0
74
61
11-75
1225
110
121
10-5
10 25
9-1
106
10-4
10-2
81
8-25
8'2
11-1
10'6
101
118
116
8-5
71
90
8-5
10-7
13-6
16-7
10-5
13 b
11-5
80
11-3
8-3
9-7
8-4
88
8-4
9-6
15-0
9'5
8-0
ll'l
80
10 25
13-7
10-23
9-4
9-0
11*1
ll'O
11-3
180
v/Vii
00
12
05
02
05
08
82
25
24
26
05
23
67
96
92
76
93
98
72
71
72
70
07
1
17
96
80
88
54
08
95
94
72
80
92
41
00
31
15
43
93
81
16
36
84
27
04
95
27
8
^
00
8
1-1
160
136
173
226
240
182
231
342
835
328
265
318
375
138
118
104
114
101
335
280
215
212
150
661
52
41
55
69
58
120
90
122
72
87
107
77
eb
96
138
180
76
49
66
200
6L
96
316
91
144
61
B
D
2-7
27
2-7
2-6
2-2
2-2
2-7
2*2
2-4
2-2
26
28
2-2
38
4 2
4-8
2-2
2b
26
4-8
3-2
2-7
26
4-2
3-6
6-4
38
6-6
7-2
4*4
12-7
10-0
18 5
12 8
10-0
3-0
30
29
3*8
22
9-8
16 0
7-0
3-6
91
4-0
2-4
66
24
4b
P
-62
•46
•52
•67
•46
•54
•54
•52
•48
•49
•50
•55
•51
•e4
•61
•64
•3>
•37
•72
•77
•65
•66
•48
•60
•49
•66
•47
•71
•86
•64
•72
•78
•85
•86
•77
•20
•22
•24
•38
•42
•69
•73
•60
•52
•49
•47
•61
•66
•44
48
H.p.
840
590
460
1120
4ia
420
240
830
350
190
106
CO
66
647
35a
620
469
376
498
160
2C0
330
260
6:0
16"0
125
680
350
310
330
113
260
266
263
186
46
160
43
46
160.
65
326
610
182
235
67
1120
616
206
622
o
O
§
•80
•73
•80
•85
•93
•88
•87
•83
•91
•65
•62
•96
•80
•85
•86
•40
•41
•88
2
2
1
2
1
1
1
1
1
1
1
1
1
2
2
2
1
1
1
1
1
1
1
2
2
2
2
2
1
2
2S
•<0
112
126
ICO
114
1G9
75
72
82
76
112
71
93
68
126
135
131
220
162
117
100
181
121
132
172
176
16T
180
109
119
129
68
V9
70
76
50
69
109
100
120
107
61
76
136
66
61
6T
76
96
98
129
Ajy-MmAJLTY UOEFFICIENTS (MOTOE-BOATSJ
. (L.Hope Esq
1.. Tnoa. I.N./
L.,19I
Class of Boat.
li
W
V
•99
^
s
D
30
i3
27
n
60
^ No. of
Screws.
•-■J
si
Ketch drifter . .
65
57
80
210
12
Lugger drifter
71
70
6-1
♦62
195
30
•24
23
J'<
Irish ketch . .
47
23
n
•89
270
2-8
•30
20
13!
Lowestoft ketch .
60
64
•80
300
27
■33
35
12i
Sailing trawler
72
114
52
•12
305
21
ai
34
101
Fishing boat . .
29
9
70
126
370
2-8
•33
12
12
it
__
— .
46
•f2
—
— .
_
7
6i
Oyster dredger
46
21
60
•89
230
3o
•29
17
9
Schooner . . .
153
735
78
•63
205
2-7
•60
190
19(
Coaster yacht . .
71
115
7-8
•ii2
320
26
•61
40
27:
Pilot ship . . .
64-5
64
60
•75
240
2-7
•30
30
in
Oil tank lighter .
60
93
55
•71
430
40
•67
45
V
yy
60
83
62
•81
155
11-3
•65
45
6<
Tunnel boat . .
59
95
70
■91
46
15-7
•66
20
7
55
12
80
103
72
78
•74
83
S\
Chain haulage tug
Cl-unt) . . .
28
49
2'5
—
220
7-8
•77
16
\
Chain towing eight
punts (100 tons)
28
105
2-5
—
4F0
7*8
•77
16
2i
Cargo and passenger
42
20
74
114
270
40
•46
45
6
Mail and passenger
94
86
102
1-08
130
36
•44
162
IK
ft*
70
18
10-6
1-28
52
65
■83
100
HI
40
5
10-6
1-6 r
78
66
•30
46
7^
Dispatch boat . .
55
11
10 9
1-72
63
31
•27
60
1()<
••
55
13
142
1-91
78
30
•31
200
*■
71
Wabships.
Ship.
ii
w
V
18-3
v/vii
^
1
3
B
D
28
•65
H.P.
B.
T.
R
No. of 1
Screws. 1
Battleship .
460
15,200
•92 237
16:400
2
230
ff
400
15,200
131
•66
237
2-8
•65
4,360
R
2
316
436
11,760
19^6
•94
142
2^9
•54
12,500
R
2
312
436
11,760
17-7
•86
142
2-9
•54
8,750
R
2
328
436
11,760
14-7
•70
142
2-9
•54
6:000
R
2
328
ff
436
11,760
100
•48
142
2-9
•54
r,730
R
2
299
580
25,000
22-1
•92
128
32
•60
32,700
T
4
282
ff
500
19,000
21-0
•94
152
31
•60
24,500
T
4
270
600
19,000
18*4
•82
162
31
•60
14,700
T
4
302
600
19,000
13-0
•68
152
31
•60
4,900
T
4
320
600
19,000
100
•46
152
3-1
•60
2,300
T
4
310
Croiser • .
355
6,270
21-3
lib
118
3-0
•50
12,500
R
2
234
99
365
6,270
19-5
1-04
118
30
•60
8,760
R
2
236
365
6,270
16-7
•79
118
3f-0
'30
5,000
R
2
282
600
14,250
24-1
108
114
2-7
•64
31,600
R
2
261
yj
600
14,250
200
•90
114
2-7
•64
14,750
R
2
318
600
14,260
13-1
•58
114
27
•64
4,040
R
2
323
600
14,250
5-6
•10
114
2-7
•54
1,690
R
2
58
660
18,760
24-3
1-04
113
31
•58
34.400
T
4
293
• A
660
18,760
22-4
•96
113
31
•68
25,800
T
4
307
660
18,760
16-4
•70
118
31
•58
8,600
T
4
361
yy
430
5,400
26-2
r26
68
32
•56
28,000
T
2
198
Destroyer
280
1,000
330
1^97
46
30
•52
15,500
T
3
231
270
950
27-0
1^65
48
2-9
•50
13,000
T
3
147
220
400
300
2-02
88
2-6
•41
6,400
R
2
229
»f
240
775
30-9
1-29
66
32
•54
19,000
T
2 1 131
SPEED.
■^— ^
Table of Tvo-thiuds Powebs.
w
w8
w
W^
w
W^
100
21-6
1.000
100
10,000
464
110
230
1,100
107
UJOOO
496
123
24-3
1,200
113
12,000
£24
130
26-7
1,300
119
13,000
663
140
27-0
1.400
125
14,000
681
160
28-2
1,600
131
16,000
608
160
29-6
1,600
137
16,006
«35
170
80*7
1,700
142
17,006
€^
180
81-9
1,800
148
18,000
«^
190
$3-0
1,900
163
19,000
712
200
84-2
2,000
169
20,0PQ
737
320
86-4
2,200
169
22,000
785
240
88*6
2,400
179
24,000
832
230
40-7
2,600
189
23,000
878
280
428
2,800
;99
28,000
922
300
44^
3,000
208
. 30,000
965
820
46*8
3,200
217
32,000
1,0?8
340
48-7
8,400
226
34.000
36,000
4,060
830
60-6
3.600
236
1,090
380
62-6
3,800
4,000
244
88,000 ■
1,1S0
400
64-3
252
40,000
1,170
440
57-8
4,400
2o8
44»000
i;246
480
61-3
4,800
235
48,000
1,321
620
64-7
6,200
300
52,000
1,393
560
67-9
6,600
316
66,0Q0
1^464
GOO
70-5
6,000
330
60,009
1,633
G50
750
6,600
848
66;000
1,617
700
78-8
7,000
336
70,000 •
1,698
750
82-6
7,600
3B3
76,000
1,778
800
86'2
B,000
400
80,000
1,867
863
89-7
8,600
416
86,000
1.933
900
93-2
9,000
433
90,000
2.006
950
96-6
9,600
448
96,000
2,082
A^ot^.— Intermediate Taluea of wf can be obtained by inlecpola-
tion where moderate acooracy alone is required, e.g, in connexion
with ihe * Admiralty coefficient ' for determiningr the horse-power of
Rhips. Otherwise fin4 w in the taUe of cubes at the end. and read off
wi in the " ngnares " column opposite.
4. From Trogressive Trials, extending the law of
comparison.
In this method it is assumed thai the form and proportions
are fairly similar ; that the H.P. of a ahip with speed oon-
stant varies as the displacement ; and that ijhe variation in
propulsive coefficient can be neglected (or it may be known
and allowed for afterwards).
SPBED. 185
Lei tiie symbols w, l, and v refer to the displaoMient,
length, and speed of the new ship, of which the power n is
required. Let w^^ L^ refer to a ship of similar type whose
horse-power (H^) over a rang« of speeds (y{) ha?e been
determined by trial.
By Fronde's law of comparison^ the data for the old ship
can be changed to Win', Li n, Vi n*, Hi n* ; where n is any ratio.
ChoqaP n 80 IJiat Lin ^s" l>, or i» = r Then seleot a speed Vi
Li
in the old ship so that Vi Vn = v, or Vi=v/ Vn.
Kead the horse-power s^ for the old ship, corresponding
to the speed v^. Then that of the new ship is H^ni, This
applies, however, to displacement w^n^, which is nsnally
different from w. On correcting for this, we have, finally, that
horse-power of new ship at speed v and displacement w =
Hin3x-^=Hi.- . V-
Witr* Wi ^ Li
Example. — ^To find the S.H.P. of a ship, 25 knots speed,
800' long, having a displacement of 36,000 tons, and generally
resembling the Lusitania.
Here L = 800, V = 25, w = 36,000. Also for Lusitania,
U = 760, Wi = 37,080.
OR
From cnrve (fig. 145) of speed and horse-power for
Lusitania, the corresponding power H^ is 57,600.
Hence power for new ship under the conditions required =
57,600 X p^ Vl^^ = 57,500.
5. From approximate Formula or Curvei.
Since it is possible to estimate the frict tonal (reaisiance
with fair accuracy, in ships of low or moderate speed a
considerable error in the value of the residuary resistance
will make only a small difference to the Hotal resistance,
which is mainly friotional. In such cases, provided the ship
be snfiSoiently well formed aft to avoid eddy-making, an
approximate formula or a simple eeries of curves may be
used to estimate the residuary irestistance.
The following data are taken from a paper by Mr. A. W*
Johns, M.I.N.A. (Trana. I.N.A., 1907).
(a) Mr. D. W. Taylor gives the formula —
Residuary resistance in lbs. per ton = 12*5 ^v^ji^ ; where
6 = block coefficient, and L is length on water-line in feet.
186
SFEBD.
(6) Mr. Taylor has also given the carves 6hown in fig. 146.
It was stated that the formula is usually more aocoraie for
low speeds (up to v/ Vl =» '9), while the curves are prefeiahle
for higher speeds. In either case, residuary S.H.P. =3
resistance in lbs. per ton x wv/326.
(0) Mx, Johns gave the curves i^own in flig, 147. These
are applicable to ships having low or moderate speeds. Take
L as the length between perpendiculars ; and in merchant
ships increase the aotual prismatio coefficient by '02 before
applying it to the curves.
Fig. 145.
SCALt OP
S.H.P
SPECD-POWER CUaveOF LU.SITANIA.
20 21 22 23 24 2$ KNOTS.
It is to be observed that Taylor's curves omit the prismatio
coefficient, and that Johns' curves omit the relation between
length and displacement. It would appear that tiie former
curves should give better results for a ship of unusual pro-
portions, and that the latter should be superior for a ship
with unusually fine or full lines.
Example, — ^To find the horse-power of tiiie ship in the
example on p. 185. Assume draught 80', L = 800 ; V <*> 25 ;
W « 86,000.
APBEt>.
187
Fio. 146.
8IDUARY RESISTANCE IN POUNDS PER TON OF DI8PLACBMENT
By D, W. Taylor, Esq.
• i V' « ■ t'o ■• i» I*! i.* ii ■ I ft* I T" ■• •■'" »ft yi f-l
9CALt PQIt SPEED — UtNOTM COEFFlClStfT 4
i
las
aPBBD.
Fig. U7.
QUBVE8 OF RE8iOUARY E.H.P.
By A, W. Johns f Esq. .(see p. 186).
A.NOJ 31V3V
.V^MO* 31V3S
Note.— In merohant Bhips increase adDal prismatic coefficient
applying it to the curves.
by -OabeM
8PEED. 189
Wetted surface by Denny's fonnala= 82800.
From the table on p. 167 it appears that / — about • 236 for
L = 800'. Frictional E.H.P. =« -236 X 82800 = 19500.
Using Taylor's curves, w/ (tqq) = 70-3 ; v/ v^L = -885,
whence residuary resistance per ton =« 3*5 approximately.
BesiduaryE.H.P. = 3'5 X 36000 X 25 -f 325 = 9700.
Alternatively using Johns' curves, prismAtic coefficient »
•62 + -02 = -64. \% = -782. Whence residuary E.H.P. «
•047 X (36000)i = -047 X 36000 X (/36000 = 9700.
In this case both sets of curves are in absolute agreement.
Total E.H.P. = 9700 + 19500 = 29200.
Actual propulsive coefficient of Luntania was '50. Take,
when estimating, a lower coefficient, say '46, in order to allow
a reasonable margin.
Henee estimated S.H.P. for new ship =^ 29200 -^ -46 = 63500.
Effeot of Shallow Water.
The following table gives the percentage increase (or
deCfeaie for fijgures in itSalios) of E.H.P. in shoal water as
compared with deep water at the ^me speed. It applies
to a destroyer form* where w/ (tqa) = ^0 ; l>ut it would serve
as a rt>i^h guide in anj otiiet ^pe of ship.
Percentage variation of Horse-power,
——■ Depth of wfttM -i- Length of ship.
•06 -1 -2 -a '4 'S -6 •? *%
207 — lO'O 9^1 60 5-3 1-9 1-9. -7 '5
I'M — il'4 94 4'S I'S '9 '7 -6 '4
1-80 — 106 7-0 — 1'8 la -5 '2 —
1-66 — 7-2 1'2 7-3 5-6 1-0 — — _
1'62 — 6 16 20 7-5 1'2 — _ ~
1-38 — 22 60 22 6-5.— — — —
1-25 — CT76 13 8 — — — —
1-11 — 160 82 2 — — — — —
•97— 2«)" 4 — — — — — —
•88 810 78 — — — — — — —
•6»400 80 — — — — — — —
•55 60 — — — — — —__
For speeds up to about 1*5 V^li thdre is a definite maximum
H.P. at a depth equal to V^ll feet (v is in knots). At Very high
speeds shoal water is usually favourable to the resistance.
• Goal Ensu]U2K}e or Badius of Action.
Distance steamed or radius of action in nautical miles
20jt.x.sp§ed i;i knots x bu^akey capacity jp tons
" toi^ns of c(Jal used ^r 24 hours.
This formula contains a marg^m for coal untrimmed, aad
for the eSect ef variation of draught.
Tons of coal used per 24 hours = horse-power at light
24
draught x lb. of coal per H.P. per hour (p. 390) x ^nrQ
190 SCREW PROPELLERS.
Economical Speed.
The economical speed of a ship is that in wKicli the radius
of action is a maximum.
Fig. lis. .
METHOD OF FINDING
ECONOMICAL SPEED.
20 25KN011
' To find, from the origin 0 of the speed— horse-power cnrre
(fig. 148) draw a -tangent touching the curve at P. Then the
corresponding 'speed OM is the economical speed. In large
warships the economical speed is usually about 10 br 12 knots.
Horse-power of Warships at Low Speeds.
For large warships, I.H.P. or S.H.P. at 10 knots = 200 -f-
8w* approximately, for small alterations of speedy vary the
coefficient of w> proportionately to the cube of the speed.
PSOPELLEBS.
Design of Screw Propellers.
(Mr. R. E. Proude, Trans. I.N. A., 1908.)
Let D = diameter of screw in feet.
p = pitch of screw in feet (see note below).
1> =p/d = pitch ratio.
H = thrust horse-power (see note below) per screw.
B = revolutions of screw, in hundreds per minute.
y 3= speed of ship in knots.
yi= speed of screw through the water in knots.
where w is the wakeef actor (see p. 163).
s = real slip ratio =
^ RP
B »a coefficient depending on the blade area (see table below).
A = area of blade In square feet, including boss.
= actual area excluding boss X 1*25 approximately.
mi. H p AAaoiAo S(1~'08S)
Then -s — j • . . ^-v = •0032162 — jz — .»
D^Vi* b(» + 21) (1-S)*
Notes. — 1. The pitch p is determined from the advance of
the screw when contributing no thrust. It may be taken to
be, on the average^ about 102 times the pitch of the driving
surface.
SCREW PROPELLEKS.
191
2. The thrust horse-power is Tv^ (p. 164). It is equal to
tho effective horse-power divided by the hull efficiency ; the
augmentation necessary to include the effect of appendages is
not usually included fos the purpose of propeller design.
3. The speed Vj = v/(l -{■ w) ; it can be estimated roughly
from the wake particulars given on p. 163.
4. The right-hand side of the equation 0032162 ^-"jy
( = y) is given in the following table : — ^ '
B
V
8
V
8
V
B
•88
V
•02
•000067
•14
•000602
•26
•001495
•003086
•04
•000139
•16
•000720
•28
•001698
•40
•003457
•06
•000217
•18
•000849
•30
•001922
•42
•003880
•08
•000302
•20
•000989
•82
•002169
•44
•004354
•10
•000394
•22
•001142
•34
•002442
•46
•004887
•12
-000494
•24
•001311
•36
•
•002746
•48
•005490
5. The coefficient b depends on the type of blade and on
the disc area ratio ; the latter is equal to the fraction a/— d*.
4
Diso Area Ratio.
SO
•0978
•104fi
•35
•iO
•45
•50
•55
•60
•65
•70
•76
•80
8 blades. elUpiical.
•1030
•1097
•1106
•1060
•1126
•1070
•1085
•1100
•1182
•1112
•1195
•1124
•1185
•U«
•1157
3 blades, wide tip.
•1148
•1166
•
•1207
•1218
'1230
•1943
4 blades, elliptical.
•1040
•1159
•1197
•1227
•1249
•1268
•1282
•1294
•1806
•ISIfl
6. Curves of propeller efficiency are shown in fig. 149.
They are correct for a 3-blade elliptical propeller, with a
disc area ratio of '45. For a 3-bladed wide tip, the efficiency
should be reduced by *02 ; for a 4-bladed elliptical by '0125.
In addition, a correction for any diso area ratio other than
*45 should be made ; this is very small for all ratios less
than '55 ; for higher ratios, reduce the efficiency by tho
foUowio^ amounts : —
Disc area ratio .
Deduction for pitch ratio .
»»
»>
it
it
•55
•65
•70
•8
•02
•04
•08
1^0
•01
•03
•05
1-2
•005
•015
•025
1-4
—
•005
•01
192
SCEEW PROPBLLERS.
Limitations of Size of 8or0U>.
Frequently a large variety of screw dimensions can be
adopted with very little variation in efficiency. In selecting
the actual dimensions^ the following considerations are ol
us*:—
1. Fouling diameter, — There should be not less than 12
inehes clearance in large vessels between screw tip and hull.
In small ships this allowance may be slightly less. Thl?
determines the greatest diameter permissible. By sloping baek
the blades a propeller of larger diameter can be fitted. With
4 screws the projections of the discs on an athwartship plane
should^ when practicable, clear one another.
Fig. 149.
•«20 '25 -^0 -as 40 -41
SUP RATIO.
CURVE OF SCREW PROPELLER EFFICIENCY.
% Oavitatton. — ^This depends on several factors, inclading
shape of Uade, velocity of rubbing, and depth of immeradbn.
As a rough guide, the ratio -.. .. : 57- should not exceed
** ** Blade aifeft m sf . ft.
'75 in ordinary vessels, or *d ia- high-speed vessels, such as
destroyers. The blade area here exchtdet the boss, or is equal
W '8 A.
The thrust T is found as described above. Negleotii^g the^
appendages T = H/68 v^. Hence, area ('8 a) should be at
least H/d'l V| in ordinary, and h/7'5 Vi in very fast vessels.
These are outside limits, and it is preferable to adopt largeir
areas, if practicable.
Dimensions of Screw*
1. Estimate the E.H.P., or that portion of it given out by
the particular screw considered^ and (taking data from i^imiiftr
SCREW PROPELLERS.
198
ships) the Iinll efficiency and wake percentage. Thence dednce
H and 7i.
2. Seleot Tariooa Talnes of diso area ratio (say, *S, '9,
'7, and *8). Note the corresponding values of B.
8. For several idip ratios * (s), read off y from the table,
and calcolate the pitch P from the equation.
p = 1-01vi/r(1-s).
Thence determine the pitch ratio p, using the modified
form of the equation above, viz. :—
FlO. 150.
IMUM AREA
CURVES OF SCREW PROPELLER DIMENSIONS.
Noie.-^ -\- 21 varies slowly with p, and it can be taken
as 22 or thereabouts without g^eat error.
4. Determine from p the diameter d ; and from the
effieiency curve the screw eficiency.
5. Plot on a base of blade area ('8 a) for various constant
disc area ratios (a) the pitch, (5) the diameter, and (0) the
efficiency ; getting a series of curves as shown on fig. 150.
6. Draw on the diagram straight lines representing the
fouling diameter and the minimum area to avoid cavitation.
* The dimensions ara usually rather sensitive to changes of slip. The
slips should therefore he chosen within narrow limits so 8,3 to give
practicahle pitch ratios.
O
194 SCREW PUOPELtERS.
^ 7. Subjeot io these limitations any dimenaiotia complying
with the curves can be given to the screws. If practical
considerations permit, the dimensions selected should Be those
corresponding to maximum efficiency.
ExaanpU. — ^Determine spot on screw dimension carves corre-
sponding to disc area ratio '6 and slip ratio '28 for a vessel of
20,000 E.H.P., four screws, each developing the same power,
three - bladed, elliptical, speed 25 knots, hull efficiency 1*02,
wake 14 per cent, revolutions per minute 275.
Ti = v/(l + t(>)==25/l-U = 21-9. R=^2-75.
20000
H = 43^37^2 = ^^^' ^^^ **^^«s ^ =" • m2, y = • 001698.
Then P = 1-01Vi/r(1 - s) = 1-01 X 21-9/2.75 X -72 = ll-lS'.
p* __ P^i^y _ (lia3)^x (21»9)»x '1112X '001698 ._„
^T2r ir~~ 4900 ®^^^-
Putting i) + 21 = 22. jp == 1.033. D = 11.13/1088 = 10-67'.
Blade aiea = -8 A = -Sx .6 x -785 x (10-67)' » 43*8. Actual
pitch = p/1.02 = 10-9'. Efficiency « -695 from curves - -02
(correction for disc area ratio) = '675. Minimum blade area »
H/5.IV1 = 43-9 sq. ft.
l^ote, — In the paper by Mr. Froude, from which these
particulars are taken, data are given from which carves may
be constructed which enable the dimensions to be obtained
with reduced arithmetical labour.
Thickness of Bladb.
i = thickness of blade at root in inches.
h ss breadth of blade at root in inches.
'<^ = diameter of hub in feet,
ft = number of blades.
Other symbols as above (see page 190).
Remember that H is only about one-half the I.H.P. per sorew.
In vessels running continuously at full speed in all
weather, t should be 10 or 20 per cent greater. In cast-iron
propellers increase t by about 50 per cent.
Thus, in example above if 5 — 40", d » 2''2,
^ 3x4900x8-47 ^,
^-3x40x2.75xl0.9'''''~^*®^-
The tliicknoBs would taper gradually from t at the root to
a very small minimum value at the tip.
SCREW PROPELLERS.
195
VaofVhsvfB Daia
FOB Tugs.
Ca8i-iron Tug Propellers. 1
Speed
laj.
parmin.
Blade
area in
8a.it.
N6.0f
Blades.
Pitch.
Diameter.
110
455
140
26
8
10' to 11'
7' 8^
10-2
416
112
27
4
11' to 12'
8'6»
90
240
143
17-4
4
8'
6'6'
9>0
190
128
20-8
3
8'
7'0^
11.76
827*
162
28
4
14'
8' or'
10-2
337
142
17-5
3
8' 6"
7'Or'
80
106
206
6-25
3
6' 8"
4' 8"
16*9
865
118
%
8
9' 9^'
7' 9^
10*2
860
120
25
8
9'^^
S'Cf'
130
1400»
118
32
3
12' 6^'
9' 6"
4
2
1
820
440
250
3-2
3-3
31
99
103
108
3*3
4-6
6-8
840
345
350
Average pull on tow rope at low speed «= 1 ton per
100 I.H.B.
Toiving trials on Thames with swim barges and screw tug of
130 tons displacement.
Number of barges
Displacement of barges in tons
Tow rope pull in tons . . .
Bevolutions per minute . . .
Speed in knots
I.H.P. .
Note. — ^The barges were towed close up to the stern, and
two abxeaat except in third trial.
Towing trial on Bhins with four barges^
• Total displaeement of barges 3,500 tons ; towed from 250'
to 950^ behind tug and' w^l staggered so as to be clear ojf
wakttk Speed .aboat.6'8 knots relative to stream. I.H.P. 970.
Tow rope pull about 8^ tons.
Conclusions derived from towing trials of ' Fulton ' and
' Froude '.
(Professor Peabody, Amer. I.N.A. and Mar. Eng.)
The diso area ratio should not be too gres^t, e.g. about '5.
A small pitch ratio ('8) is best for pulling, and a large
pitch ratio (1*5) is favourable when running free. Actually
a moderately high pitch ratio is generally used.
Tugs can be powered from the Admiralty coefficient. That
for a small model was about 22 wheny/VL was '65.
As great a length of tow-line as practicable should be
used. If L represents the length of the tug, increasing the
length of line from 2l to 6}L saved 10 per cent of the power ;
* Twin screws. Total power given.
196
PADDLES
but the power when towing abreast was 10 or 12 per cent more
than when towing with a length 3l.
Paddles.
Area of FloaU (D. W. Taylor).
A — Combined area of two floats (one on eaeh side) in square feet.
D "ii mean diameter to centres of floats in feet.
I = indicated horse-power.
B = revolutions in hundreds per minute.
V = speed of ship in knots.
8 s slip ratio » (Vp — .v)/Vp (where Yj, is the peripheral speed
in knots of centres of floats) =» (vrd — Y)/iriu>.
A = (212.6~375s)^
8 should not greatly exceed '15 for feathering floats and
'20 for fixed floato ; a large slip leads to a low e£3cienoy of
'propulsion.
Number, Hze, and position of paddles.
When fixed to the wheel, the floats are spaced about 3 feet
apart, or in fast ships slightly less. The spacing of feathering
paddles should be 4 to 6 feet. Excessive spacing is liable to
cause vibration.
The width of float is about half the beam of the vessel
for smooth water, or one-third in seagoing steamers. The
depth is determined from the area. Thickness of wood float
«^width ; of steel float in iiiches = '15 -f (*16 X width in
feet). .
Paddles should be placed longitudinally so that they run on
or near the crest oi the wave, which can .be determined
approximately from experience in similar ships. They must
also be near amidships, so as not to be effected by changes of
trim, except in stern-wheelers.
The upper edge of the lowest float should be immersed at
mean draught about 18 to 20 inches in large sea-going vessels,
and about 12 to 15 inches in smaller vessels ; about ft toi
6 inches is sufficient in river steamerSc
—)^:^
BPSSD TBIALS. 197
Construction Jor mechanism actuating feathering paddles.
Let WL (fiff. 151) be ihe water-line and o the centre of
paddle. It is desirable that the paddles should enter and leave
the water without shock ; and that in the lowest position they
should be vertical. Let B, o^ D be centres of paddles in
these positions, D being vertically below o. Make kd = od
X slip ratio. Join KB, EO ; and miUce the psiddles at b and
0 perpendicular to these lines. Draw on the positions of the
st^-lever ends B% d^ o'. Find a the centre of a circle
passing through D^ q\ d'. Then A is the centre of the
eccentriCj and all the radius rods must be of length ad'.
SPEED TBIALS.
ICeasured Mils.
To determine the true ^ean speed of a vessel when the runs
are taken on the measured mile, alternately with and against
the tide, with approximatelyt equal intervals of time between
each run.
Rule. — ^Multiply the apparent speed in each run by the
factor A given in the table below ; divide the sum of thie
products by the number b ; the quotient is the required speed.
Note. — ^This process gives the same result as that obtained
by the ordinary 'mean of means' method.
Ar.<«M7.««. /»/ Mf^i* Faeiora A which multiji^ the »..».,»..« i»
Number of runs. epeeds in ordar. Number B.
3 12 1 4
4 18 3 1 8
6 1 5 10 10 5 1 32
EoMimple, — The speeds dedaced iroiA. the times over the mile
are 15-4, 101, 14-8, 11-0, 13-2, 11*8 knots. Determine the
mean speed.
Mean s d « (^5'44-ll'8)+5(10-l'H3'2)4-10(14'3+ll'0)
32
= an = 12-397 knots approximately.
Note, — ^The revolutions and I.H.P. or S.H.P. observed
during the several runs should be meaned by the same method
as the speed.
Speed of the Cuebent.
To find the speeds of the current in the lifie of the ship^s
course during her speed trials.
Rule. — ^Find the differences between the real speed of the
ship, as above determinedi and her observed speeds on the mile
during the several runs.
198
8PSBO TKtAlM,
JSxample,
Buna
ODaerred
Speed
Real Speed
DiffOMIlMB
1st
2nd
3rd
4th
5th
6th
15-4
10-1
14-3
110
13-2
11-8
12-397
12-397
12-397
12-397
12-397
12-367
3003
2-297
1-903
1-397
•803
-597
Knots with the ship
,f against „
„ with „
„ against ,,
„ with „
,, against „
Sea Trials.
To determine the true mean ipced of a vetsel when the dittance
run is great.
Rule 1st.— Calculate the apparent speed of each ran as
usual, by dividing the distance by the time, and group them in
sets of three ; for example, 1, 2, 3 ; 2, 3, 4 ; 3, 4, 6 ; &c.
2nd.— ^Eaeh set of three is to be treated as follows : — ^Find
the two intervals of time between the middle instants of thefir^t
and second, and of the second and third rang of the set; reduce
those intervals to the corresponding angular intervals by the
following proportion :—
As 12^ 24*" (t^e duration of a tide) : is to a given interval of
time : : so is 360' : to the corresponding angular interval.
3rd. — Multiply the Jir$6 apparent speed by the co^secant of
the firgt angular interval, the teeond apparent speed by the sum
of the co-tangents of the two angular intervals, the third
apparent speed by the co^secant of the seoo-nd angular interval.
4th. — Add together the products and divide their sum by the
sum of the before-mentioned multipliers ; the quotient wiU be a
speed from whidh tidal effects have been eliminated.
5th. — Add together the velocities deduced from the sets of
three runs, and divide by their number for a final meaji.
Note, — When an interval elapses of more than a quarter of
a tide, or S** 6"*, between the middle instants of the two runs of
a sel, certain multipliers and products must be subtracted.
The following example will determine whether these oertain
multipliers are to be taken as positive or negative.
Time.
Between O*"
and 3''
Between 3''
and &"
Between 6*»
and O**
Between S"*
and 12''
0™ \
6- /
6- "l
12" /
12'» \
18* J
18™ \
24™ /
Exarnple,
Angles.
r Between 0°
I and 90"^
Between 90®
and 180**
Co-9ecant9. Co tiingmts.
I Positive Positive.
KNOTS TO MILtS iND MILES TO KNOTS.
1(
Tablb of
COMPABISON Of ADlitBALTT KNOTS AND STATUTB MILBS*
Knots Miles
1-1515
1-4394
1-7273
2-0152
2-3080
2-5909
2-8788
3- 1667
3-4545
3-7421
4-0303
4-3182
4-6061
4-8939
5-1318
5-4697
5-7576
6-0455
6-3333
6212
?
Miles Knots
•8684
1-0855
1-3026
1-5197
1-7368
1-9539
21711
2-3882
2-6053
2-8224
30395
3-2566
3-4737
3-6908
3-9079
4-1250
4-3421
4-5592
4-7763
4-9934
Knots
600
6-25
6-50
6'76
7-00
7^25
7-50
7-75
8-00
825
8-50
8-75
9-25
Miles
19-0000 21-50
19-2879|21-75
0606il2-00ll3-8182|1700|l9-5758 22-00
17-25 19-8636 22-25
6-9091
7-1970
7-4848
7-7727
8
8-3485
8-6364
8-9242
9-2121
9-5000
9-7879
10-0768
9-00110-8636
10-6516
9-60 10-9394
11-2273
11-5152
11-8030 15
9-75
10-00
10-26
10*501120909
10-7512-3788
Miles Knots
600
625
6-50
6-75
700
7-25
7-50
7-75
800
8-25
8-50
8-75
900
9-25
9-50
9-75
1000
10-25
10-50
10-75
5-2106
5-4276
5-6447
6-8618
60789
6-2961
6-5132
6-7303
6-9474
7-1645
7-3816
7-6987
7-8158
80329
8-2500
8-4671
8-6842
8-9013
9-1184
9-3355
Knots
U-OO
11-25
11-50
11-75
12-25
12-50
12-75
1300
13-25
Milos
12-6667 16-00
12-9545|l6-25
16-50
5303116*75
13-2424
13
14-1061
14-3939
14-6818
14-9697
15-2576
13-60 16-5465
13-75
14-00
14-25
14-50
14-76
15-00
25
16-50
15-8333
16-1212
16-4091
16-6970
16-9848
17-2727
17-5606 20
17-8486 20-50;23-606 1126-60
15-7518-1864
MUes Knots
U-00
11-25
11-50
11-76
1200
12-25
12-50
12-75
13-00
13-25
13-50
13-75
1400
14-25
14-50
14-76
1500
15-25
15-50
15-75
9-5526
9-7697
9-9868
10-2039
10-4211
10-6382
10-8553
11-0724
11-2895
11-5006
11-7237
11-9408
12-1579
12-3750I19
12-5921
12-8092
13-0263
13-2434
13-4605
13
Knots
Miles h Knots
18*424221-00
18-7 12l!2 1-25
17-6020-1616
17-75 20-4394 22*76
18-00;20-7273 23-00
18-25,21-0152 23-25
18-5O;21-5O30 2360
18-75:21-5909 2375
19-0021-8788 24-00
19-26 22-1667 24-25
19-5022*4546
19-75,22-7424
2000,23-0303
2523-3182
20-76:28-8939 26*75
Miles Knots
1600
16-25
16-60
16-75
17-00
17-25
17-50
14-7632
14-9803
151974
17-7515-4145
1800 15-6316
18'2515-8487
18-5016-0658
18-7516-2829
19-0016-5000
2516-7171
19-5016-9342
19-7517-1513
200017-3684
20-25117-5855
20-50ll7-8026l25-50
22-50
24-60
24-76
25-00
25-26
Miles
24"l8i
24-461
24-751
25046
25-33)
26-62t
25-90$
'26-197
26-484
126-772
27-060
;27-348
27-636
27-924;
28-212
28-500<
28-7871
290751
29-363<
29-661i
Mfles
Knots
13-8947 21-0018-236^
14-1118 21-25!l8-453I
14-3289 21-5018*671]
14-5461 21-75 18*888i
21-25
21-50
21-75
22-00
22-25
22-50
22-75
23-00
23-25'
23-50]
23-75'
24-00'
24-25
24-50
24-75
25-00
25-25
6776 20-7518-0197
25-76
19-105i
19-322:
19-539.'
19*756(
19-973:
20190^
20-4071
20625(
20-8421
21-0591
21-276.-
21-493
21-710i
21-927(
22144'
22-361;
N.B. The Admiralty lt&ot= 6,080 ft. ; 1 statute mile = 5,280 ft.
00 KILOMBTEES TO KNOTS AND KNOTS TO KILOMETRES.
Table of II^ilometres to Adkiraltt IlNOTs and Admi-
BALTY IlNOXS TO KlLOUETRES.
Kilos.
Knots
Kilos.
80
Knots
Kilos.
150
Knots
Kilos.
220
Knots
Kilos.
Knots
1-0
•540
4-317
8094
11-872
29-0
15-649
1-25
•675
8-25
4-452
15-25
8-229
22-25
12-006
39-25
15-784
1-5
•809
8-5
4-587
15-5
8-364
2^-5
12-141
29-5
15-019
1-75
•944
8-75
4-722
15-75
8-499
22-75
12-276
29-76
16-054
20
1079
90
4-867
160
8-634
230
12-411
30-0
16-188
2-25
1-214
9-25
4-991
16-25
8-769
23-25
12-646
30-26
16-323
2-5
1 -349
9-5
5-126
16-5
8-904
23-5
12-681
30-6
1<(*458
2-76
1-484
9-75
5-261
16-75
9039
23-75
12-816
30-75
16-693
3-0
1-619
100
5-396
17-0
9173
24-0
12-951
310
16-728
3-25
1-754
10-25
5-531
17-25
9-308
24-25
13086
31-25
16-863
3-5
1-889
10-5
5-666
17-5
9-443
24-6
13-221
31-5
16-998
3-75
2024
10-75
5-801
17-75
9-578
24-75
13356
31-76
17-133
4-0
2158
110
5-936
18-0
9-713
250
13-490
320
17-268
4-25
2-293
11-25
6071
18-25
9-848
25-25
13-625
32-25
17-403
4-5
2-428
11-5
6-206
18-5
9-983
25-5
13-760
32-5
17-638
4-75
2-563
11-75
6-340
18-75
10-118
25-75
13-896
32-75
17-672
50
2-698
12-0
6-475
190
10-253
260
14030
330
17-807
5-25
2-833
12-25
6-610
19-25
10-388
26-25
14165
33-26
17-942
5-6
2-968
12-5
6-745
19-5
10-623
26-5
14-300
33-5
18-077
5-75
3-103
12-75
6-880
19-75
10-657
26-75
14-435
33-75
18-212
BO
3-238
130
7-015
200
10-792
27-0
14-570
34-U
18-347
5-25
3-373
13-25
7-150
20-25
10-927
27-25
14-705
34-25
18-482
5-5
3-508
13-5
7-285
20-5
11-062
27-5
14-839
34-5
18-617
5-75
3-642
13-75
7-420
20-75
11-197
27-75
14-974
34-75
18-752
r-o
3-777
140
7-555
21-0
11-332
280
15-109
36-00
18-887
r-25
3-912
14-25
7-690
21-25
11-467
28-25
15-244
35-25
19021
r-5
4-047
14-5
7-824
21-5
11-602
28-5
15-379
35-5
19-156
r-75
4-182
14-75
7-959
21-75
11-737
28-75
15-514
35-75
19-291
InotB
Kilos.
Knots
4-75
Kilos.
Knots
8-5
Kilos.
Knots
I Kilos.
Knotf;
16-0
KUos.
LO
1-853
8-803
15-752
12-25
22-701
29-651
1-25
2-316
50
9-266
8-75 16-215
12-5
23165
16-25
30-114
L'5
2-780
5-25
9-729
9-0 16-679
12-75; 23-628
16-5
30-577
1-75
3243
5-5
10192
9-26i 17-142
13-0
24-091
16-75
31041
20
3-706
5-75
10-656
9-5
17-605
13-25! 24-554
170
31-504
2-25
4170
60
11-119
9-75
18068
13-5 25018
17-26
31-967
2-5
4-633
6-25
11-582
10-0
18-5.32
13-75 25-481
17-6
32-430
2-76
5-096
6-5
12046
10-25 18-995
14 0
25-944
17-75
32-894
30
5-560
6-75 ; 12-509
10-5 19-458
14-25! 26-408
18-0
33-357
3-25
6-023
70 ,12-972
10-75 19-922
14-5
26-871
18-25
33-820
3-5
6-486
7-25 13-435
U-O ! 20 385
14-75
27-334
18-5
34-284
3-75
6-949
7-5 il3-899
11-25 20-848
150
27-798
18-75
34-747
i'O
7-413
7-75 14-362
11-5 21-311
15-25
28-261
19-0
36-210
4-25
7-876
80 ;14-825
11-75 21-775
15-5
28-724
19-25
35-673
4-5 18-339
8-25 15-289
12-0 122-238
15-75
29-187
19-5
36137
2
Effect ok Speed aitd Fowsb of Iitobbasb of Resistan
If H.P. «a xyn (aee p. 169), and k is increased moderat
through fool bottom, or increaised draught, or any other sou
of a&itional resistance, the speed v is decreased and 1
power H increased by the percentage in the following tal
which should be multiplied by the percentage increase in
The rate of revolution of the propeller is assumed constant.
Real Slip
Ratio
•90
•35
•80
•36
•40
-°/oV
•17
•20
•2.1
•20
•29
n = 3
+ °/oH
•50
•40
•31
•22
•14
-°/oV
•14
•17
•19
•21
•22
n = 4
+ °/oH
•48
•33
•95
•18
•11
-°/oV
•12
14
16
17
18
n = 5
+ °/oI
•38
•29
•21
•15
•09
Principal Measured Distances off the British Isles
Approzima
depth of
water at loi
Place.
Measured
distance
True
in feet.
oourae.
water sprin
tides in
fathoms.
EcLst Coast of England.
Tyne Biver ....
6080
341<>
11 to 12
Colne Biver ....
,f # . . .
! 3036
3048
333°
846°
1 7 to 13
.^-^'-fa^"^:
6080
6080
25°
67°
4i to 41
6to8
South Coast of England.
Stokes Bay ....
6080
295°
10 to 13
Southampton Water
•Portland (Chesil Beach) .
5888
126°
2to2i
8678
314°
17
*Polperro ....
6990
266°
17 to 20
Plymouth (Outer) .
6080
273°
11 to 15
,, (Inner) .
4562
275°
5to6i
Falmouth ....
6989
347°
a to 5
West Coast of England.
Barrow
6080
829°
3i to 5
West Coast of Scotland,
1
Skelmorlie . • • .
6080
0°
36 to 42
Gare Loch ....
6080
336°
15 to 22
East Coast of Scotland.
St. Abbs Head
6084
291°
24 to 28
TayBiyer ....
6080
266°
4i to 6i
Ireland.
Belfast Lough
6080
265°
5Ho6
* In order to nae the following speed tables for the Portland ooone, first dedi
10 percent. (WM aeenrately) from the times: for the Polperro course add U per cv
(14*97 McnraMr) to the speedn dedaced from the tmVloB.
202
SPEED TABLES.
Speed Table for Mbasubed Coubse of 6,080 Feet.
•
1 Minute.
0
90
40*000
36
37-895
40
36-000
45
34 286
60
32-727
55
31-304
•1
39*956
•855
35-964
•253
•698
•277
•2
•911
•815
-928
-221
•668
•250
•3
•867
•775
•892
-188
•638
•223
•4
•823'
-736
•857
•166
'609
•196
•5
•779-
•696
•821
•123
•579
•169
•6
•73cr
•667
•785
•091
•660
•142
•7
•691-
•618
•750
•069
•620
•116
•8
•648
•678
•714
•026
•491
•066
•9
d
31
L -604
36
•539
37-506
41
•679
"46"
33-994
-462
66
•061
39-560
SS-G'W
33-962
51
32432
31-034
•1
•617
•461
•608
•930
•40?>
•OOB
•2
*474
•422
•573
-898
-374
30-981
•8
•430
-383
•533
•866
•345
•954
•4
•387
•344
•503
-836
•316
•928
•5
•344
-306
•468
-803
-287
•901
•(>
•301
•267
•433
-771
•258
•875
•7
•258
•229
•398
-739
-229
•848
•8
•216
•190
•363
-708
-200
•8-ffi
•9
0'
•173
87
•152
37113
42
•329
47
•676
•172
57
•796
32
S9130
35-294
33-645
52 321431
90-76^
•1
•088
•075
•260
-613
-114
•743
•2
•046
•037'
-225
•682
•086
•717
•3
•003
36-999
•191
•661
'
-057
•691
•4
38-961
•96L
•166
•620
-028
•664
•5
•919
•923
•122
•488
•000
•638
•6
•877
•885
•088
•457
31-972
%
•612
•7
•835
•847
•053
•426
•943
'586
•8
•793
•810
•019
•395
•915
•660
•9
0
33"
•761
38
•772
36-735
43
34-985
48'
•3^
•887
•634
38-710
34-951
33-333
53
58
30-508
•1
•668
•697
•918
•302
•830
-483
•2
•627
•660
•884
•272
•8f2
•457
•3
•585
•623
•850
•241
•774
•431
•4
•544
•585
•816
-210
-746
•406
•5
•503
•648
•783
•180
-718
•380
•6
•462
•511
•749
•149
-69D
•354
•7
•421
•474
.716
•119
•662
•329
•8
•380
.437
•682
•088
•634
•303
•9
0
34
•339
-400
36-364
44
•649
49
•058
54
•607
59
•278
38-298
39
34-615
33-028
31-679<
30-252
•1
•267
-327
•582
32-997
-651
•227
•2
•217
•290
•549
•967
•624
•201
•a
•176
•254
•51'5
•937
•496
•176
•4
•136
-217
•483
•907
•469
•151
•5
•095
•181
•460
•877
•441
•126
•6
•056
•145
•417
•847
•414
•100
•7
•016
-108
•384
•817
•386
•076
*8
37-976
•072
•351
-787
•389
•060
•9
•935
-036
•318
•757
66
•332
•o»
0
»
37-895
40
36-000
46
34-286
60
32-727
31-304r
60
80-000
SPEED TABLES.
203
Speed T^ble fob Measubed Coubse of 6,080 Fbct.
i 1 2 Xiautes.
0
0
90'00a
10
27^692
20
26-714
30I24OOO
40
22 600
60 21 176 1
•2
29-950
•650
•678
28-968
-472
•152
•4
•900
•607
•641
•93B
•444
•12
•6
'860
•666
•606
•904
•416
.10:
•8
•801
•523
•668
31
•873
23-841
41
•388
If
•07
0
1
29-7S2
U
27-481
21
25 532
22-360
2106;
•2
•703
•439
-496
•810
•333
•021
*4
•664
•397
•460
•778.
•30^
'00
^
'606
•366
'424
•74T
•277.
20-$7il
•8
•ra&
12
•314
22
•388
-W
•716.
42
•250
•965
0
2
29'506(
27-273
25-352
28-684
22 222
^
2093d
•2
•460
•232
•316
•663
•196
•906
•4
'412
•191-
•281.
•622
•167
•88-3
•6
•361
•150
•245
•691
•140
.857
•8
•316
•109
•210
26175
•660
•113
2t086
53
•833
20-809
0
8
29268
13
27068
23
33
23-529
43
•2
•221
•027
•140
•499
•059
•785
•4
•173
26-987
-105.
•468
•032
•761
'6
•126
•946
•070
•438
•005
•737
8
4
•079
14
•906
•035
34
•407
21-978
61
•713
a
29032
26-886
21
25-000
23-377
44
2i-951
20-690
'9.
26*986
•826
24-965
•346
•924
•666
■[
•939
•786
•930
•316
•898
•642
■■"y
•892
•746
•896
•286
•871
•619
•'*
5
•84e
•706
•862.
35
•256
46
•84b
•695
'J
28-800
15
26-667
26
24^828
23-226
21-818
55
20571
ft
4tf
•754
•627
•794
•196
•792
•648
«
•708
•662
•688
•549
•760
•726
•166
•13&
•766
•739-
.626
•601
.
6"
•617
28-571
16
•510
26
•692
24^658
36
•107
46
•713
lie
.478
26-471
23077
21-687
20-455
i
•626
•432'
•624
•047
•661
•431
•4
•481
•393
•590
•018
•635
•408
•6
•436
•354
•557
22-989
•609-
•as5
•8
7
•391
•316
•623.
37
•959
•583
57
•362
0
28-346
17
26-277
27
24-490
22-530
47
21-557
20-339
•2
•302
•239
•457
•901
-531
.316
■4
•257
•201.
•423
•872
•506
•293
•6
•213
-163
•390
•848
•480
•270
•8
'169
•125
26087
28
•357
38
•814
•454'
•247
0
8
28125
18
24-324
22-785
48
21-421)
58
20.225
•2
•081
•049
•291
•756
•403
.202
•4
•037
•012
•269
•727
•378
.179
•6
27-994
25-974
•226
•699
•352
•157
i-B
•960
19
•937
29
•194
24161
39
-670
22-642
49
•327
~59
•134
20112
0
9
27-907,
26-899
21-302
•2
•864
•862
-129
-613
•277
-089
•4
•821
•825
•096
-586
•251
•067
•6
•778
•788
•061.
•556
-226
•045
•8
•736
•751
•032
•528
•201
•023
0
.10
27-692;
20
26-714
30
24-000
40
225J0.
60
21-176
60
20003
8FBED TABLES.
Speed Table foe Measuked Coubse
or
6,080
Peet.
3
3 KluiiMi.
A.
0
Q
20000
10
18.911
20
IBOOO
30
17-113
40
16 864
19.9IB
■927
n-983
■127
U^.
;*
•D66
908
•364
.946
•091
•334
'S
'8CS
16,818
IT
^
ST
"^
IT
-304
IfllSO
-2
1
;868
■828
-809
■8JS
^867
■013
1
"r
■803
15710
TT
la-^TBO
83
■889
If^
w
1C^981
■D66
7F
-B31
IOI6
.202
i
.fl»t
'693
■7G9
■761
■936
■173
■158
TT
■fl
■8
1
T
■esa
■608
■6OT
■833
15
:676
11
33
"1
.683
w
■8H
■838
"is
16 -U3
■100
31
17-647
.080
ie'822
■807
M
IBO.T
SI
■fl
■602
:499
■6''H
■8
■z
T
i9T60
I
'35-
^627
"35-
16744
.713
■«
"1
TS
.124
■8
■37a
.40B
!49a
■682
-943
1'6:939
I
T
19356
TT
^
26
ITTTff
.469
Sfl
'*-6Gl
46
■fl
•S
T
■m
TT
Mi
■293
37
■4^5
■408
iFsJi
87
.636
C20
lell
1
if
■845
57
i?i
!368
1
T
Aw
Tb
319
.2C0
15182"
w
■311
38
16 61 1
l8
■803
I6l59
"BF
■I
li;l
rraSB
-a
■1
■6
TT
-f
"■m
■108
-027
19
■163
.109
is'oao
■073
■39-
■291
■253
irm
■193
■^
-484
-76B
»
t
"■e
'"1
49
■70T
■8
IS'SBT
•086
■176
■169
■K13
'666
-0
'5"
TB-
19:947
15-
WOOO
"M
1I"11?
■40
IZ
60-
1F6M
_
Bfl
SPEED TABLES.
206
Speed Table for Measured Course of 6,080 Feet.
4 KinuteB.
0
1
2
3
4
6
6
7
8
T"
10
15*000
14-988
•976
•963
•96')
10
11
.12
13
14
15
14-400
•388
•377
•366
•364
20
il
18-846
•886
•826
•814
•804
30
133'i3
•328
•314
•804
•294
40
12-867
•848
•88)
•830
•821
60
12-414
•406
•897
•388
•380
14-988
•925
•913
•901
'888
14*343
•331
•320
•308
•2^7
18-75 3
*78b
•772
•761
•761
dl
13-284
•2T4
•266
•266
•245
41
12^811
•802
-793
-784
•776
(Si
12-3^
•3'*8
-364
•816
•387
14-876
•864
•861
•8S9
•827
14-236
•274
•253
•252
•2*1
22
23
13^740
•783
.720
.709
•699
32
13-285
•22>
•216
•206
•196
13-187
•177
•167
•163
•148
13133
•129
•120
•110
•ICO
4i&
12-766
•767
•748
•739
•783
62
12329
•320
•312
•88
•295
14*816
•803
•790
•778
•766
14-22)
•218
•207
•196
•184
13-6^8
•678
•667
•66T
•647
33
43
44
12721
•712
•703
•694
•686
63
64
12*287
•278
•270
•262
•253
14*764
•742
•780
•718
•706
14173
•162
•151
•140
•129
24
13-686
•6«6
•616
•606
•606
34
36
12*676
•667
•663
•649
•640
12*6?2
•623
.614
-606
•696
12-246
•237
•52^
•221
•212
14*6^4
•682
•670
•668
•646
14118
•107
•096
•066
•074
25
26
i7
-676
•664
•561
•644
13091
•081
•072
•062
•0«3
45
66
12-908
•196
-187
•179
•170
r
t
14*634
•622
•610
•699
•687
16
14*063
-002
•041
•030
•019
13-581
•624
•614
•503
•493
36
13043
•031
•025
•016
*0O3
46
47
48
49
12587
•579
-570
•561
-682
66
67
68
69
60
12*162
•161
•146
•133
•129
I
!
[
1
14-576
•663
•681
•640
•62S
17
14-003
13997
•986
•976
•964
13*488
•473
•463
•463
•443
37
12^996
•987
•978
•968
*969
12-544
•536
•626
•517
•609
i2-5C0
•491
•483
•474
•465
12^457"
•448
•440
•431
•428
12121
•113
•106
•097
•089
)
I
I
•
14-616
•601
•493
•481
•4<«>
18
13963
•943
•982
-921
•910
28
13-48?
•423
•413
•403-
•803
38
12-950
•940
•931
•922
•912
12081
•072
•oei
•066
•048
12040
•03?
•024
•016
•003
f
I
[
\
\
14-4Kr
•446
•485
•423
•41-?
19
139 0
•889
•878
•837
•867
29
133^3
•378
•8S8
•858
•8^3
39
12-903
•894
•885
•876
•863
U'isto
20
13-846 1
30
1
138^
40
12857
60
12*414
12000
206
SP££D TABLES.
Spfed Table for Measueed Couese of 6,080 Feet.
S 1 5 MinuteB.
1
^1
io
0
12-000
10
11-613
20
11-260
30
10-909
40
106g8
60
10*286
•2
11"992
•606
.
•243
'902
•682
•284,
H.
-984
•698
-283
'896
:
•676
-274
•ft
k
•976
•690
-2^^
•88J
^
•570
•268
•8
-968
"u
f
-222
31
•883
-568
•
•262
=0
1
11960
11-676
tl
11-216
10-876
41
10691
51
10-26d
♦2
'963
•668
-2oa
•870
•561
•261
•4
-944
•661
^
•201
•863
•615
-915
•6
.
'983
•563
.
194
•863
\
•639
•28»
•8
•928
1
12
-616
22
•181
■•
•860
•682
52
•2c3
0
2
11921
11688
11180
32
10-843
.42
10*626
10*k2
•2
•913
•631
•178
•837
•620
•22.
•4
906
•624
-166
•880
-614
•216
•6
•897
-616
'169
-
•82t
•
■608
•210
•8
'
•889
•609
fe3
•162
•817
.
•603
53
*20l
0
3
11-881
13
11602
11116
to
W811
■^3
10-496
1019-5
•2
-873
•494
•1»)
•804
'
•490
•193
•4
■866
•487
•132
•798
1
•483
•187
•6
*8«8
•480
•125
•791
•477
•181
•8
•86J
•472
11-466
ti
•U8
-786
.
•471
•
•175
0
4
11-842
14
11111
84
10778
44
10^466
m
l016.
•2
•881
•468
•104
•772
•
•45J
•164
•4
-827
•460
I
•097
•766
•463
•
•16S
•6
•819
•443
•091
•769
»
•44T
■
•152
•8
-8U
•433
•081
35
•763
•
•441
.
•147
0
5
11803
15
11429
^i,
110^7
10-746
45
10-436
;66
lOlil
•2
•796
•421
'070
•740
,
•439
•135
•4
•788
•414
•063
•733.
•423
•129
•ft
•780
•407
•067
•727
•417
•134
•8
•772
•400
•050
•721
46
•411
'56
•118
■0
6
Il7d5
16
11392
26
11043
te
W714
10-405
10112
•2
•757
•886
•036
•708
•899
•107
•4
•749
•378
•029
•702
•393
•101
•6
»
•742
•371
•023
•6»5
•387
•096
•8
•731
•384
27
•016
ft7
•689
-381
67
•090
0
7
11-726
17
11-356
1100.)
W683
47
10-376
100!*4
•2
•719
•819
•C02
•676
•369
•078
•4
•711
•3t2
W996
•
•670
•363
•
•073
•6
•704
-
-335
•939
•664
•367
-
•067
•8
0
8
•696
18
'328
28
•982
33
•667
•351
68
•061
11-688
11321
10-976
10-631
48
10-316
lOOoo
•2
•681
•814
•969
•645
•333
•050
•4
•673
•307
-962
•635
•333
•(H5
•6
•666
•299
•956
-632
•3*37
•03)
•8
0
9
•663
19
•29«
29
•949
•626
•321
•033
11-66J
11286
10-942
39
10-619
49
10^315
69
1002i<
•2
•643
•278
•933
•613
•839
022
•4
•636
•271
•929
•607
•338
•017
•6
-628
•264
•922
601
•297
•Oil
•8
•620
•267
•916
•6»4
-2D2
•003
d
10
11-613
26
11-250
30
10-903
40
10-658
60
10-286
60
10000
SPEED TAKLKS.
207
Speed Tabi,e foe M{:asubf.t> Coubse of 6,080 Feet.
OB
6 IKinuteB.
0
lo-o:»
10
9-730
20
9*474
90
9*231
40
9-000
GO
8*789
1
9-972
U
9*704
21
9-449
31
9*207
41
8-978
51
8*759
2
9*945
12
9-677
22
9-424
32
9*184
42
8-955
52
8*738
3
9*917
13
9-651
23
9-3SI9
33
9-160
.43
8-933
63
8717
4
9-890
14
9-626
24
9*316
84
9137
44
8-911
54
8*696
6
im
16
9-600
25
9*351
35
9*114
45
8-889
56
8*675
6
16
9*574
26
9-336
86
9*091
46
8-867
66
8*6J4
7
9-809
17
9*649
27
9-302
37
9*068
47
8-846
67
8*633
8
9-788
9*766
JB
9-524
28
9-278
38
9*045
48
8-824
58
8*612
9
19
9*4§9
?9
9*264
39
9*023
49
8-802
59
8-692
7 Xinates. |
0
8-6Z1
10
8-372
20
8*182
80
8*000
40
7-826
60
7*660
1
8-661
11
8-353
21
8-163
31
7-982
41
7-809
51
7*643
2
8-631
12
8-333
22
8-146
32
7-965
42
7-792
62
7-627
3
8*611
13
8-314
23
8*126
33
7-947
43
7*775
63
7-611
4
8*491
14
8-296
24
8*108
84.
7-930
44
7-769
64
7-596
6
8*471
16
8*276
25
8*090
35
7*912
45
7-742
65
7-679
6
8*461
JO
8*257
26
8*072
36
7-895
46
7-725
66
7*563
7
8-431
17
6*238
27
8*054
37
7-877
47
7*709
67
7-647
8
8*411
18
8*219
28
8-036
38
7-860
48
7*692
68
7*631
9
8*892
19
8*200
29
8-018
39
7-843
49
7-676
69
■
7-616
SXinuteB. |
0
7*500
10
7-347
20
7-200
30
7*059
40
6-923
60
6-792
1
7*484
11
7-332
21
7*186
31
7*045
41
6-910
51
6*780
2
7*469
12
7-317
22
7171
32
7*031
42
6-897
52
6*767
3
7*463
13
7-302
23
7167
33
7*018
43
6-883
53
6*764
4
7-438
14
7*287
24
7*143
34
7-004
44
6*870
54
6*742
6
7*423
16
7-273
26
7-129
35
6*990
45
6*857
65
6*729
6
7*407
16
7-268
26
7*115
36
6-977
46
6*844
66
6*716
7
7*392
17
7*243
27
7*101
37
6*963
47
6*831
57
6*704
8
7-877
18
7*223
28
7*087
38
6*950
48
6*818
68
6-691
9
7*862
19
7-214
29
7073
39
6*936
49
6*805
59
6679
9 Minutei. 1
0
6*667
10
6-645
20
6*429
30
6*316
40
6*207
to
6*102
1
6-664
11
6-534
21
6-417
31
6-305
41
6*196
5i
6-091
2
6-642
12
6*522
22
6-406
32
6-294
42
6*186
62
6*081
3
6-630
13
6-510
23
6-394
33
6*283
43
6*175
53
6*071
4
6-618
14
6-498
24
6*383
34
6*272
44
6164
64
6*061
6
6-606
16
6*486
25
6-372
35
6-261
46
6*154
55
6*050
6
6-693
16
6*475
26
6-360
36
6-260
46
6*143
56
6*040
7
6'68X
17
6*463
27
6*349
37
6-239
47
6*133
67
6*030
8
6-660
18
6*452
23
6-388
88
6*223
48
6*122
58
6*020
9
6-657
19
6*440
20
6-327
39
6-218
49
6*112
69
6*010
208 SAILIKO.
SAILING.
Centre of Latebal Resistance.
The centre of lateral resigtance is the centre of application of
resistance of the water ; and as this varies in position with the
speed of the ship, &c., it is not determinate, bat a point is
generally taken at the centre of the immersed longitudinal
vertical middle plane of the vessel as sufficiently accurate.
Centbe of Effobt.
The point in the longitudinal vertical middle plane of a vessel
which is traversed by the resultant of the pressure of the wind
on the sails is termed the centre of effort ; its position varies
according to the quantity of sail spread, &c., but its position is
determined approximately for purposes connected with design-
ing the sails, all plain sail only being taken — that is, the sails
that are more commonly used, and which can be carried with
safety in a fresh breeze (see table, p. 210). They are as
follows :—
In square-rigged vessels : the fore and main courses, fore,
main, and mizen topsails, fore, main, and mizen topgallant
sails, driver, jib, and sometimes the fore topmast staysail.
In fore and aft rigged vessels : the main sail, fore sail, and
sometimes the second or third jib.
In calculating the position of the centre of effort the salU are
taken braced right fore and aft.
The centre of gravity of the whole sail area is calculated bv
the ordinary rules for the C.G. of a geometrical area (p. 59
and after).
Ardency.
Ardency is the tendency a ship has to fly up to the wind,
thus showing that the position of her oentre of effort * is
abaft the oentre of lateral resistance.
Slackness.
^ Slackness is the tendency a tbiy has to fall off from the
wind, thus showing that the position of her oentre of effort
is before the oentre of lateral resistance.
Belatite Position of Centre of Effort and Cbntbb of
Lateral Resistance.
The calculated centre of effort lies usually between *01 l
and *03 l before the oentre of lateral resistance, L being tho
length of ship. With a large fine deadwood aft, this diatanoe
* Thii nfers to her real, not her calculated, centre. The latter may be
slightly before the centre or lateral resistance.
SAiLiNa. 209
should l>e slightly diminished. In many sailing boata the two
centres are coincident lateraUy. If the oentre of effort be too
far forward, the vessel becomes ' slack ** and will not readily
go about ; if too far aft, the tevel may be too ' udent \
PowEB TO" Causy Sail.
The vertical distance between the centres of effort and
lateral resistance (ss h feet) is arranged in accordance with
the following formula :— *
w = displacement of vessel in tons.
OM = metacentric height in feet.
. A = sail area (plain sail only) in sqivi>re feet.
^ ^ ., 2240W.GM
Power to carry sail « — r —
The power to carry sail is about 3 to 3'5 in sailing boats,
about 3 for yachts, and about 15 in auxiliary ships such as
doops*
Nate* — ^The power to carry sail is approximately the re-
ciprocal of the angle in radians to which the vessel will heel
under a wind pressure of 1 lb. par square foot* of sail
(cozrespondii^ to a breeze of about 16 knots on the beam).
Real and Appabbnt Motion op thb Wind.
By the real motion of the wind is meant its motion relatively
to the earth, and by its apparent motion its motion relatively
to the ship when she is sailing.
The apparent motion being the resultant of the real motion
of the wind and of a motion equaland directly opposite to that
of the ship.
Fig. 161a. ^ ^' ^^^^ ^®*' ^^ represent
■^ ■ in magnitude and direction the
^ 4 ^?^N ^^^^ motion of the wind, and
Vll ^ \v^ \^S<r ' -^^ *^ direction and velocity
\7^'**v^^^ o^ t"^® motion of the ship;
\ ^^<s^ through B draw BD parallel
**^ J^B and equal to AC ; join DA: then
DA will represent in magnitude and direction the apparent
motion of the. wind.
Sail Ajusa.
The sail area in regard to size of ship may be determined
approximately from die driving power, which is about the
stone in similar vessels.
A » 8£dl area in square feet.
w = dlsplacemeiit in tons.
Then a/wI «= driving power (see p. 184 for table of wl).
This is approximately 1^ in sailing boats, 200 in yachts,
mJkd 80 in sloops where sails are carried as auxiliary to steam.
p
210
Specification of the Beaufobt Scale with fbobabl:
{From Report of the Advisory Committee fo
I
n
S
5
6
7
8
9
10
11
12
Admiral
Beaufort's
general
description
of wind.
Calm . . .
liight air .
Slight .
breeze \
}
Gentle
bieese
Moderate
breeze
Fresh
breeze
Strong
breeze
Moderate
gale {high, i
wind)\
Fresh gale
Strong gale
Whole gale .
Stoxxn • • •
Hnrricane .
AAnlxal Baaofort'a
specification, 1806.
Calm.
Just sufficient to give'*
steerage way.
That in which
a well • Qon-
ditioned man-
of-war with ,
allsailsetand'^
'clean full'
would go in
smooth water
from
/I to 3
knots.
8 to4
knots.
6 to 6'
knots.
\
/Royals, etc.
Single-reefed top-'
sails or top-
gallant sails.
Double-reefed
tope«ilStJib.etc.
Triple-reefed
topsails, etc.
Close-reefed top-
sails and
courses.
That which she could'\
scarcely bear with I
close-reefed main top- 1
sail and reefedforesail V
That which would re- 1
duce her to storm |
stay-sails. J
That which no canvas
could withstand.
Description
of wind.
Light
breese
Moderate
breeze
Strong
wind
Gale
forcei
Storm
forces
Hurricane
Mode of eatimatiav
aboard sailiac
Sufficient wind foi
working diip.
Forces most adyan-
tageoQs for sailinf
with leading wind
and all sail drawing
Reduction of sail
necessary with
leading wind.
Considerable redi
tion of sail nc
sary even
wind qi
dose-reefed sail
ning, or bore
under storm-
No sail can stand <
when running.
* The fishing smack in this column may be taken as representing a trawl
allowance must be made.
-I- It has recently been decided that for statistical purposes winds of force
of the term moderate gale' for force 7 the Beaufort description has bl
')f the descriptions in italics for forces 7 and 8.
J
21
EQUIYAIiENTS OF THE NUMBEBS OF THE SCALE.
Aeronautics, 1909-10, By Dr. W, N, Shaw, FM,S.)
Speeiflcation of Beaufort Scale.
Nil
** 8
For coast aae liaaed on
obHervatioos made at
Scilly, Tarmonth, and
Holyhead.
For nse on land, tNued on
observations made at
land stations.
Mil
Cakn.
Calm, smoke rises
vertically.
0
0
0
Fishlnfir smack * just
Direction of wind shown by
•01
2
2
has steerage way*
smoke drift, but not by
wind vanes.
1
Wind fills the sails
Wind felt on face; leaves
•06
6
4
of smacks, which
rustle; ordinary vane
then move at about
moved by wind.
1-2 miles per hour.
Smacks begin to
Leaves and small twigs in
•28
10
9
careen and travel
constant motion; wind
about 8-4 miles
extends light iSag.
per hour.
Good working
Raises dust and loose paper ;
•67
15
13
breeze ; smacks
small branches are moved.
carry all canvas
with good list.
Smacks shorten sail.
Small trees in leaf begin to
sway ; wavelets form on
inland waters.
1-81
21
13
Smacks have double
Large branches in motion ;
2-8
27
23
reef in n^insail.
whistling heard in tele-
Care reanired
graph wires; umbrellas
when fishing.
used with difficulty.
Smacks remain in
Whole trees in motion;
8^6
85
30
harbour, and those
inconvenience felt when
at sea lie to.
walking against wind.
All smacks make for
Breaks twigs of trees;
5-4
42
87
harbour if near.
generally impedes pro-
gress.
Slight structural damage
ooours (chimney-pots and
slates removed).
7-7
50
44
V ^^^^
Seldom experienced inland ;
trees uprooted ; con-
siderable structural dam-
age occurs.
10-5
59
51
Very rarely experienced:
accompanied by wide-
spread damage.
140
68
59
—
—
Above
Above
Above
170
75
65
verage type and trim. For larger or smaller boats and for si>ecial circumstance
lan 8 shall not be counted as gales, and to avoid the ambiguity implied by the us
todilled for use in connection with the daily weather service by the substitutio
JZ
DISTANCES DOWN THE THAMES.
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CONSTANTS FOE ESTIMATING WEIGHTS.
WSI0HT8 AND BIXSN8I0KS OF XATESIALS.
Tons X 2340slhe. Tons x 20 = cwts. Lbs. x -000446428= tona,
Weight qf JRoiind or Elliptical Bars.
Diameter x diameter x length in feet x constants weight in lbs.
Weight of Square or Rectangular Ban,
Width X thickness X length in feetx constant = weight in lbs.
WeigJd of Plating or Planking.
Thickness x breadth in leet x length in feet x constants weight In lbs.
Values op Constants for Round ob Elliptical Babs.
Material
Diameters taken in |
Ins.
iln. 1 ^In.
|In.
iVIn.
A In.
Brass, sheet
Iron, wrought .
Lead, sheet
Steel, soft .
Elm, American .
Mahogany, Honduras
„ Spanish *
Oak, Dantsdc
„ English
Pine, red ,
„ yellow
Teak, Indian
2^905980
2^61800
3-88778
2-67036
•261800
•196350
•287980
•261800
•307615
•196360
•157080
•287980
•726495
•654500
•971933
•667590
-065450
•049088
•071995
•065450
•076904
•049088
•039270
•071995
•181624
•163635
•242983
•166898
•016363
•012272
•017999
•016363
•019228
•012272
•009818
•017999
•045406
•040906
•060746
•041724
•004091
•003068
•004500
•004091
•00480/
•003068
•002454
•004^00
•011351
•010227 .
•015186
•010431
•001023
•000767
•001125
•001028
•001202
•000767
■000614
•001125
-00383$
•003557
•0O3797
•0Q2608
•000366
•000192
•00QS81
'000856
•OOO900
•000192
•0OO153
•000381
Values of Constants foe Square or Rectangular Bars.
Material
Width and Thickness tnken in |
Ins.
iln.
iln.
Jin.
•057813
A In.
^In.
Brass, sheet
3-70000
•925000
•281250
•014453
-003613
Iron, wrought .
333333
•833333
•208333
•052083
•018021
-003255
Lead, sheet
4-95000
1-23750
•309375
•077344
•019336
•004834
Steel, soft .
3-40000
•850000
•212500
•053136
-013281
•OO8320
Elm, American .
•333333
•083333
-030833
-005308
-001302
-0003-26
Mahogany, Honduras
•260000
•062500
•016636
•003906
•000977
-000244
„ Spanish .
•366667
•091667
•023917
•005739
•001483
•000358
Oak, Dantzic .
•833833
•083883
•030833
•005208 •001203
-000836
„ EngUsh .
•891667
•097917
•024479
•006120 •001680
•000S8S
Pine, red .
•350000
•062600
•015625
•003906 ^000977
•000244
„ yellow
•300000
•060000
•012600
•003126 '000781
•000195
Teak, Indian .
•866667
•091667
•023917
•006729 •001433
•000858
Values of Constants for Plating or Planking.
Material
Thickness taken in |
Ins.
iln.
iln.
*In.
tVIn. 1 AIn-
A In.
Brass, sheet
44^4
23-3
11-100
5-550
2^7750 1^38750
•68S75
Iron, wrought .
40^0 .
30-0
10-000
5-000
2^5000 1-2500O
•62500
Lead, sheet . .
59-4
39-7
14-80
7-425
3-7136 1-85635
•93813
Steel, soft .
40-8
30^4
10-30
5*100
3-5500 1-37500
•63750
Elm, American .
4-00
3-00
1-000
•5000
-35000 -13600
•63600
Mahogany, Honduras
8'00
1-50
•760
•3750
•18750 •09875
•04688
,, Spanish .
Oak, Dantzic •
4-40
3-30
1^100
•5500
•37600 •18750
-06875
4-00
3-00
1^000
•5000
•35000 ^135000
•06350
„ English .
4-70
3-85
1*175
•5876
•29375 •14688
•07844
Pine, red . •
300
1-50
•760
•8750
•18750 •09376
•04688
„ yellow
3-40
1-30
•600
•3000
•15000 ^07500
•0S760I
Twk, Indian .
440
3-30
1100
•6600 ^27500 •18760 1
•0687S|
CONSTANTS FOB ESTTMATINQ WEtOHTB.
228
WBiaHT OP Pipes.
Waswelgfat per lineal foot in lbs.
KsooDfltant from below.
W=(Di-*)K.
Dss outside diameter In ins.
<fs inside M
>f
B2-90e0.
C!oppersS*»94$.
Values of K for Pipes,
Iron, cast =2*4383.
„ wroughts2-6180.
Weight of Ai^ole Iron.
Lead»8-8877.
Steels 2HS704.
Wairriglit in lbs. per liDeal foot. 8=gwn of tbe widths of flanges in in&
T=: thickness of flanges in ins.
W=T (8-T) 8-33338.
Relative Weights of Different Substances.
Wxooght ironsl.
Brass, sheet:
Copper „ i
Iron, cast
Lead, sheet :
Steel, soft !
Tin-
Zinc
t-llOO.
: 11488.
: -9975.
a-48fiO.
:10S00.
: -ftftOO.
: -9494.
Beech =-0896.
Elm s'lOOO.
Fir.spmoe s'OSSS.
Mahogany, Hondara8s*0750.
„ Spanish =fiiOO.
Maple ='1021.
Oak, Dantsic ='1000.
Oak, English :
Pine, zed
„ yellow I
Sycamore
Teak, African:
„ Indian :
WiUow 1
:-07fiO.
'•0600.
:-0806.
:ai4ft.
:1S7T.
Weight, &;c., op Fresh Water.
A cubic foot =-0379 ton =:62*89 lbs. =998*18 ard. ozs.=6*3331 galls.
A cubic inchs-OSei lb. =*5776 ard. oz. =-0836 gall.
A gallon =*0046ton Bl(H)001bs. =100-16 avd. ozs.= -16044 on. (t
A ton =85-900 en. ft.=2240 lbs. =838-76 galls.
Weight of fresh water= weight of salt water x -9740.
Weight, &c., of Salt Water.
A cubic foot = -f '286 ton = 6406 lbs. = 1024*80 ard. ozf. «6*2831 galls.
A cubic inch=-0371 lb. = *6&6Q avd. os. = •0086 gall.
A gallon =-0046 ton =10*276 lbs. =164*41 ard. ozs. = 16044. on. fi
A ton =34*973 en. ft.=2240 lbs. =217-96 gaUs.
Note,— A. cubic foot of salt wato: is usually taken at 86 en. ft. to tbe ton
and 64 lbs. to the cubic foot, fresh water being taken at 86 cu. ft. to the ton and
62*26 lbs. to the cubic foot.
^llSCELLANEOUS FACTORS.
A ton
tonnean.
An ard. lb.
Afoot
Asq. foot
Asq. indi
metres*
Acu. ft.
A cubic yard
A mile
Knot per hour
»» n
second.
Mile per hour
A gallon
=: 1*01606 tonne or
=•45869 kilogram.
=•304797 metre.
=-092901 sq. metre.
=646-148 sq. milli-
=;'0283I6cu. metre.
='764534 cu. metre.
=1*60938 kilometre.
=1-688 foot per second.
=*6144 metre per
:1*467 foot per second.
r4-64102 litres.
A tonne or tonneau:
A kilogram
A metre
A sq. metre
A sq. millimetre
A cubic metre
»
A kilometre
Foot per second
hour.
Metre p3r second
hour.
Foot per second
hour.
A litre
: -984206 ton.
: 2-20462 lbs.
: 3 2808693 feet.
: 10-7641 sq. feet,
r 00166003 sq. in.
:85*8166 cu. feet.
: 1*80799 cu. yd.
: 621877 mUe.
:*592 knot per.
B 1-944 knot per
= •683 mile per
s '320216 gallon.
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^
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oor-i^h-o»«»oaoSssac5
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WEIGHT OF AMGLB AMD T StEEL.
OQ
Flanges
(ins.)
^oTSf© S^^^,-4;;2?c??f«
Snm of 1
Flanges!
1
i
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1
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ill
280 WEIGHT OF SHEET METALS PEB SQUARE FOOT.
H
O
o
OQ
525
W
o
M
w
H
OQ
O
M
>
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to 00 CO I-l bo
tp to b- op t^ CO
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cq o<i oq (N 9) <ft
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53
-4 s
^•i
f^ 9. 3 S'S 2
WEIGHT OF BOUND AND SQUARE STEEL.
21
TaBLB Of THI WXtCfttT OF ROUKD AlTD SqUABB BaB SuSBI
or Lbs. nm La:NBAL Foot.
i4
4
iff
Weight in Lbs.
Bound
042
094
167
261
376
511
667
845
043
262
502
762
2*044
2-347
1
1
1
1
2-670
3-380
4172
5-049
6008
7051
8178
9-388
10^681
12058
13-519
16062
16-690
18*400
20-196
22*072
24-033
26078
28-S06
30-417
32*712
Bonnd
Square
•053
•120
•213
•332
•478
-651
-860
1-076
1-328
1*607
1-913
2-245
2*603
2*988
3-400
4-303
6-313
6-428
7-650
8-978
10-413
11*953
13-600
15-353
17*213
19*178
21*250
23*428
28*713
28-103
30-600
33-203
35-913
33-728
41-650
Square
^5
5
t
6
t
I
la
Weight in Lbs.
Round
35-090
37-552
40097
42-726
45-438
48-233
61*112
64076
57-121
60-250
63-463
66*759
70139
73-602
77-148
80-778
84-492
88-288
92-169
96133
100-18
104-31
108-52
112-82
117-20
121-67
126*22
130-85
135-56
140-36
145-24
150-21
155-26
160-39
Weight In Lbs. I^.H
Square
44-678
47-813
61063
54-400
67-853
61-413
65078
68-850
72-728
76-713
80-803
85-000
89-303
93-713
98-229
102-85
107-58
112-41
117-36
122-40
127-55
132-81
13818
|l43-65
149-23
154-91
160-70
166-60
172*60
178*71
184*93
191-25
197-68
204*21
BoiJtid
Siq[aare
n
8
9
i
10
i
\
11
12
'il
Weight in Lbs. 1^3
Weight in Lbs.
Bound
166*60
17090
176*29
181*76
187-30
192-93
198-66
204-45
210-33
216-30
222-35
228-48
234-70
241-00
248-38
253-85
260-40
267-04
273-75
280-56
287*44
294-41
301-46
308-59
315-81
32311
330-60
337*97
345*62
363-15
360-87
368-68
376-56
384*63
Square
21085
217-60
226-26
231^41
238^48
245-66
252 93
260-31
267*80
276-40
283-10
290-91
298-83
306-85
314*98
323*21
331*56
340-00
348-65
357-21
365-98
374-85
383-83
392-91
40210
411-40
420-80
430-31
439-93
449-65
459-48
469-41
479-45
489-60
Bound Square
Weight in Lbs.
^32
WEIGHT OF MALLEABLE IRON PIPES.
Tabib of xhs WEieHi of Maixbabub Iron Pipes in
Lsa. PBR LiNBAL Foot.
Bore
(ins.)
i
i
h
3
i
I
r
i
I
8
i
9
10
i
11
12
I
Bore
(ins.)
Thickness in Inches
8-27
8*93
4-58
6-24
5-89
6-56
7-20
7-85
8-51
9-16
9-82
10-47
11-13
tU-78
12-43
18-09
13-74
14-40
15-05.
16-71
16-36
17-00
17-67
18-38
18-98
19-63
20-28
2094
21-60
22-25
22-91
23-56
24-21
24-87
25-52
26-18
26-83
27-48
28-15
28-80
29-45
80-75
82-07
}
5-40
6-38
7-36
8-34
9-33
10-31
11-29
12-27
13-25
14-23
15-22
16-20
17-18
1816
19-14
20-12
2M1
22-09
23-08
24-05
25-03
26-01
27-00
27-98
28-96
29-93
30-92
31-90
32-89
33-87
34-85
85-83
36-81
37-79
38-78
89-75
40-74
41-72
42-71
43-69
44-66
46-62
48-60
7-85
9-16
10-47
11-78
13-09
14-40
16-71
16-02
18-32
19-63
20-94
22-25
23-56
24-87
26-18
27-49
28-80
30-11
31-41
32-72
34-03
35-34
36-65
87-96
39-26
40-57
41-88
43-19
44-51
45-81
47-12
48-43
49-73
61-05
52-35
53-66
54-98
56-28
57-60
58-90
60-20
62-82
65-45
10-63
12-27
18-91
16-64
17-18
18-82
20-45
22-09
23-72
25-36
27-00
28-63
30-27
31-90
33-54
35-18
36-82
38-45
40-08
41-72
43-36
44-99
46-63
48-26
49-90
61-63
58-17
54-81
56-45
58-08
69-72
61-35
62-99
64-62
66-26
67-90
69-64
71-17
72-81
74-44
76-07
79-35
82-63
f i
H U
16-71
17-67
19-63
21-60
23-66
25-62
27-49
29-45
31-41
33-38
36-84
87-30
89-27
41-23
43-20
46-16
47-12i
49-08
61-05^
53-01
54-97
56-93
68-90:
60-86J
62-82
64-79
66-75
68-72
70-68
72-64'
74-61
76-66
78-58
80-50
82-46
84-43
86-38
88-35
90-31
92-27
96-20
100-18
21-76 —
24-06 28-80
26-34' 31-41
28-63! 84-03
30-92 $6-66
83-211 89^7
35-60, 41-88
37-79 45-50
40-08 47-12
42-37 49-74
44-67 62-36
46-96 64-98
49-25 67-69
61-54 60-21
53-83 62-83
56-12 66^6
58-41 68-06
60-70 70-68
62-99 73-29
65-27 75-91
67-57 78-53
69-86 81*16
72-15 83-77
74-44 86-38
76-73 89-00
79-02 91-62
81-32 94-24
83-60] 96-86
85-90 99-47
88-18102-29
90-47104-71
92-77,107-88
95-06 109-96
97-36112-66
99-64115-18
101-92117-79
104-22 120-42
106-61 123-04
108-80 125-66
113-88 130-88
117-15 136-18
36-81
39-76
42-70
46-65i
48-69
61-54
64-48
67-43
60-88
68-82
66-26
69-21
72-16
76-10
78-04
80-98
83-93
86-87
89-82
92-77
96-71
98-65
101-60
104-24
,107-50
illO-43
118-38
116-33
119-27
122-22
126-16
128-10
130-05
133*99
136-95
139-89
142-88
147-96
154-61
Bore
(ins.)
ft
1*
45-81
49-08
62-35
55-63
58-90
62-17
65-45
68*72
71*99
76*26
78*64
81-8
85-08
88-34
91-62
94-89
98*16
10W4
104*71
107-98
111-25
114-62
117-80
121-07
124-84
127-62
180-89
184-16
187^8
140-70
143-97
147-25
150-521
153-80
15707
160-88
166-88
178*43
U
ThicknefB in Inches
10
11
i
12
Bore
(Ini.)
WEIGHT or CAST-IRON PIPES.
288
Table op the WstGHT of Oast-iron Pifis ni Lbs.
PER Lineal Foot.
Bore
(ins.)
i
8
k
i
I
i
h
\
I
5
\
I
■♦
I
8
10
11
12
i
I
i
I
I
\
I
Thickness in Inches
Bon
Urn.)
i
8*06
8-69
4-29
4-91
6-63
6*14
6-74
7-36
7-98
8*69
9-20
9-76
10-44
IMO
11-66
12-27
12-88
13-50
14-11
14-73
15-34
15-96
16-57
17-18
17-79
18-41
19-03
19-64
20-02
20-86
21-69
22-09
22-71
23-81
23-93
24-55
26-16
26-77
26-38
27-00
27-62
28-84
80-06
5-06
6-98
6-90
7-83
8-76
9*66
10-68
11-60
12-43
18-34
14-21
16-19
16-11
17-08
17-94
18-87
19-78
20-71
21-63
22-56
23-47
24-39
25-31
26-23
27-15
28-08
2900
29*98
30-83
31-74
32-90
83-59
84-52
35-48
36-86
37-28
38-20
89-11
4004
40-96
41-88
43-71
45-66
7-86
869
9-82
1105
12-27
18-50
14-72
16-95
17-18
18-35
19-64
20-86
22-10
23-37
24-54
25-77
26-99
28-23
29-45
80-68
81-91
33-18
34-36
85-59
36-82
38-06
39-05
40-50
41-71
42-95
44-40
45-40
46-64
47-86
4909
60-82
51-64
52-77
54-00
56-22
56-46
58-90
61-86
9-97
11-61
18-04
14-67
16-11
17-64
1917
20-70
2219
23-78
26-31
26-86
28-38
29-97
81-44
32-98
34-51
36-05
37-68
39-12
40-65
42-18
43-72
46-26
46-79
48-10
49-86
51-38
62-92
54-45
56-21
57-62
5907
60*69
62-13
63-66
66*20
66-73
68-26
69-80
71-33
74-39
77-46
14*73
16*56
18-41
20-25
2209
23-92
25-71
27-62
29-46
31-30
33-18
34-98
36-87
38-65
40-50
42-33
44-18
46-02
47-86
49-70
51-64
58-39
55-23
56-84
58-91
60-74
62-59
64-42
66-26
68-88
69*95
71-80
78-63
76-47
77-32
79-16
80-99
82-84
84-67
86-52
90-19
93-60
20-4
22*66
24-7
26-81
28-93
31*14
83-29
85-44
37-58
39-73
41-88
44-08
46-17
48-32
50-46
52-62
64-76
56-91
69-06
61*21
63-86
65-28
67-65
69-79
71-96
7409
76-28
78-88
80-76
82-68
84-84
86-97
89*13
91-28
93-42
95-67
97-71
99*86
1020
106*3
110-6
I
2700
29-46
31-86
34-86
36-81
89*28
41*72
44-18
46-63
4909
61*60
53-99
66-45
58-90
61-86
68-81
66-27
68-73
71-18
78*41
76-09
78-63
81-00
83-46
85-90
88*35
90-81
93*49
95*72
98-18
100-6
103-1
106-6
108*0
110*4
112-9
1 .6-4
117-8
122-7
127-6
H
84-46
37-28
4003
42-80
45-66
48-82
51-08
53-84
56-61
69-42
62-12
64-89
67-64
70-41
7317
75-94
78-70
81-23
84-22
86-97
89-74
92-50
95-26
98-02
100-8
103-6
106-6
109*1
111*8
114-6
117-4
120*1
122*9
125-6
128-4
131-2
133-9
189-4
145-0
U
r.
H
42*96
4602
49*08
62-16
56'«2
58-29
61-36
64*43
67-65
70-56
73-63
76-69
79-77
82-84
86-91
88-75
92-04
9510
98*18
101*2
104-3
107-4
110-5
118-5
116-6
119-9
122-7
125-8
128-9
181*9
135*0
138-1
141*1
144-2
147-3
150*3
156*4
162-6
U
Thickness in Inches
Vr
Bon
(Ins.)
Bort
(insO
234
WEIGHT OF LEAD PIPE AND COPPER EODS.
Tablb of thb Weeght op Lead Pips in Lbs. peb Lineai,
Foot, and Lengths in which it is usually Manttfactttbed.
•^7
Us
12
t
go
I
2
2i
Weight per Foot in Lbs.
•933 1-07 1-2
1-2 1-47 1-67
1-47 1-60 1-73
1-87 2-4 2-8
3 00 3-17 3-60
3-50 400 4-67
5-837007-33
700 8-00 9-33
10-5 —
1-47
1-80
1-87
300
4-33
5*08
8-00
1-73 1-87
213 2-4
3-60 3-93
5-08
6-00
5-25
700
2-33
3-00
4-20
t
0) ^>
Wght. per Ft. in Lbs.
12
10
8
H
4
4^
5
6
9-0
130
9-6
11-6
13-6
13-5
200
23-4
330
10-5 120 —
12-013-4:!l5(
15-0ll6-6;i8-4
16-0
21-6
25-4
18-4 2OOI
23-4t —
28-0 —
• Also in 60-feot coils.
t Also ill 36-feet coils.
Table of the Weight of Round Copper Rod in lbs. per
Lineal Foot.
Diam.
Weight
•1892
•2956
•4256
•5794
•7567
•9578
M824
14307
Diam.
(ins.)
Weight
lA
1
?
1^7027
1^9982
2-3176
2-6605
3-0270
3-4170
3-8312
4-2688
Weight
4-7298
5-2140
5-7228
6-8109
7-9931
9-2702
10-6420
12-10821
Diam.
(ins.)
Weight
13-6677
15-3251
17-0750
18-9161
20-8662
22-8913
25-0188
27-2435
Diam.
(ins.)
Weight
29-5694
31-9723
34-4815
37-0808
39-7774
42-5680
45-4550
48-4330
Table op the Weight of Oast-tron Balls.
Weight
(lbs.)
12-55
13-62
14-76
15^95
17-21
18-54
19-93
21-38
22-91
24-51
26-18
27-92
29-74
31-64
S8'«2
Diam. Weight
(ins.) (lbs.)
35-68
37-81
40-04
42-36
44-75
47-23
49-80
52-47
55-23
58-09
6004
64-09
67-24
70-50
77-32
Diam.
(ins.)
Weight
Obs.)
H
9
- *
10
lOK
lOj
10}
11
111
12
84-57
92-25
100-39
108-99
11806
127-63
137-70
148-29
159-40
17106
183-28
196-06
209-42
223-38
t287-94
1
WfitGHT OP OOPPES PIPE, BTC.
285
SHttlNKAGE Of CASTlKGfi.
The muAl allowanee for eaeh foot in length U
In large' cylinders. *»^inch. Inline
InsmaU „ . «A »»
lA beams and girders ^ ^ „
In thick brass • c»^
In thin „ . *=^
»
In lead
In tin
In copper .
In bismuth
In cast-iron pipes Ȥ- inch.
follows : —
Tabes of tub WfiroHT of Ooppbb Pipb ik Lbs. pbr
Lineal Foot.
li
35
5
3?
8
?
39
i
4
BM« OfPIpft Iki TfAsnes
1
4
•11
•24
•89
•87
•77
•99
1-24
1-51
*
•13
-28
•46
•66
•89
•14
•41
•70
•16
•33
•58
•76
1-01
1-28
1-57
189
3
•20
•43
•67
•95
1-24
156
1-90
2-27
•25
•52
•82
114
1^48
184
2-23
2-66
9
•30
•61
•96
1-32
1-71
213
■^57
303
i
•34
•71
110
1-61
1-96
241
2-90
3-41
•39
•80
1-24
1-70
219
2-70
3'23
3-78
i*ta
■
1
To
3
Iff
1.
4
Bore of Pipe in lAclidft
H
•90
1-89
2^98
4-16
U
•99
208
326
4-54
If
1^09
2-27
3-55
4^91
H
1^18
2-46
3-83
5-30
If
1*28
2-65
4-12
5-67
1}
1-37
2-84
4-40
6-06
n
1^47
803
4'68
6-43
2
^1^
1-56
8-22
4-97
6-81
Bore of Pipe in Incbes
H
166
341
6-26
M9
Urn
2i
2i
2|
2|
2i
2J
1-75
1-84
1-94
204
2^13
222
359
3-78
398
416
4-35
4-54
5-63
5-82
6-10
639
6*67
695
7-67
7-94
8-33
8-70
9^08
9-46
Bore of Pipe in Inches
n
3
2-41
4-92
7-52
10*22
H
2^51
5-11
7*81
10-60
3|
2-60
6-30
8-09
10*97
•^1.
2-70
6-49
8-^7
1185
H
2-79
6-68
8-^6
11^73
H
2-89
5-87
8-94
1211
H
2'98
605
&-22
12-49
286
WEIGHT OF HOOP IRON, WJEB, AND BOLTS.
Table op the Weight of Koop Iron in Lbs. peb
Lineal Foot.
Breadth (ins.)
i
i
*
4
i
1
H
17
Thickness (B.W.G.)
23
22
21
20
19
18
Weight (lbs.) .
•0313
-0466 ^0666
•0875 •
1225
•1633
! •217fi
Breadth (ins.)
n
1»
H
If
2
2t
H
Thickness (B.W.G.)
16
16
15
14
13
13
12
•9083
1 Weight (lbs.) .
•2708
•3300 ^3600
•4842 •
6333
•7126
• -m,
Table op the Weight op Iron, Steel, Brass, aito
Copper Wire in Lbs. per Lineal Foot.
n
0
1
2
3
4
5
6
7
8
9
10
Lbs. per Lineal Foot 1
6
•
■
11
12
13
14
15
16
17
18
19
20
21
Lbs. per Lineal Foot
Iron 3teel Brass
Copper
Iron Steel Brass | Copper
•3058
•2676
•2134
•1802
•1611
•1246
•1146
•0926
•0729
•0660
•0496
-.3092
•2604
•2157
•1822
•1528
•1259
•1157
•0935
•0737
•0668
•0502
•3343
•2816
•2332
•1970
•1652
•1362
•1251
•1011
•0797
•0722
•0543
•3517
•2962
•2454
•2072
•1738
•1433
•1316
•1064
•0838
•0759
•0671.
•0413
•0314
•0234
•0169
•0137
•0105
•0080
•0061
•0047
•0032
•0017
•0418
•0318
•0236
•0171
•0139
•0106
•0081
•0062
•0047
•0033
•0018
•0452
•0343
•0256
•0185
•0150
•0115
•0087
•0067
•0051
•0034
•0019
•0475
•0361
•0269
•0195
•0158
•0121
•0092
•0070
•0054
■0037
■0022
J
Table op the Weight op Nuts and BOlt-hbads en
Lbs. per Pair.
Diameter of bolt (ins.)
J.
•060
i
•100
1
•200
•240
•366
i i
1
-a .
1-26
1-31
Hexagon head and nut
•600 ,-770
Square head and nat
•062 '^121
•400
•560 -880
Diameter of bolt (ins.)
1*1 U
H
3^75
442
1*
676
2
8^76
n
17-QO
21-00
Hexagon head and nut
1-75 213
3^00
Square head and nut
2-10 2^66
3-60
7-i
OO
10-6
LIMITING SIZES OF PLATES.
237
LIMITING Sizes of Plates (Adiobalty).
Mild Steel,
Thickness.
Width . .
Area
101b.
ft. in.
33 0
610
sq. ft.
145
121b.
ft. in.
33 0
7 0
sq.ft.
155
141b.
ft. in.
36 0
7 3
sq.ft.
170
171b.
ft. in.
40 0
a 0
sq. ft.
200
201b.
ft. in.
50 0
8 3
sq.ft.
240
22i lb.
ft. in.
50 0
8 8
sq. ft.
240
251b.
ft in.
45 0
8 3
sq. ft.
240
Thickness.
Length .
Width .
Area
27i lb.
ft. in.
45 0
8 3
sq.ft.
240
301b.
ft. in.
45 0
8 3
sq. ft.
240
351b.
ft. in.
45 0
8 3
sq. ft.
210
40 and
451b.
ft. in.
45 0
8 3
sq. ft.
210
50 and
551b.
ft. in.
40 0
8 0
sq. ft.
200
601b.
ft. in.
40 0
8 0
sq. ft.
200
70an<
801b.
ft. in
30 0
7 6
sq.ft.
150
High Tensile {H.T,) Steel
Thickness.
Length .
Width . .
Area . .
101b.
ft. in.
40 0
6 0
sq. ft.
150
121b.
ft. in.
40 0
6 6
sq. ft.
170
141b.
ft. in.
40 0
7 3
sq.ft.
190
171b.
ft. in.
.40 0
8 3
sq.ft.
220
201b.
ft. in.
50 0
9 6
sq. ft.
280
22ilb.
ft. in.
50 0
9 6
sq.ft.
280
27jlb
ft. in
50 0
10 0
sq. ft.
280
Thickness.
Length . .
Width . .
Area . .
301b.
ft. in.
50 0
10 0
sq. ft.
280
351b.
ft. in.
50 0
10 6
sq. ft.
300
40 and
451b.
ft. in.
50 0
10 6
sq. ft.
300
50 and
551b.
ft. in.
50 0
11 0
sq. ft.
350
601b.
ft. in.
50 0
11 0
sq.ft.
350
70 and
801b.
ft. in.
40 0
9 0
sq. ft.
280
•».
IS
' 16
J
. 1^*
17*
18
IS
19
IS^
iX*
—
22
—
24*
u*
13
25
—
2^*
12
27**
11
28
10
31*
9*
34*
a
36*
7
..
4«
..
6
^—.
4»
^—^
3
2»»
—
1
—
OP**
000
,
55
54
52
51
49
47
45
43
42
41
38
34
31»
29
23
19
13
6**
2
C»
G
K
M»
P*
8*
Y»
X**
iV^tf, — Sizes which differ from those in the first cdimm
b^ more than '002 of an inch are marked thus ** ; those of
which the difference exceeds *001, thus \ All others either
Gorrei^KJtnd exactly, or ax^e within ^001 of an ioch.
•m^-grmmfit^m^npim^fr
^U^E OAUOE, BOABD
OF TBAPE STANDARD. 2.
Legal Siandasd Wib£ Qaugb.
Descriptive
Number
Equivalents
in Parts
Metric Egoi-
valent in
Descriptive
Equivalents
in Parts
Metric Equ
Talent Ip
of an Inch.
Millimetres
of an Inch
Millimetre
No. J
0500
12-700
No. 23
0024
0:610
tt
•464
11-785
24
•022
•659
i
•432
10-973
25
-020
•608
4
1
•400
10-160
26
•018
•467
•372
9-449
27
•0164
•4166
2
•348
8-839
28
•0148
•3759
0
•324
8-229
20
•0136
•3464
1
•300
7-620
30
•0124
•3160
2
•276
7-010
31
•0116
•2946
3
•25a
6-401
32
•0108
•2743
4
•232
5-893
33
•0100
•2640
5
•212
6385
34
•0002
'2337
6
•192
4-877
36
•0084
•2134
7
•176
4^470
36
•0076
•1930
8
•160
4064
37
•0068
•1727
9
•144
3-668
38
•0060
•1624
10
•128
3-251
39
•0052
•1321
11
•116
2946
40
•0048
•1219
12
•104
2-642
41
•0044
•1118
13
•092
"2-337
42
•0040
•1016
14'
•080
2-032
43
-0038
•0914
15
•072
1-829
44
•0032
-0813
16
•064
1-626
45
•0028
•0711
17
•056
1-422
46
•0024
•0610
18
•048
1-219
47
•0020
-0608
19
•040
1016
48
•0016
•0406
20
•036
0-914
49
•0012
-0805
21
•032
•813
60
•0010
•0264
22
•028
•711
240 TABLE Ot WBIOHT AND STRENGTH OP WIttB.
Table of Weight and Strength of Steel Wire.
i8>
7/0
6/0
6/0
4/0
3/0
2/0
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Diameter
Inch
500
464
432
400
372
348
324
300
276
252
232
212
192
176
160
144
128
116
104
092
080
072
064
056
048
040
036
Mm.
12-7
11-8
11-0
10-2
9-4
8-8
8-2
7-6
70
6-4
59
5-4
4-9
4-6
41
3-7
3-3
3-0
2-6
2-3
20
1-8
1-6
1-4
1-2
1-0
•9
Boo*
Weight of
Approximate
Length of
lOwt.
tional
Area
100
Yards
IMile
Square
Inch
•1963
Lbs.
193-4
Lbe.
3,404
Taivis
68
•1691
166-5
2,930
67
•1466
144-4
2,541
78
•1257
123-8
2.179
91
-1087
107-1
1,886
106
•0951
93-7
1,649
120
•0824
81-2
1.429
138
•0707
696
1,225
161
-0598
589
1,037
190
•0499
491
864
228
-0423
41-6
732
269
•0353
34-8
612
322
•0290
28^5
602
393
-0243
240
422
467
•0201
19^8
348
566
•0163
160
282
700
•0129
12-7
223
882
•0106
104
183
1,077
•0085
8^4
148
1,333
•0066
65
114
1,723
•0050
6-0
88
2,240
•0041
4-0
70
2,800
•0032
3-2
66
3,500
•0025
24
42
4,667
-0018
1-8
32
6,222
•0013
1-2
21
9,333
•0010
1-0
18
11,200
^st
Lbs.
43,975
37,854
32,823
28,144
24,354
21,302
18,464
15,831
13,398
11,169
9,467
7,904
6,486
5,460
4,603
3,648
2,882
2,368
1,903
1,489
1,126
912
721
662
406
281
228
%
^
7/0
6/0
5/0
4/0
3/0
2/0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
BRITISH STANDARD SECTIONS. 241
DIMElfSIOlfS AND P&OFERTIES OF BBITISfi ftTAlTDABD
SECTIONS.
Explanation of Tables,
The particulars in the following tables are taken by
permission from publication No. 6 of the Engineering
Standards Committee.* They are in many oases amplified
and arranged ih the form given in the Pocket Companion of
Messrs. Dorman, Long & Co., Ltd., and reproduced here
by permission.
Thicknesses, within limits, other than those given can
generally be rolled ; but in angle bulbs and Z bars the
increa»e of flange must be about one-half that of the web
thickness ; while in bulb tees and channels the web thick-
ness alone can be varied.
For intermediate thicknesses the radius of gyration is
approximately constant in angles, angle bulbs, tees, and
Z bars, so that the moment of inertia varies directly as
the sectional area ; but in bulb tees and channels the effect
of varying the web thickness is best allowed for by
directly calculating the moment of inertia of the web.
^he least radius of gyration (-s- Vle^t moment of
inertia —- sectional area) has been determined for all sections.
In sections such as I beams, channels, tees, tee bulbs, and
equal angles, the corresponding axis is either the axis of
d3rmn]etry or the one bl right angles to it. In the case
of •unequal angles, bulb angles, and Z bars, the position of
the axis about which the Radius is least is given and
marked 'minor axis'. The maximum moment of inertia
is always aboufc an axis perpendicular to that for the
minimum, these forming the 'principal axes ^ (p. 73) ; its
value in unsymmetrical sections is found by subtracting the
minimum m.i. from the sum of the m.i. about xx and Tir.
From the maximum and minimum m.i. that about any
other axis is found by the rule on p. 73.
The section modulus about xx is equal to the moment
of inertia about xx divided by the greatest distance of any
part of the section from xx ; symbolically it s= I/y =s
bending moment for unit stress.
* Report No. 6. Properties of British StAncIard Sections, published by
Messrs. Crosby Lockwood & Son. price 2«. 6el.
242
BRITISH STANDARD
SECTIONS.
Dimensions and Pbopebties of Bbitish Standabd I Beaks n
Inch Units.
'
DlAORAM.
■
Reference
Mark.
Size.
Weight
per
,
BSB.
Inches.
Foot-
Web
Flange
Radius.
lbs.
t
T.
Bl.
Bl
30
24x7i
100
6
107
•7
■35
29
20x74
89
•6
101
•7
•3S
28 .
18x7
75
55
•928
•65
32
33
27
16x6
62
•55
•847
65
26
15x6
59
6
•88
•6
•3
FlQ* 152.
25
15x5
42
•42
•647
•62
•26
1
24
14x6
57
•5
•873
•6
•3
ft 1
23
14x6
46
•4
•698
•5
•25
• Hi
22
21
12x6
12x6
54
44
•6
•4
•88?
•717
•6
•5
3
•25
■:^^R.^^^.^^iiJ^^
■ ' 1
20
12x5
32
•85
•65
•45
•23
^
•«
^ ,.y^^
19
10x8
70
•6
•97
•7
•3i
f
18
10x6
42
•4
•736
•5
•21
X
\ X
17
16
10x5
9x7
30
58
•36
•55
•652
•924
•46
•65
■Ol
i 1 J|
~
3!
1
15
9x4
21
3
•46
•4
2
14
8x6
35
•44
•597
•54
•2:
^^^^&!K
13
8x6
28
•35
•575
•45
•S
J
'^^^^ ^
12
8x4
18
•28
•402
•38
•n
11
7x4
16
•26
•387
•35
•17
10
6x5
26
•41
•62
•61
•25
4
9
6x4i'
20
•37
•431
•47
•28
-
8
6x3
12
•26
•348
•36
18
«'
7
6x4i
18
29
•448
•39
19
6
5x3
11
•22
•376
•32
•16
^ b %
5
4|xl|
6-0
•18
325
•28
11
4
4x3
9-5
•22
•336
32
•15
3
4xl|
5
•17
•24
•27
•13
2
3x3
8-6
•2
-332
•3
•15
1
3x14
4
•16
•248
•26
•13
The properties of British Standard Sections in above table
arepiabl
ishedby
permission of the Engineering Standards Comm
ittee.
BRITISH STANDARD SECTIONS.
248
Dimensions And Pboperties of British Standard I Beams in
Inch Units.
Area.
SQTiare
Moments of Inertia.
Radii of
Gyration.
Inches.
Section
Modulus.
About
Centres
of Holes
c
Inches.
:Beferenc<
Mark.
Inches.
About
About
About
About
BBB.
x-x.
T-Y.
x-x.
Y-Y.
x-x.
294
2654
66 92
95
15
2211
45
30
26 17
1670
6263
7-99
164
1670
45
29
22 06
1149
4704
7-21
146
127-6
40
28
1823
725-7
27 08
631
121
90-71
35
27
1735
6289
28-22
602
127
83 85
35
26
12-35
428
1181
5-88
•978
6706
2-75
25
16 76
5329
27 96
563
1-29
7612
35
24
1353
4405
216
57
1-26
6292
35
23
1588
3755
283
4-83
1-33
6258
35
22
1294
3153
2227
4-93
131
5255
35
21
9 41
220
9753
4-83
101
36 66
2-75
20
20 6
3449
71-67
4-03
1-86
68 98
475
19
12 35
2115
22-95
413
1-36
42-3
35
18
8 82
1456
9 79
40e
105
2912
2-76
17
1706
2295
468
36C
1-64
51-0
40
16
6176
811
4-2
362
•824
1802
225
'^^
1029
1105
17-95
32?
132
27-62
35
14
18 24
8932
1026
329
111
2233
2-75
13
(5 294
5569
3578
3-24
•822
1392
2-25
12
4 706
3921
8-414
2-88
•851
11-2
225
11
7 35
4361
0116
2-43
111
1463
275
10
6-88
34 62
6415
242
•969
1164
2-5
9
3 53
2021
1339
2-39
•616
6-73«i
15
8
529
2269
5664
207
103
9076
26
7
3235
1361
1462
205
•672
6-444
1-5
6
1912
6-77
•263
1-87
•37
2833
_
6
2794
752
1-281
1-64
•677
376
16
4
1-47
3668
•186
1-58
•355
1-834
—
3
2-5
8-787
1-262
1-23
•71
2624
1-5
2
1176
1*659
•124
118
•324
llOf
*"^
1
Thedii
practice <
frhis tabic
without 1:
nension c,
>f Messrs. I
i has been t
loles.
jiving the i
>onnan, Lo
aken. The
position of the ho
ng & Co., Ltd., fi
1 areas and propi
les, is in accordance
•om whose Pocket Co
ytUea apply to the f u
with the
rnpanio^i
U section
24-1
BRITISH STANBARD S
ECTIO
NS.
BBTTiaH Standard Channelb.
DiMBNUOHS *NB Pbopeetzes in Inch Units.
Relerenw
Size
Standard
EidU.
Ego.
'
V
T
»
'
"-J
27
25
IS x4
12 » 4
13 X3J
'525
■525
■500
630
'625
'600
630
625
600
■440
■425
■425
4194
36 47
32 88
Fro.lSS.
2i
22
12 x3J
11 X 4
11 x3i
375
'500
475
500
600
■575
500
■800
575
■350
■425
■400
26-10
33 22
29 82
"^
21
20
19
10 x4
10 x3i
10 x8!
475
475
375
'57S
576
5WI
■575
■675
■500
■400
400
S60
3016
2821
23 55
!?
18
17
16
9 x4
475
450
375
■575
'550
•500
■576
■550
500
400
375
350
28 55
23-39
22 27
15
li
13
g x3
8 x4
8 x3i
375
450
425
■437
'550
■525
437
560
525
350
375
•375
19 37
25' 73
22 72
1 V
12
11
10
6 xS
8 x2i
7 xsS
37-1
312
400
'500
437
'500
500
437
500
350
30O
360
1930
151!
20 23
9
B
7
7 x3
6 x3i
6 x3
375
375
375
'475
476
'475
175
475
476
326
825
325
1756
17-9fl
1629
6
6
4
6 x3
'312
312
312
'437
■375
■376
■437
375
375
■30O
620
260
1449
12'M
10 93
3
2
1
4 x2
31x2
3 xlS
250
250
'250
'575
812
312
■375
■312
312
260
aao
'220
79S
6 75
62:
The propertiea or British at
sndnd BeoHons in aboos table m» pnbL
Engineeclng Btandirfls Oommltte*.
-hedbT
t
BSITISH STANDAED SECTIONS.
245
British Standard Channels.
Dimensions and Properties in Inch Units.
Area.
Square
Inches.
a
o
'3
Momenta of
Inertia.
12*334
10727
9671
7676
9771 1063
8771
8871
8296
6 925
8-396
7469
6550
5696
7669
6682
5675
4 448
5 950
5 166
5 266
4791
4261
3542
3230
2-3411
1-986
154^
•936
1031
•867
•860
About
zx.
SiectioB Moduli.
•896
1102
•93d
-933
1151
•971
•976
•754
1-201
1011
-844?
•666(
10$1
•874
V119
•928
•93a-
•704
•757
636
'645
-484
3770
2182
1907
1586
170-5
I486
1307
1179
1026
1017
8807
79-90
$518
7402
6376
53 43
4109
4455
3763
29 66
2603
24611
1876
1213
6-709
3 701
1-994
About
About
zx.
__i4
14-56
1365
8922
7572
12-812
8421
1202
8194
7187
11635
7660
6963
4 021
10790
7 067
4 329
2-288
6 498
4017
5-907
^822
3503
1880
1-774
•843
•713
•296
5027
36 36
3179
26-44
30 99
p702
^6-14
0359
2052
^2-59!
19-57
1776
14-48
1850
16-94
1336
1027
12-73
1075
9885
8-678
8008
6254
4 854
2855
2-115
1-329
About
Tr.
Radii
of Gyration.
Inches.
Reference
Mark
About
XX.
4-748
4599
3-889
2-868
4-362
3-234
4147
3192
2800
4-084
3029
2759
1-790
3855
2 839
2008
1245
2-664
1889
2-481
1845
1699
1047
1018
•627
•626
•291
663
451
4 44
4 55
418
412
8-84
377
385
348
343
349
338
3 13
3 09
3 07
304
2-74
2 70
236
233
2-37
2-302
1-94
156
1-37
1135
About
109
113
-960
993
1145
■980
116
•994
1-02
1177
101
103
-840
1194
103
-873
-716
104
•882
106
893
-907
•729
-741
-600
•599
•437
BBO.
27
26
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
5
4
3
2
1
246
BKITI8H STANDARD SECTIONS.
British Standard Equal Angles.
Dimensions and Properties in Inch Unitj
1.
Fig. 154.
Minor AxfsVf 5?.
1
1
•
\
'^rCcntrttff Gravity
^r-
• T"
— X
„*..•
^ t \ \
•->< \
Beferenoe
Mark.
8izeand
Thickness.
Area.
Sqnare
Inches.
Weight
per
Foot-
lbs.
Badil.
•
§
-a
S
Moment
of
Inertia.
Seotion
Modalus.
13 SI
Root.
Toe.
Hi
PI3
BREA.
J
zx
47-4
XX
16
8
x8
xj
775
26 36
■600
•425
2*16
... — 1
81C
1-58
16
1 1
ti
9*609
3267
-600
426
2-20
58-2
1003,
1-57
16
1 1
tf
1
11-437
38-89
-600
•425
225
685
1191
156
16
»»
})
i
13*234
45-00
-600
•425
2-30
78-41
1376
1-56
14
6
x6
xf
4*362
14-88
-476
326
1-61
1499
341
119
14
II
)»
i
6*763
19*56
-476
•325
1-66
19-52
4-60! 118
14
»i
1 1
i
7112
2418
•476
-325
171
23-8
5-5fi' 118
14
f )
1 )
i
8*441
28-70
•475
•326
1-76
27*8
656
117
14
i»
))
1
11003
37*41
•475
325
1-86
3509
5-4b
116
13
5
x6
^A
3*028
10-30
•426
•300
1*34
7*18
196
•99
13
«)
i»
1
3610
12-27
•425
•300
137
8*61
234
•98
13
'»
)i
i
4*750
1616
-425
•300
142
11*0
307
•98
13
1 1
)}
5-860
1992
•425
•300
1-47
13*4
380
•93
13
»»
)}
i
6938
23-69
-425
•300
151
16*6
441
•96
12
4i
x4j
x|
3-236
1100
•400
•275
1-22
614
1-87
•88
12
ft
)i
i
4*262
1446
-400
275
1-29
7*92
247
•87
12
1 1
i f
6*236
17-80
•400
•275
1-34
9*66
303
•87
12
♦ »
>»
i
6*189
2104
•400
•276
1-39
HI
367
-87
11
4
x4
xA
2-402
817
•350
•250
1-10
3*61
1*24
•78
11
1 1
} i
i
2-869
9 72
•350
•250
112
4*26
1*48
•73
11
t )
it
i
3-749
12-75
-360
•260
117
5-46
193
•77
11
M
>»
t
4*609
1667
•360
•260
122
6*66
2-36
•77
Thif
1 table has been taken by permission from Messrs. Dorman,
Cjongft
Co.Hi
Pocket
page, i{
Companion. An additional British Standard Angle, not ind
J :— BSBA 15-7' X T' X .5" to •OTff*.
luded on thii
BRITISH STANDARD SECTIONS.
247
British Standard Equal Angles.
Dimensions and Properties in Inch Units.
9
9
9
9
9
7
7
7
7
6
6
6
5
5
6
5
4
4
4
3
3
3
2
2
Size and
ThicknesB.
5J X 3i X ^fif
I
i
5 x3 x}
A
» »♦
»
1 »
I)
»»
ft
II
i
2i X 2) X I
» »
II
II
II
2i X 22 X 2
A
II
II
2 x2 xA
>i II 4
II
II
II
II
A
lixifxA
i
A
11
II
lixlixA
i
A
Area.
Square
Inches.
II
II
IJxlixA
II
2091
2485
3251
3*985
144
1779
2111
2762
3362
1187
1464
1-733
2-249
1063
1309
1547
•716
-938
1153
1-36
•622
•814
•997
•526
•686
•839
433
•561
Weight
per
Foot-
IbB.
711
845
1105
1355
490
605
718
9*36
11-43
404
4-98
589
765
361
4-45
5-26
243
3-19
392
462
211
2-77
339
179
233
2-85
1-47
191
Radii.
Root.
325
325
325
325
300
300
300
300
300
275
275
275
275
250
250
250
250
250
250
250
225
225
225
200
200
200
200
200
Toe.
225
22a
225
225
200
200
200
200
200
200
200
200
200
175
175
175
175
175
175
175
150
160
150
150
150
150
150
150
•a
•975
1-00
105
109
•827
•853
•877
-924
•970
•703
•728
•752
799
•643
•668
•692
•554
•581
•605
629
•495
•520
•544
•434
•468
•482
-371
•396
9 33
S'Sfe
239
2-80
8-57
4-27
1-21
147
1-72
2-19
2-59
•677
•822
•962
1-21
•489
•592
•686
•260
•336
•401
•467
•172
•220
•264
•106
•134
•159
•058
•073
83
xz
•95
112
146
177
•66
•68
•81
105
128
•38
•46
•55
•71
•30
•37
•44
•18
•24
•29
•34
•14
•18
•22
•10
•la
■16
•07
•09
o a
3l
t
•68
•68
-68
•68
•59
•58
•58
•58
•58
•48
•48
•48
•48
•44
•43
•43
•39
•39
•38
-38
•34
•34
•34
•29
•29
•29
•24
•23
Seenoteonpreoeding page. Additional British Standard Ay^^^^
in this page, are i-BBEAS-arx^xy toy; gndBBHAl-l"xrxy k»y.
'^48
BRITISH STANDARD SECTIONS.
British Standard Unequal Angles.
DiBfKNSIONS AND PROPERTIES IN INCH UNIT^.
Pig. 165.
McnorAxIs^
\
1'
1
►■_^» ••* "^ ^» ^» m^ ^w ^» mm ^ ^
X — -y
•
i\Ctntr« Of Gravity
-*__ .
' 1
iilYV
a>'
2S
^t • 4
u
•13^
Radii
Dimen-
" sioos.
Moments
of Iii^rtia.
Section
Moduli.
it
£<i
Size and
m-g
•si
12-91
49
^3
^
1^
s s
tf 1
98UA.
Thickness
•
•425 3
0 2-45
p
¥
d .
4-24 119
15
25
7 XflSxa
3-797
•713
m3x>.
332
•75
25
S-000
l7-0<^
•425 -3
0 2-60
•764
251
4-28
6-5S
1-66
1^
•74
•25
" *' B
61i72
20-98
425 -3
0 2-55
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3055
Biff
6-86 1-92
1^
-74
25.
It tt 1
7-313.
24-86
•425 -3
0 2-60
•862
36 63
695
8-11
2 26
1?
•73
24
QhyXhx
3-8182
1364
•45 -3
26 203
l-04
17^08;
676
3-82 r95l
25i
-9S
24
5$4B
17-84
•45 -3
26 2^08
109
22-2
8-75
5-02
2-57
W
■97
24
tt tf
6-482
2204
•45 -3
25 213
114
2709
10-60
6-20
316
25
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24
tf tt
1
7-686-
2613
46 -3
25 218
119
3166
12-32
7-333-72
26
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22
6)x8ix
3*610
12-27
-425 -3
0 2 22
•741
15-7
3-27
3-67 flS
4-83 1-56
161
•75
22
4-750
16-15
•425 -3
0 22^
•792
20-4
4-20
iSI
•75
22
1 1 1 1
5-860
19-92
•425 -3
0 2 33
•841
24-83
606^
6-961.90
3-235-54
UB
•74
21
6 x4 X
3-610
1227
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0 1 91,
•923
13^2
4-73
2{3i
•87
21
ti tt
4-750
1616
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0 1-96
•974
171
6 10
4232^02
2a
•86
21
tt tt g^
5-860
19-92
•425 -3
0 202
102
20-8
7-36
5-23 5-47
2Si -85
21
tt tt ^
6-938
23-69
•425 3
0 206
107
2421
862
6152^91
23
-^5
20
6 X3^x4
3-424
11-64
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75 201
•773
12-6
3-22
3161-18
19
•76
20
ft ft 9
4-502
15-31
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75206
•823
16-4
414
416 1-65
19
•75
20
" " a
5-549
18-87
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75 211
•872
19-8»
4-97
6-111-89
181
•75
20
I* tt <t
6-564
22 32
•40 2
•40 -2
76216
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2314
6-74
603 2-2-2
18
•74
19
6ix8ix|
3-236
11-00
751-80
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993
3-16
2-68117
2&
•76
19
tt tt 8
4-252
14-46
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751-85
•867
12-80
405
3-51 1-63, 22
•75
19
It, tt 8,
5-236
17-80
•40 -2
75190
•905
156
4-86 4-33 187 21%
•75
18
5ix8 xA
2 562
8-71
•375 2
5 1-87
■636
8-00
1-72 220 -73
202 2-62 m
17
•65
18
ti «t B
3-050
10-37
•375-2
6 1 90
•662
945
17
•64
18
ti tt a
4-003
13-61
•375-2
5 1 95
•711
12-2
2-58 3-44 113
164
•64
18
tr tt B
4926
16-74
-375 -2
6 200
•759
14-7
3-08 4-20l^37l
16|
•6S
17
6 X4 >^|
3-236
11-00
•40 -2
751-61
101
7-96
4-53
2-28
1-62
32
•85
17
tf It 4
4-252
14-46
•40 -2
751-56
1-0^
10-3
5-82
2-99
1-98
32
•M
17
L* J.N Tt
5-236
17-80
-40 -2
751-60
111
12-4
701
3-66
2-41*
32
•B:>
16
6 xa^xA
2-562
8-71
•875 2
6 156
•822
6-47
2-63;i-8$
-98
254
•76
16
•f II 1
3050
10-37
•376 2
6 1-59
•848
764
309 ,2-24
117
36
•7n
16
II ti a
4-003
13-61
•375-2
6 1 64
•897
986
3-96 2-93
1-62
26
'7r»
16
It It 8
4-925 16.74
•376-2
5 1 69
•944
11-9
4-75 3-60
1-86
25
•74
This table has been taken by permission from Messrs. Dorman, Lioner ft C
o.'s
^oekH Companion. An additional RritiBh Standard Angle, not included on this
pwe. is :— BSUA ag-6i"x4"x -6^".
BBITISB STAMDABD 8BCTION8.
249
Bkitish Stakdabd Unequal Angles.
Dimensions and Pbopebtibs in Incb Unitci.
is
&5
Badii.
Dimen-
sions.
Moments of
Inertia.
Section
Moduli.
9 s
9c
S S*
Si^eani
«>S
sf
1
•6S
S.5
SUA.
Thickneis.
1^
u
•
•36
•
1
»26
J
' P
1^
1-
184
•72
20
15
5 x8 x^
2-409
8-17
1-66
•667
6-14
1-68
15
»* *t w
1^-868
9^^T2
•86
•2&
168
•693
724
1-97
218
•85
19*
•65
16
tt *• 9
8-749
12-75
•86
•26
173
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Seenoteorfpreoedingpage. Adgtional British StotodAr«l^^
on this pa^e,Me:-B8UAl8-4rxar'^;8^'to-f : BsUAa-ir xii' x.i76^ to -ff ;
BBUAa-Wx1**^rtor? BmiAl-Uf'xl"xif' tor.
250
BRIT
SH STANDARD SECTIONS.
B&nisH Standabd Bmx Angles.
Dimensions and Pbopebtibb in Inch Units.
1
.r..
II
^1
BkdiL
I*.
A
BBBA.
T.
Via. ice.
SO
IS
Ilk
■soo
■BBO
s
36-46
30-44
88-87
■676
■636
■676
■450
■4£S
-40D
'
IT
B
i
IS
11
■sso
■660
-660
■a7B
•360
11
-ioO
G'S39
I5S
19-6S
■KB
-626
-600
■SSO
■3M
!i
B
■*2E
If
II
-600
■S!6
-sa
»
S
7;jl
s
4M0
1B;29
■460
■426
■sc«
■300
•ajf
t
3
B
■m
8S32
11-33
1
-!7i
•2iO
i
an
■8M
ISS
s
s.
■250
■»
BEinSH STANDABB SECTIOKS.
251
BbITIBH STAin>ABD BlTLB ANGLB8.
Dimensions and Fropebttrs in Inch Units.
Badii.
Centre of
Qravity.
Moments of
Inertia.
Section
ModnU.
11
ii
§.!4
BgBA.
r«
r<
r«
J
P
About
zx
About
About
zx
About
1-126
1050
•975
•675
•626
•675
•560
•625
•600
6-586
6*188
4-622
•778
-686
'693
191-443
133-866
98-228
8-366
5-170
4*828
29-843
23-031
18*265
2-693
1*837
1*720
i'
6
•821
•716
•724
90
19
18
•950
•900
•900
•660
•660
•660
•476
•450
•460
4-361
4095
4-238
-694
-695
•603
82*418
66-383
64*712
4-586
4-336
2-792
16-038
13-941
13-589
1-634
1*646
1*166
4
-729
•736
•618
17
16
16
•850
-850
•825
•626
-626
•600
•425
•426
•400
3-798
8-956
8-643
•706
•598
•712
67-726
52*686
47-072
4*266
2-603
4-031
12-277
11-694
10-661
1*626
1084
1-446
7
6
8
•740
•621
•746
14
18
12
•826
•800
•800
•600
•475
•475
•400
•400
•400
3-698
3*290
3*419
-600
•717
-612
42-863
37-824
36-726
2-449
3^772
2405
9-964
8*984
8-754
1-020
1-856
1007
P
•627
•750
•632
11
10
9
•750
•760
•700
•460
•460
•425
•375
•375
•360
2*998
3141
2-723
•737
•614
•747
30-914
28063
23-943
3-7a)
2-260
3-494
7-726
7*272
6-339
1-360
•943
1-269
10
7
11*
•768
-638
•764
8
7
6
•700
•675
•650
•425
•400
•375
•360
•3S5
•326
2-866
2-597
2-346
-619
■638
•649
21-677
17*360
13*082
2*008
2*057
1-909
6-963
6098
4132
•881
•871
•812
81
10
Hi
•644
-648
•653
6
4
8
•600
•626
•360
aoo
-aoo
-250
2193
1-661
•638
-677
8-802
4-461
1-021
•916
3-136
1*907
'520
•476
lU
•540
•648
2
1
252
9KITI9H STANDARD SE^IONS.
British Standard Tees.
DIMENSKWB ASSDf: I^BO^BRVHSfik J!N DfCS T^NITS.
^ ^
Sisetnd
Thickness.
i
BVt,
71
21
I 21
20
20
19
19.
17
17
15
IS
14
U
13
13
11
11
10
10
8
8
8
7
•7
6
6
5
6
4
4
3
8
6 -9<4 xj
II II
II If
6 X8 X|
II II
6 x4 Xj
•I II
6 x8 x|
II II
4 x4 xj
•I i»
4 x3 XI
a^xsjxj
t< II 1
a x3 Xj
(I •».
3 xajxi
2jx2ijx;
II II,
•I •»
2 X2 X;
tt iq
3-634
4'771
5f878
4-272
5*266
a* 267
4-268
2*875
3r762
2f872
3*768
2*498
3r262
2-496
3P259
ari2i
2-76
1-929
2-506
1-197
L-474
1-742
1 1-071
1,-564!
'»«Z
l;367
•820
11-003
-820
-999
•631
•692
'♦-
I
I
I
Si
^
s
1^
1236
16-22
19-99
14-53
17-87
1107
14-61
9-78
12-79
$•77
12-78^
8'49
1108
8-49
11-08
7-21
9-98
6-56
8-62.
4-07
6-01
6-92
3-64
5-28.
3-22.
4-64
2-79
3-41
2-79
3-40
1-81
2-35
RadiL
425
8
425 -300
425 -300
300
400 -275
400-276
400-275
40q -276
350-250
360
350
350
325
325
325
325
300
300
275
275
273
275
275
-250
-250
-250
•250
•225
•225
•22S
•223
-200
•20Q
250
250
260
226
21^
225
225
200
200
200
200
200
200
200
•176
•176
•175
•150
•150
•180
•150
•IfiJ)
•150
g
5
J
•916
•%8
•02
•684
•998
•05
-691
-741
-11
•1€
•767
-816
-988
-04
-668
•G95
•742r
•697
•724
-750
•638
•689
-679
•628
•648
•674
•519
•644
-43^,
'460
Mo||ieutBOf
Inertia.
l»
f
4700
6070
7-SSO
2635
8rl44
4-471
772
973
2516
41^9
6'40e
l-8flP
2-366
2^768
354P
1^706
2'16B
VOIB
V276
•6T7
•823
•959
•488
•686
•337
•409
-307
•369
•221
•2$5
•106
•136
Is
6344^
8-621
10-912
8-649
10'988
a-691
6,-017
3-716
6031
1-901
^690
1914
2-599
1-284
1-752
•816
1115
-814
1109
•302
•387
•473
-224
•349
•157
-246
•068
•088
•107
-137
-048
-067
Seetion
ModuU.
0
1-52
2-00
247
114
139
1-49
1-96
•85
111
1-45
f90
-83
1-08
110
144
•80.
104
•56
•73
•38
•46i
•5S
•30
-44
-24
•34
-23
-2»
•18
•22
•10
•13
l«
2-W
2-87
3-64
2-88
3-66
1-48
2-01
1-49
201
•95
1-29
-96
130
•73
100
•54
•7.4
•54
•74
-24
•31
•38
•20
•31
•16
-25
•09
•12
•12
•16
•06
-09
Raaiiof
Gyration.
•137
•128
•118
•785
•773
•172
-163
'828
•818
•208
•199
•863
•861
•053
•043
•897
•886
•725
•713
•752
•747
•742
•675
•664
•697
•686
•612
•607
'620
•515
•447
•442
•321
•344
•36i
•43
•4i3
065
•084
-137
-156
•814
•830
•8X5
•893
•717
-733
•620
•63S
'650
•665
■so;
•512
•5-21
•457
•414
•4071
•4:4
•2^
•2^
'S6\\
•370
•301
•31i
___ — » I » ■ — ^ —
the Moperties of British Standard Seotions^m above table are publiahed
by t>eci&ia8i<m ol the Engu^eerins Standards Committee, and ot Messrs.
Dorftian, Long 4 Co., from w^ose Pocket Connpanwn tine table bee been
copied. Qther sections included in the compl« '
Sre :-B8T. 22-r x sT x Jf' ; BSI. 18-6" x ^x
BST, l-l"Xl"xJ"toft".
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£3:
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€ompoHti<m,
F«rcentao$
of Iron,
FeaOi
72
Fe,0»
70
2(Fea08).3HaO
60
FeCOs
48
FeCOs and clay
17-48
Fe8s
46
MATERIALS. 256
H0TS8 OH XATSBIALS.
(For tesU $99 pp. 263 U 285.)
ISON-OSB.
Iron-ore is found prinoipally in ihe following conditions:-^
Name.
Magnetic
Specular ore or led h»matite
Brown hiematile
Spathic iron-ore
Clay ironstone
Iron pyrites
Clay ironstone is the principal soaree of iron in England.
Pia Ibon.
This is produced from the ore by heating it with a suitable
flux in a blast furnace. It is graded according to its
appearance. The darkest grey iron Is termed No. 1, then
the lighter irons are termed Nos. 2, 3, 4 (foundry), 4 (forge),
mottled, and white. The white iron has a hard silvery-white
fracture, and is extremely brittle. The mottled and Ko. 4
(forge) are used for the manufacture of wrought iron, the
remaining numbers being used for making cast iron. The per-
centages of carbon are as follows : —
Grey, Mottled, Wliite.
Graphite ... 8'4 ... 2*2 ... 0*1
Ck)mbined Carbon . 01 ... 14 ... 8*2
Cast Iroii.
Produced by melting down and purifying pig iron. It is
a material having ereat compressive strength (40 tons per
aquare ineh) but we&k in tension or under shear, and lacking in
ductility. Its properties are greatly influenced by the presence
of minute quantities of silicon, phosphorus, and manganese.
Grey cast iron is the most suitable for small castings
where fine definition is required, but not great strengtl:^
JFhiU or light-grey cast iron is used for large castings.
Malleable Cast Iron.
Ordinary iron castings are heated in contact with iron
oxide ; iihe materii^ of me casting is thus rendered malleable
and much lesa brittie. A special quality, made by the *' Blaok-
heart" system, Is much supemor and is used for eastings
requiring moderate tensile or pressure tests.
Wbought Iron.
Is nearly pure iron, produced by abstracting the greater
portion of the carbon from cast iron.
The puddling process, which is that generally employed,
consists of heating the cast iron (fairly white) with a basic
slag which oxidizes and removes the principal impurities.
The iron obtained Is hammered and rolled. It is then out up.
"25^ MATfeUIALS.
piled, re-heated, and again rolled. In the best qnalitiei
(No. 8) this process is again repeated.
Wrought iron is extremely ducUle and malleable, and can
be readily welded. It Cannot be greatily hardened by quenching
likfe steel. It has a fair tensile strength.
The longitudinal strength is increased by rolling, and the
tensile is greater with the grain than across.
Strength and toughness are indicated by a fine olose-grain
nniform fibrous structure, free from all appearance of
crystallization, with a clear bluish-grey colour and silky lostre
on a torn surface ; its tenacity is not appretiiably diminished
at a temperature of 395^ Fahrenheit, but at a dull red heat it
is reduced to about three-fourths.
Steel.
A compound of iron with fr(mi '1 to 1*5 per cent Of carbon;
the varieties containing teas carbon can be welded and forged
(although not so readily as wrought iron), and ftre termed mild
steel, and used for plates and forgings. The presence of
manganese increaded the toughness and makes it easier to weld.
Bessemer steel is produced by removing the carbon by
a strong blast from molten cast iron, leaving mainly pure
iron, into which a certain amount of carbon and manganese
is introduced by adding spiegeleisen.
The metal is then run into large ingots, and hammered
and rolled like wrought iron.
When fractured slowly it presents a silky fibrous
appearance, but if suddenly a granular appearance, nearly
free of lustre and unlike the brilliant crystalline appearance
of iron.
The open-hearth protJess is now usually employed for
making mild steel and certain qualities of cast steel.
A mixture of piff iron and h^avy fifcrap are heated to-
s^her in a reverberatory furnace. The addition of pore
nsematite completes the process of oxidation. Ferro-man^anese
(miM steel) or spie^leisen (hard steel) is finally added.
Cast Steel has a high tensile strength ; castings must be
carefully annealed to aroid excessive brittieBiesB.
Quenching steel is the process of hardening it by heating it
to a sufficiently high temperature, (depending on the quality
— aboat 700° C.) and then suddenly cooling by qaenching.
Tempering consists of re-heating a qnenohed steel to a
moderate temperature, and then quenching in water or oil.
The hard steel is softened or tempered by this process ; oil
softens more than water. The temperatures and colours for
tempering are : —
Bazors, taps, dies, etc. Straw. 230° C.
Punches, chisels, etc. « Purple. 276* C.
Swords, springs, eto. Light blue. 288** 0.
Hand saws , « , Nearly black. 316" C,
MATERULS. 2^7
Annealing is heating to a low fed heat, and then coolinff
very elowly, e.|f. nnder ashes. This removes the inteniftl
stresses set up in oa£[tings or in mild steel after moch woiiing
or punching.
Copper.
Very tough and elastic, of considerable strength, maUeable
and ductile, suitable for hammering into forms requiring
strength and elasticity oombined with lightness,, but doea not
make good castings.
It is hardened by hammering or rolling, but can be restored
to its normal condition by annealing, whieh is performed br
liyeatiiis and quenching. It is easily brazed, and mixed with
other metals it forms very valuable alloys, and oorrodes but
little under the action of sea water.
Tin.
Very malleable but only slightly ductile, and when bent
gives a peculiar cracking noise. Principally used with other
metals to form alloys, or as a protective covering to other metals
liable to rust, as it is, little affected by the action of the air or
weak acids. ...
Zinc.
Brittle when cold, malleable when hot. forms with other
metals valuable alloys. It is little affected by the air or weak
acids generally, and is theref(»re much used in coating metals to
pioteot ^em from the action of the air or sea waAer.
Bronze or Gun Metal.
Strictly an alloy of copper and tin, but a little zinc is often
added to increase the fusibility. Tin increases the hardness and
mixes Well in all proportions. * With 2 parte of copper to 1 of
tin an alloy is formed which cannot be cut with steel tools.
Qtinmetal oAnnot be rolled owing to the high proportion of tin ; but ife Is
rnoeh Qied ia fehe fonn'ol'cMtiiias*
3RASS.
An alloy of copper and zinc, with a small quantity of tin
som/eiktsies addOdi:o. increase the -hardness or vary the colour.
Lead may be added to increase the dactility and make it more suitable for
turning or filing. It is rerj maHeabie and easily worked cold, but not fit for
<oi8iltf l^t a z«d heat
A good mixture for fine or yellow Imtoss is 2 parts copper, 1 part zinc ; nsed for
ornamental castings, &c.
Admiralty brass maalt oontflin at least 93 per qept. ol oopper, and not more
than 8^ 1^ Qtnt. of lead.
MuNTZ's Metal.
Composed of 3 parts copper and 2 parts zinc. Has a very
high tenacity, very ductile, and can be forged hot, and if
hammered or rolled cold can be used for springs.
Much used for sheathing slitps, and for engine bolts;, «&c., liable to rust.
8
1»
dl
,-- *-^
jes
h
_. ^. .» »
I
.-•r^
. 1^ . — ht
••Ait ZTtt"^ •
«f^
MATERIALS.
25
Quality,
A
B
C
D
YifAd 9trw,
Umslin.*
19-3
150
290
26*0
Ultimaie strefioiK pBreeniage
tonn I in.^ eUninatifni on
2S-4 18
28-5 24
37-2 6
381 2
Table of Alloy&
Allot
Soft gun-metal .
Metal for toothed wheels
9*
n
>»
Hard bearings 'for machinery
Orvax metal. Admiralty
Speculum metal .
Sound copper castings
Tombac, or red brass
Red sheet brass .
Brass that solders well
Ordinary brass
Muntz metal
Extremely tenacious metal
Bearings to stand great strains
Extremely hard metal
Government standard metal
Articles for turning
Bearings, nuts, &;c.
Bell metal .
Statuary bronze .
Oomponent Farts
Oopper
16
10|
16
8
88
24
1
8
ii
2
16
16
16
144
16
90
Tin
1
1
2i
1
10
1
14|
2
2i
5
2
SSino
2
32
Brass
5
12
a
Table of Solders.
SOLDSRS
Component Parts •
-
Copper
Tin
1
1
n
3
2
1
Lead
3
2
1
4
2
Zinc
3
1
1
Blsmntb
Flux
Coarse solder for
plumbers .
Fine solder for
plumbers .
Solder for tin
„ pewter .
„ bismuth
Brazing, soft
y, hard
„ hardest .
4
1
0
2
1
Resin
„ or chloride of zinc
»» »»
»» »»
1 Sal ammoniac or
f chloride of zinc
258 materials.
Kayal Brass.
Is Muntz'8 metal with about 1 per cent, of tin added,
the action of sea water whilst retaining all the other properties.
Can be forged hot, has a very high tenacity. It can be rolled into bait,
and is used for bolts and studs where a non-msting material is required.
Admiralty composition. 62 per cent, of copper. 87 per cent, of sine, and
1 per cent of tin.
Phosphor Bronze.
Very hard, tongh, close-grained alloy, composed of copper
and tin with a small amount of phosphorus.
Very raperior for bearings, wheels, ete.» bat if made hot is liable to
crack.
Admiralty composition for bolts, etc.. 90 per cent, copper, and 10 per
cent, phosphor tin containing abont ( per cent, of phosphoms.
Manoanesb Bronze.*
Very uniform closo'grained bronze, with a proportion of
ferro-manganese ; can be rolled either hot or cold, very tough
and strong, largely used for propeller blades, etc.
Aluminium Bronze.
Has nearly double the tenacity of gun metal, is not liable
to rust, and can be forged either hot or cold ; composed of
90 parts copper and 10 parts aluminium.
Babbit's White Ketal.
Used for bearings ; composed of 10 parts tin, 1 copper,
and 1 antimony.
Fenton's White Metal.
Used for stern bushes, bushes for paddle wheels, etc.;
fairly tough and hard, contains 8 parts zinc, 1*66 tin, and
•44 copper.
Chrome- Vanadium Steel.
Contains 1 per cent chromium and *15 per cent vanadium.
Tensile strength 50 tons per square inch. Used for springs
and for the protection of ships in lieu of thin armour.
Properties op Light Alloy 'Duralumin' (Hard Drawn).
(2V. Walter Jtcsenhain.')
Specific gravity -2*8 ; weight per cubao foot 175 lb. ;
Young's modulus, 4,650 tons per square inch. Composition —
copper 4'6 per cent, iron *5 per cent, manganese '5 per cent ;
magnesium *5 per cent ; silicon '5 per ceot ; aluminium 98'4
per cent.
* This material resembles brass rather than bronze in its workiof
Qiialltios.
i
MATERIALS.
259
QtuUity,
A
B
C
D
Tidd »tre88,
tons fin.*
19-3
150
29-0
26*0
Xntimaie atrenifth. PereentaoB
toTM I »n.* elofigation on ST,
28-4 18
28-5 24
37-2 6
381 2
Table of Allotb.
Allot
Soft gun-metal .
Metal for toothed wheels
ft
It
it
Hard bearings 'for machinery
Gtxm metal. Admiralty
Speculum metal »
Sound copper castings
Tombac, or red brass
Red sheet brass .
Brass that solders well
Ordinary brass .
Muntz metal
Extremely tenacious metal
Bearings to stand great strains
Extremely hard metal
Grovemment standard metal
Articles for turning
Bearings, nuts, &;c.
Bell metal .
Statuary bronze .
Component Parts
Table of Soldebs. 1
SOLDSBS
Component Parts •
Flnx
Copper
Tin
1
1
n
3
2
1
Lead
3
2
1
4
2
Zinc
3
1
1
Bismntb
Coarse solder for
plumbers .
Fine solder for
plumbers.
Solder for tin
„ pewter .
„ bismuth
Brazing, soft
„ hard
„ hardest .
4
1
3
2
1
Resin
„ or chloride of zinc
»» »*
» »»
1 Sal ammoniac or
r chloride of zinc
260
WEIGHT AND STRENGTH OF MATERIALS.
Table op the Weight and Strength of MaTbrxai^.
Metai^.
Name
n
Aluminium, cast
„ sheet
Antimony, cast
Arsenic
BisraUith, oast
Brft88,. cftst .
„ shedt
„ wire.
Bronze
Cobalt, cast
Copper, bolts
cast
sheet
wire
GK>ld, pur« .
„ nammeced
„ Btaodard
Gun metai .
Iron, cast> from
„ to
„ average
wrought, from
„ to
„ „ avoTftge
Lead, cast .
„ sneet
Mercury, fluid
„ aolid
Muntss's metal
Niekel, oast
Pewter
Phosphor bronze
Platinum, pure
„ sheet
Silver, pure
standard
a
a
II
ff
Steel, cast .
Steel, hard .
„ soft .
Tin, cast .
Type metal
Zmc, cast .
„ sheet .
Specific
Gravity
2-5«0
2-670
6-702
5*768
8.000
6*625
8-544
8-222
7-811
8-850
8-607
8-785
8-878
19-258
18-862
17-647
6-153
6*955.
7-29&
7-125
7*660
7-eoo
11*852
14-400
13-568
li6-682
8*200
7-807
1X600
8-600
19*500
20-387
10474
10-584
7-829
7-818
7-884
7*291
10*450
7028
7*291
Lbs. in
a Cubic
Foot
160*0
166*9
418-9
860-2
618*9
524-9
582-8
5880
513-4
488-2
581-8
587-9
549^1
548-6
1208 6
12101
1102-9
509-6
484-7
4559
446'8
472*8
487«
480 0
709-^
712-8
8480
977-0
511-0
487-9
702-5
586*8
1218-8
12710
654-6
668-4
489*8
488*6
489*6
456*7
658*1
489*8
455-7
Tearing
Force
Lbs. on
8q. In.
11.500
2,798
18,000
81,860
49,000
86,000
19,066
80,000-
66,000
20,400
86,000
18,400
29,ooa
16,500
86,000
68,000
60)000
1,792
8,828
49,000
88,000
265,000
42,000
f58,S
(67,(
,240
to
,000
108,000
121,700
4,600
8,600
7,111
Crushing
Force
Lbs. CD
10,800
82,000
145,000
112,000
40,800
8ft,000
86,000
6,906
IC^dnlus of
Elasticity
Lbs. on
9.800 000
QiT/i Ann
,X rv,VOTr
14260^006
9.000.000
}=
14^600
I7.M9M0
9^7a«000
14,000,000
22,900,000
17,000,000
88^060,006
SM6.660
24^40,000
42,000,000
29,000,000
t6.006.0iJ
18^r60,(
19,650,(
WEIGHT AND STRENGTH OF MATERIALS.
2
TIMBEK.
Specific Gratitt and
DSNiUTY.
Nune.
Spee.
Gzmy.
Lb.
per
eu. ft.
Name.
Spec.
6»T.
Ll».
per
ea. ft.
Name.
Spee.
OxmT.
A-caeia . . .
'71
44
Greenheark
1-00
63
Mahogany,
Alder . . .
•M
85
Hawthorn .
•91
67
Mexican .
•68
Apple . . .
••»
50
Hazel . . .
•86
64
Oak, English
•82
Ash Englifih .
•74
46
Hornbeam .
•76
47
Oak, Russian
•84
Ash, AxDrerioKB
*48
90
Jarrah . .
101
68
Oak. Spanish
104
Beech . . .
•TO
44
Kanri, New
Oak. White
BiTch . . .
•76
47
Zealand .
•54
84
(U.S.) . .
•98
BOK . . . .
1-00
m
Labumiun .
'93
67
Pine, ^d .
•55
Cedar . . .
•49
8t
Larch,
%
Pine, VeUow
•60
Chestnut . .
•54
33
Rofisiaa .
•65
41
Pine, Oregon
•60
Cypress , .
•66
41
Lancewood .
•68
42
Pine. Pitch .
•66
Ebony . . .
1-ao
75
Iiignum-vitae
1-25
78
Sabion . .
•92
glder . . .
•TO
43
Lime • • •
•76
47
Teak . . .
•80
Elm, EnglUh .
•66
85
Mahogany,
\^alnut . .
•CT
•75-
47
Cuba . .
•77
48
Willow . .
•40
Jif.Riga . .
•66-
85
Mahogany,
Yew . . .
•81
Fix. e^niM .
•48'
30
Honduras .
•66
41
—
^
• •
SntENCFTB AMD ELAE
JTICTl
«
l&x
irwsieh
. frofln
I^aOeWa '* T\
vmber
aiid 3
7tt»6er.rr»M"
t
Name.
tsh, English «
b. American
Sla. English \
tlm, Canada .
(ir.Riga . .
'iff Spruce .
rraenheari
lonbeam . .
hxtth . . .
ivri. . . .
Mth . , .
lakogaoy,
Cuba . . .
Tons per Same Inch.
^•fs
1
2'
1
2
4
1
1
7
4
8
4
1
6
7
89
2-9
8
0
9
1-7
t
: Nof«.~.i%e bending
pn the Wnding tests,
3
2
2
2
4
8
2
1
4
0
5
0
0
2
68
8-7
2
8
'7
3^2
6^2
8-8
2-4
5*5
4-5
4-0
8-0
42
3^8
5-3
^1
640
890
280
700
Wd
8TO
490
880
780
850
Name.
Tons per Sanare Ii
li
Mahogany,
Honduras
Mahogany,
Mexican .
Oak, English
Oak, Russian
Oak, Spanish
Oak. White
Pine. Red .
Pine, Yellow
Pine, Pitch
Sabicu . .
Xeak « • •
13
1-5
S-4
1-9
8<
1
I
2
2
1
27
2*6
3-4
8-4
81
2*1
varia
2-9
89
2^8
5*2
4-
4-
2*9
84
4-9
8-9
bleS-;
5*3
7^8
5^8
strength and modulus of elasticity of woods are detc
using the usual formulae. All these data are liable t
im
WEIGHT OF MATEBIAL8.
Table of the Weight and Strength op ;
MATERIAI3
(concluded).
Miscellaneous Substances. 1
o >>
4^ ^
n tk.
.4
Name
Weight 0
un. Foot,
|8^
2 S c
Name
Specific
Gravit:
Weight 0
nb. Foot,
Orushin
Force.
bs. on Sq
2-50
0
156
^
2-79
178
a
Aspbalte .
_
Mica
Asbestos .
8-07
191
—
Mortar
2^48
165
— .
Basalt .
2-72
170
16,800
Peat, hard
1-83
83
_
Brick, common
2-«0
125
Plumbago
2-27
139
— .
„ red . .
2-16
134
808
Porcelain, China .
238
149
— .
„ Welsh fire .
2-40
150
Portland stone
2-67
161
6,856
Cement, Portland .
1-35
84
5.984
Pumice stone .
•914
57
—
Clay . , .
1*98
120
—
Piurbeck stone
2-60
163
9,160
Coal. • •
1«27 79-4
—
Rag stone ,
2-47
154
— ,
Concrete . •
S-00
124
— •
Rotten stone .
1-98
134
_
Cork . .
•25
15
— -
Salt ....
213
133
^_
Glass, flint
3-U78
192
27,60«'
Sand, fine pit .
1-52
95
— .
„ orown
2-52
167
31,00G
„ ooareepit .
1-61
100
— .
„ common green
2-528
168
31,876
„ riyer
1-88
117
—
„ plate
2-76
172
—
Slate
2-62
164
1S,000
Gypsum . .
2-17
135
Sugar
1-61
100
-^
Granite . .
8-76
169
12,800
Sulphate of soda .
2-20
137
— — .
Grindstone
214
134
—
Sulphur, native
2^03
127
-_
India rubber .
•934
58^4
—
„ fused
1-99
124
*^
Lime, quick .
•843
63
— .
Tallow .
•94
59
__
Limestone
2-96
184
9,160
Tar ....
1-02
68
__
Marble .
2-72
170
9,21s
Tile» common .
1-83
113 — 1
LiQI
tJIDS.
1
«i
* ^
tf •" tf -T
2-^
|j.
^^
IJ^I.
Name
1-06
§1
0
66-4
Weight
Cnblc I
Ozs
Name
11
•916
o
67-2
Weight
Oubio I
OSB,
Acetio a<*id
•615
Oil of olives .
•530
Alcohol, proof .
•928 57»6
•684
„ turpentine .
•870
54-9
•506
Ether, acetic . ' .
•866
64
•501
„ wlmle .
•928
67-7
•634
„ muriatic
•730
45-6
•422
Oils, average .
•880
65-0
•610
„ snlphuric .
•740
46*3
•428
Petroleum
•878
64-8
•506
Muriatic acid .
1-20
75
•694
Sulphuric acid
1-84
115
1*066
Nitric acid .
1-27
79 4
•786
Vinegar . ,
1^01
68-1
•685
Oil of anisf^ed .
•9^7
Hl'6
•570
Water, rain .
l-OO
62-5
•679
„ caraway seed
•905
66-6
•624
„ sea .
I 025 64-01 •693 1
M hempsecd .
•926
57-8
•536
Wine, champagne .
•998
62-^
•678
„ layender
•894
65-9
•517
M burgrundy
•991
62^0
•673
„ linseed .
•940
58 8
•544
M madidra
1-04
65<0
-•01
M rapeseed
•913
67-0
•528
„ port .
•997
62-3
•677
ADMIRALTY TBST8 268
Proof Bpirit hM » Bpeoiflo vtviiij of H or •988. Spirit k nM to te
SB per cent over proof when 100 purU of Bpirii yield on dilation 100 -l> a
parte of proof spirit ; it is s per cent under proof when 100 parts contain
100 — « parts of proof spirit «&d » parts of water.
ABHIBALTT TE8T8, ETC., 70& MATXBIAI8.
Gbrebak.
'AQ. gtlvBxnziDtt io be done by the 'hot process', anless
otherwise epecifiea, and tests should generally be oarried out
before gfal^anizing. The specified weight of plates, etc., shall
be that before galvanizing.
All steel is to be free from lamination, surface and other
defects. It is to be made by the open-hearth process, either
acid or basic.
Hull material, whatever thickness, which is to be used
for purposes where no structural strength is involved, need
not be tensile tested, and only such bending and other usage
tests as are considered necessary need be made.
Steel casting are to be dean, sound, out of twisty and
SB free as possible from blowholes ; steel forgings are to be
perfectly sound, dean^ and free from all flaws. Where re-
quired, all castings and forp^ings must admit of being
machined to the required dimensions ; and no piednfi^,
patching, bushing, stopping, or lining will be general^
permitted.
Ko cast iron to b^ used except such as is permitted by
specification, or as may be specially allowed.
All important shackles, links, eto.> are generally to be made
of Admiralty quality cable iron, and the iron is to be tested
as specified for cable iron. The securities for receiving the
el^iches of the cables, all shackles, ring and eyebdlto, stopper
bolts and slips, all blocks, including eyes, eyeplates, or hooks
for cables, boats' davits, and similar work, on tiie efficiency
of which the safei^ of the vessel or of life directly depends,
are to be tested by stress and fire-proved.
Davits of all kinds are to be bent hot and fire-proved in
the presence of the overseer after bending, and tested when
in place - with a dead load equal to twice the estimated
working load. Heel sockets, clamps, collars, lugs* and other
work to be tested by the dead load tests in the davits, the
forged parts being previously fire-proved as may be considered
necessary.
Derriohi to be tested with a dead load equal to twice the
working load. The test load to be stamped on i^e derrick;
The above regulations apply to cable clenches, evebolts,
stopper bolts, and eyeplates for cable gear, as wdi as to
davi| and derrick fittings.
301 ADBtlRALTT TtSYS.
iK^TAltOTIONS Foil TbEATMENT OF KtLD AK1» HlOH THNSIIB
SfSEL.
All plates or bars ^liieh oan be ^nt ^oid aftt 1^ 4]« m
treated ; and if the whole length cannot be bent cold, the
portion to be bent hot mtust be of a nniform temperatarel
throughout ; the varyiBg tempwatuie from bet to «aid portion
to be extended so as to avoid tn abrupt termination of the heat.
In cases where plates or ^bars ha'Ve to be heated, the greatest
care, should be taken to .prevent any work being done . upon
the niAterial after it has.failen to tiie dmgerong lintH of
temperature known as a "blue-beat *'-—«iy from 600" to 400*
P, Should this liffiit be reaiched during working, the plaltea
or bars should be re-heated..
Where plates or bars hate been heated tHrotighout for
bending^ flanging, etc., and the work has been ootnpleted at
one heat, subsioqueivt annealii^ is unnecessary, but oare should
be taken to' pi^vent, ta far as iK>s8ible any stidden cooling of
the material.
Special Quality steel plates H.T. should not h^ Iteated in
any way after delivery at the shipyard.
Special Quality steel plates H.T, are to !>e planed ( In.
on 1^ edges and butts before curving; bending, or working.
Where simple forge-work has been done, such as the m-
mation of joggles, eomers, and easy curves or bends, on
portions of plates or bars, and ilie material has not been much
di^lnresde^ (mbsequent annealing is tmneeessary.
Plates or bars which have had a large amount of wt>rk
tnt vpen them while hot, and have to be re-heated, diould
e subsequently annealed. It is preferred ^at this annealing
shotild be done simultaneously orer the whole tof each pl^te or
bar when this oan be done conveniently. If it is incon-
venient to perform the operation of annealing at one ^hne
for the whole of a plate or bar, proportions may be annealed
separately, prop^ eare being taken to pr&vent an id)i:itipt
termination of the line of heat. If the severe working had
been limited to a comparatively small part of a plate or bar
annealing may be limited to the parts which have been heated,
the same care being taken t6 prevent hn abrupt termination
of the line of heiit.
If desired, exceptionally long or quickly eurved bars, such
a9 frames, may be formed of shorter pieces wii^ the butts
suitable shifted and strapped.
It Is not nspessary generally to anneal plates or bars after
punching as a meansf of making good damage done in
punching. Fo;r plating that forms to important featam in
the general strustural strength, suoh as the inner and outer
bottom plating, deck plating, deek stringers, plating behind
side armonr, etc., the butt stl^ps shonld have the holes* drilled
or be annealed after the holes are ^nched. In ^uch {^luting
I
▲DMIRALTT TESTS. 266
the eonBteitenk holes ehould be pnnohed aboat ) in. less In
diameter than tke rivete which are nsed, the enlargedMnft of
tlM h0l«e }miig «Md« m tke eennteridnking, whiok thoold ia
all oases be earned tiifoo^ the whole thiekness of tte ptsies.
AU coaaleiainkiBg is to be eai«lally done^
It is imiportant that 'the wkoie surface of Ihe bottem
plating skbold be thoroaghly oleared ef tke scale fof med in
mannfactare before any paint or oompoiitioai is ftit Qp^ti it*
TfiNsiLE Test Pieces.
3%nsile test pieoes are of one of three trpes : A for flat and
B swd 2 for oireular pieees of material. The actual
diameters of test pieees of the forms B and Z shoald be alt
approtred by the oveneer.
T^H Piece A.-^To be of rectang^nlar section, planed down
to tke widUi speciied below over a length of at least 9 inches.
Gan^ tiiftrks to be 8 inehes apart.
For test pieces over ¥' thick, maximum width allowed . « IV
,, ,, ff' to I" inclusive, maiimum width allowed » 2"
„ „ under i'' maximum width iJlowed • • *» 2^'
Nw^e.-^AM test pioees may be ent in a planing' machine
and have the sharp edges taken dff.
Tefil Piece B. — ^To be of xiircular section with enlarged
end). To be parallel for a length of not less than nine times
the reduced diameter. Gauge length not less than eight tines
the diameter^
Test Piece Z, — ^To be of circular section with enlarged
ends. To be parallel for a length of not less than four and
a half times the reduced diameter. Gauge length not less
than four times the diameter. . .
Ship Plates, Okdim aky Quality.
The jplates will be ordered by weight per superficial foot.
The weights named must be adhered to as nearly as possible
for eadi plate, but the latitude stated below will be permitted.
Plates 20 lb. per square foot 5 per cent below the
and upwards. specified weight. None
above.
Plates under 20 lb. per From 5 per cent below to
sqaai^ foot. 5 per cent above the
specified weight.
Strips 1} inches wide, out eros9wIee or lengthwise, mus%
stand without eraoking bending doublei in a press, or'
hamnaring double over a blodi^ td a curve of which the inner
fftdivB ia net i^reater than one and A half limes the thiekness
266 ADMIRALTY TESTS.
of the steel tested. The strips to be te3tod as the overseer
may wish, either cold from the plate, or after being heated
unifonnly to a blood red, and cooled in water of about 80^ F.
The ihiciness of strips oat for bending tests to be equal to the
thickness of the plate, except that they need not bo more than
f in. thick in the case of plates of over 30 lb., in which case
the strips are to be so bent that an original outside surfac^
is always in tension.
Strips must also stand such hot for^e tests as neoessary to
show that the plates will stand sacn heat treatment and
bending as they may be subjected to in the shipyard.
Ther pieces of plate cut out for tensile testing are to be in
accordance with test piece A when cut either lengthwise or
orosswise, they are to have an ultimate tensUe strength of not
less than 26, and not more than 30, tons per square inch of
section, with an elongation of not less than 20 p'er ooit on
a length of 8 inches. (Te^ piece A.) For plates of 101b.
and under the elongation may be not less than 18 per oent.
Special Quality Steel Plates, H.T.
The Specification for these plates, including the general
working qualities, is exactly the same as for ordinary quality
plates, except as follo\v;8 :~>
Each special quality plate is to be Btampad H.T. ; and
the ultimate tensile strength is to be not less than 33 tons
and not more than 88 tons per square inch.
Bending Tests. — ^Test pieces IJ inches wide to bend
6old without cracking through an angle of 180**, the inner
radius of bend being not greater than twice the thickness
of plate tested.
No sample must, on analysis, show more than 0 15 per cent
of silicon for plates up to and including 40 lb. par square foot,
nor more than 0*2 per cent for plates over 40 lb. per square
foot. It is particularly desired that this steel should oontain
not less than 0*1 and not more than 0*15 per cent of silicon,
and not less than 0*25 and not more than 0*35 per oent 0|jE
carbon for plates up to and including 40 lb. per square foot ;
and the same percentage of carbon with a slightly higher
percentage of silicon for plates over 40 lb. per square foot.
Special Quality Steel, H.H.T. (Destboybb Quality).
All plates and sheets shall be carefully annealed in a proper
annealing furnace after rolling and before test pieces are
taken. The test pieces shall be taken before galvanizing.
(a) Tensile T&et. — ^Plates and sheets of 3^ lb. per .sqoare
foot and upwards are to have an ultimate tensile 8treii|^ of
ADMIRALTY TESTS. 267
37 to 48 tons per square inch, and sheeU iuid«r 8| lb. pei;
square foot are to have an ultimate tensile strength of from
35 to 45 tons per square inch, with an elastic limit of not less
than one half the ultimate tensile strength, with a minimum
of 20 tons.
This material is to have a minimum elongation on a length
of 8 inches (test piece A) of —
15 per cent for plates 10 lb. and above ; 15 per cent
for sheets 7} lb. up to, but not including 10 lb. ; 14 per
cent for 5 lb. up to, but not including 7} lb., and 12 per
cent for less than 5 lb. per square foot.
(6) Bending Test. — ^Tesb pieces to ' bend cold without
cracking through an angle of 180% the inner radius of bend
being not greater than twice the thickness of the plate tested.
Each plate or sheet is to be marked with the letters H.H.T,
The general conditions to be as for ordinary quality steel
plates, except that the weight is subject to a latitude of
10 per cent above the weight specified, but nothing below.
Nickel Steel Ship Plates.
The specification for these plates is the same as for ordinary
quality plates, except as follows :—
Each nickel steel plate is to be stamped N.I. The steel
must contain not less than 3 per oent of nickel, and is to
be such that it can be sheared, punched, bent, etc., with
ordinary shipyard appliances. The ultimate tensile
strength is to be not less than 36 tons and not more than
40 tons per square inch, with an elongation of not less than
18 per cent on a length of 8 inches (test piece A).
The special forge test on strixts is not required for tluis
c'ass of plats.
Ghequebed Steel Plates.
To be of diamond patt'^m, measuring in the clear Ij" x J"
along the diagonals between 'lie ridges, which should be ^" wide,
andproject from ^' to A" above the upper surface of plate.
For plates demandea of rectangular form the edges are to
bo sheared parallel to the diagonals. .
The plates will be ordered by weight per sup3rficial foot,
which weight is always to be token as inclusive of the rib,
but exclusive of the galvanizing when galvanizing Is ordered.
The weights named must be adhered to as nearly as practicable
with the same latitude as with ship plates.
A strip, sheared or cut lengthwise or crosswise from the
plate and not less than 1| in(3ies wide, must stand withoulb
fracture being doubled over when cold until the internal radius
of bend is not greater than 1^ times tiie thickness of the test
piece and the sides are parallel*
t68 ADMIRALTY TESTS.
The piec^ oi pUta cut out for tcnnie tetttng are to b« in
QiMOtdanoe with test piece A. Either lengthwise or eronwise
Aejr are to hAvt an ultimate tensile strei:^^ of not lesB tkan
26 and not more than 30 tcms pev aquare ineh, with a miniiitiin
elongation of 20 per cent on a length of 8 inches. f or 'platea
abort 12i lb. per sqfoare fooit, aixd 16 per cent for plaits l^ lb.
per square foot and below.
Mil© Steel foe Angles, Bulbs, etc.
Strips cut IJ inches wide^ or pieces of full section of the
bar as rolled, must stand bending double in a press, ^or
hammering double over a block to a curve, of which the inner
radius is not greater than 1^ times the thickness of the
steel tested. The samples to be tested either cold from
the bar, or after being heated uniformly to a blood red, and
cooled in water of aboat SO** F. Hie steel is to stand such
forge tests, both hot and cold, as may be sufficient, to prore
soundness of material and iHness for the service.
The pieoes of beam, angle, etc./ cut out for tensile iesiltig
are to have an ultimate tensile strength of -not less than 26
and not more than 30 tons per square inch of section, with an
elongatiini «f nt^t leas tluin 20 per oent on a lengA of 8 inches
(test piece A) or full section ol the bar as rolled.
SectftOnal material will be ordered by weight per fool run ;
a latitnde of 5 per mat below thisse we^hts^ bnt nutiking
above, being allowed for roUing.
HiLD Steel Baes (Flat, Rqund, Segmental, Squaue, and
Hexagonal).
, Test pieces from the bars are to have an altimate tensile
slfemgth of not less than 2d tons per square inch, and not
more than 32 tons per square inch, with an elongation of not
less than 20 per cent measured on test pieces A or B, or
23 per cent on test piece Z.
Strip cut not less than 1 in. square, or pieces of the foil
thickness or sections of tho bar as rolled, must stand bending
double in a press or hammering double over a block to a carvo
of which the inner radius is . not greater than 1^ times
the thickness of the steel tested. The sa^iples tested are
to be bent with the rolled or outside surface in tension ;
either cold from the bar, or after being heated uniformly to a
blood red and cooled in water of about 80^ F.
The steel is to stand such forge tests, both hot and oold, as
may be sufficient to prove soundness of material and fitness
for the service.
Steel Bivets.
Ordinary QualHy.—Tht whole of the rivets are to be
properfy heated in making, except tiiat rivets not gretAet
than A of an inch in diameter may be made cold, Car« ia to
ADMIRALTY TBST9. 2B9
be ialceii thai, t^ ceM made rireto mp« |(rop«flir anaeaM^
and those made ho^ ace to be allowed to oool gradoallT.
The tensile breaking ttrengiik of Munples eeleeted from
mild steel rivet bars when reader for rh'et makiii|^ akali be aot
less than 26 and not more than 30 tons per sqvare inoii iM
section, with an elengatton of not leas than S0 jper emt on 8
diameters of tiie test pieee; (Test pieoa B.) TIm bact sajr
be tested the full size, as rolled.
BiTots are to stand the fetlewiiit^ teite':*^
(«) Beadinr the ^ants o<^ and hamfterinit ^>*'^ ^ ^*^
paiis of the wank toneh, in tiie manner dion ki fig. 14%.
'v^ithoQJb fraoti^e on tiie outside of the bend.
(^) FhiiteBing> oif tile rtret head, while tot, la the nnmner
shoini in %. 169-, witihont oraokingf at the edges. The head
to be flattened ontil Ite diameler is not Imw than 2i tiaiei the
diameter of thcf shank.
1^0.169; TM.-m.
Certain of the finished rivets may be aabjeeted l». a tentiiB
test, and ef the eerewed- :pi¥ete to a cotd bending tei^ aft«r tim
serew- fs cni.
Speeiea QmUt^y BJT, ^md ^.Jf.r.-^Bivets for use in Ugh
tensile eteet plates are to- be made ef steel of special malS^
and are subject to tile same eonditioiis and tsets as eroiimi^
quality rivets esoept as nndet z-^
(A) H.T,, 3 J?i'^.— The tensile stipeagth of the bars op to
and inoluding { in. in diameter is to be not less than. 3A tooM
and not more than 88 ioMS per- s^viar^ sneh, wiith an
e.loogatton of not less tiiaa 74 per eent on a lengtii ai
eig%t ^ameters (test piece H). For bars over fin.
in diameter- the tensile strength shall not be less thaa
82 tons and' not more tban 86 tons per square indi, with att
dottj^tion of 20^ per omt (test pieee B). Bach pan4iead and
snap-head rivet shall be marked in the usual manner for snoh
riviKS with three equidistant rifts en the side of the head ;
eountersonk ri vets shaU be marked with a triangsiar pjrramidid
recess. No ohemieal' Mialysts is required for tiiese rivet hasm
(B) M.ff.T., 4 sib, — ^The tensile strength of the bars is to
be not less than 37 tons* and not more than 43. tons per square
inch, with an elongation of sot less than 18 par cent
en a lengtii ef eight dtameters (test pieos B). For bars
less than ^in. diameter a minimum eiloiigatibB. of 2fi pdr
cent on four diameters (test piece Z) will be accepted in lieu.
Sttoh pBn-head and snap-heaa rivet shnli be marked with four
270 ADMIRALTY TESTA,
equidistant ribs <m the side of the head ; oountersnnk rivets
sluill be marked with a rectangular pyramidal recess.
Nickel Steel RiveU. — ^To be subject to the same graeral
conditions and tests as for ordinary rivets, except that the
tensile strei^^ of the bars to be not less than 36 tons and not
more than 40 tons per square inch, with an elongation of not
less than 20 per oant on a length of eight diameters. (Test
piece B.^
The nvets to be branded with a single rib on the ade of the
head. To contain not leas than 3 per cent of nickel and to
admit of being worked in the sjiipyard like ordinary steel
rivets.
Steel Bolti, Nuts and Studs,— Tbe bolt heads bhall be
forged from the solid and the nuts shall be made from the
solid bar, except in cases where bolts, studs and nuts am
manufactured by machinery from drawn bars ; the bars in the
latter case are to be carefully annealed beforehand.
Test pieces for bars over 1 an. in diameter to be 1 in. in
diameter, and for bars under 1 in. diameter to be the fuU
section of the bar. To have an ultimate tensile strength of
not less than 30 tons per square inch and not more than
35 tons per square inch, with an elongation of not less than
23 per cent measured on test piece B. Pieces from the selected
bars shall be capable of being bent cold without f raotore
ihrongh an aogle of 180^ the internal radius of bend being
not greater than 1) times the thickness or diameter of the
bar ; or after being heated uniformly to a blood red and
cooled in water of about 80° F., must stand bending doable
without fracture in a press, to a curve of which me inner
radius is not greater than 1^ times the thickness or diameter.
The tests to which the bolts and studs will be subjected
are as follows :«^
(a) Kicking on one side and bending to show the quality
of the material, which must be satisfactory to the overseer.
(b) When the bolts and studs are of sufficient length in
the plain part to admit ol beini^ bent cold they shall stand
bendmg double without fracture in a press to a curve of which
the inner radius is not greater than 1^ times the diameter of
the bolt or stud.
When the bolts and studs are not of sufficient length in
the plain part to admit of being bent cold the screw^ part
shall stand bending cold without fracture, as follows : —
)in diameter and under ... through an angle of 35*
Above i in. and under 1 in, „ „ 80*
1 in. diameter and above ... „ » 25*
Samples of the nuts must stand such drift tests aa may
be considered necessary.
StKEL POIKHNOS.
The ingot steel Tor forgings is to bo made by the acid
open-hearth process.
ADMIRALTY TESTS. 271
The forgings are to be j^radually and aniformly forged
from solid ingots, from which at least 40 per cent of the
total weight of the ingot is to be removed from the tdfp end
and at least 5 per cent of the total weight of the ingot from
the bottom end. These ends of the ingot may be removed
either before or on the completion of the forging, and are
not to be nsed for any forgings for H.M. Service.
When finished the sectional area of any part of a steel
forging (as forged) shall generally not exceed one-sixth of
the mean sectional area of the original ingot used for the
forging, and no part of the forging (as forged) shall exceed
one-h^f of the mean sectional area of the original ingot nsed
for the forging. The finished forging must be perfectly sound.
All steel forgings shall be thoroughly annealed in a
properly constructed annealing furnace, which must permit of
the whole forging being uniformly raised in temperature
throughout its whole extent to the necessary intensity required
for annealing purposes. If the forging be subsequently
heated for any further forging it shall again he similarly
annealed if required.
Tests, — The tensile strength and ductility shall be deter-
mined from test pieces prepared from sample test pieces out
lengthwise from the finished forging from a part or not less
sectional dimensions than the body of the forging. Such test
pieces shall be machined from the sample piece without
forging down, and the sample piece shall not be detached from
the forging until the annealing of such forging has been
completed.
One tensile and one bend test are to be taken from each
forging, and they are to be out from any part of the sample
piece as nearly as practicable midway between the centre and
outer edges, and are to be tested as follows :— >
Tensile Test, — ^Test pieces prepared in accordance with test
piece Z are to satisfy the following conditions :—
nn»i{4» Ultimate Tensile Strensth t?i«««««a-
QuaUty. p^ g^^^ ^^ of section. Elongation.
Not more Not less
than than The elongation measured on
the test piece must not be
less than 19 per cent for
A. 38 toPA B4 tons. 38 ton steel, and not less
than 29 per cent for 28 ton
steel, and in no case must
B. So ions^ 3X tons. the sum of the tensile
breaking strength and
corresponding elongation be
C. 32 tons 28 tons. less than 57.
272 ADMIRALTY TBST9.
C^ld Mend Tests.— The cold beud tesits are to be mad& itpon
teet pieees of reetangalar section 1 in. wide and f in. thick.
The test pieoes are to be machined to these dimensioas and
the edges rounded to a radius of ^ in. The test pieces shall
be beat over the thinner seotion.
The test pieoes are to be bent by pressure or by blows, and
must withstand without fracture beioff bent through an angle
ol 180% the internal radiu3 of the hend being hot greater
than that speoified below.
Speciflea Tensile St««gtholP«rdng. ^"^^ag^5ln«^*
< 28 to 32 tons per square inch . . } inch.
Above 32 and up to 36 tons per square
inch '. I inch.
Above S6 and up to 88 tons per square
ineh ...... g Inch.
ForgingfS; after completion, are to be subjected to the nsnal
lire-proving tests, if considered, necessary.
Special Quality StfiEL fob ShiTavb Pins, etc.
The bars to be forged or rolled from ingots of aeid open-
hearth or crucible steel.
The material to stand the following tests after all heat
treatment, if any, has been carried out on the material for the
purpo^ of converting it into pins or otherwise.
Ultimate tensile strength . • 40 to 50 tons per aqnare
inoh.
£k)iigation on a length of i
diMueters (test piece Z). • Not less than 15 per cent.
Bend test, cold , • . . Through 90** without
fracture, the internal
radius of the bend
being not greater than
the diameter of the
test piece*
Steel Casmnos.
•
Steel castings for purpose of ship construction are divided
into thre^ classes, viz. ' Quality A ', ' Qualitjr B ', and
'Quality C ', in accordance with the following classification: —
Quality A. — Boss' castings, if forming part of stmcture of
ship, and stern tubes, capstan gear, cut up of keel, deck
compressors (where formed of castings), head and heels for
derricks, riding bitta, rudder frames, rudder head castings,
rudder crossheads, shaft brackets, stems, sternposts and atern
castings, steering gear fittings, bracket on mast to take heel of
derridk.
ADMIRALTY TESTS. 278
Quality B. — Sheaves to blocks for derricki, coalingr fiouttlea
an proteotiTe ^oeki, bearing rings, bosa castings, tioUiffds,
cleats, deck pipss, fairleads, hawse pipes, link plates fop
scouring guns, net defence — ^roller fairleads, boom heel sockets,
pafeking rings, h-ames for watertight doors, watertight iouttles,
stem mooring pipes, b^s to coaling winches, shoes for davits,
brai&ets for helm signal gear.
QtiifUity C. — Articles except those marked * may be made in
special malleable east iron in lieu of quality C. Brackets,
eto., for valve gearing,* brackets, etc., for watertight doors,*
dismottntlng gear, mitre wheels and spur wheels for work
other than steering or capstan gear, mud boxee, scuttles,^
coaling, hand up and side, scuppers, universal joints,* hand
wheeb, weed boxes, hawser reel castings^ link plates for net
defence.*
Ail castings to be clean and free from defect, and to be
annealed. Quality A to be of open-hearth steel, either acid or
basid.
Te9t9, etc., for Quality A Caaiings. — (a) Pieces of
suitable section and lengpth to be formed on each casting for
providing test pieces, or test pieces may be out from the head
of tiie casting. Where a number of castings are made from one
oharge> test pieees to be provided as required.
(6) One piece to be turned in accordance with test piece Z,
and to have a minimum tensile strengtii t>f 26 tons per square
ineh. l^e elongation to be at least 13| per cent on a length of
4 diamet^t.
(0) A second piece to be planed to a section 1 inch square^
and to admit of being bent cold witikout fracture in a press,
or on a slab or blo^ through an. angle of 4d°, or if preferred
turned 1 inch diameter and bent through an angle or not less
than 60'', the internal radius of bend being not greater than
1 ineh in either ease.
(«() Additional ][»eoes to be ataiiable for repeating either
of me above tesis, it. eaSe Of any dispute or doubt as to the
result ]^rei»nUng the qulili^ Of the material of th^ easting.
(^) m^dk tMiil^ to be raUed to »& angie or height ohbsen
by th^ overseer and allowed to fftll on hard grouiid of the
hardness of a good maeadamiaed road, or on an iron or steel
plate, tho casting to show no mgns of fracture after -this test*
After the above tests, each casting is to be aubjeoted to such
hammering tests as may be considered necessary to ]^ove the
soundness ahd efiieiency of th& casting for the intended service,
and carefully eiramined f6r any sttrlace defeots or flawe.
Testi, ete.y for- ^HaUiy B Castings. --^^-Ti} reoeive the same
tests ttB A quality eastings, eteept that in (d) the elongation
to be at least 10 per cent on a length of 4 diameters.
Tests, etc., for Quality 0 Castings^— {a) Each casting to
ho raised to a height named by the overseer, and allowed
. 274 ADxMIRALTY TESTS.
to fall on hard ground of the hardness of a good macadamized
road, or an iron or steel plate ; the casting to show no signs
of fracture after this test.
(6) After the above test, each casting to be subjected to
such hammering tests as may be consider^ necessary to prove
the soundness and efficiency of the casting for the intended
service^ and carefully examined for any surface defects or
flaws.
Certain of the articles may be selected and te^Aed to
destruction with a view to ascertaining the efficiency of the
casting.
Note applying to all Grades of Castingi, — ^In exceptional
oases, such as in castings of light eection, large aree^ and
intricate form, the drop test may be waived.
Steel Tubes foe Magazine Coolers, Pillars, etc.
The ends of the tubes must admit of being expanded hot,
without injury, to an increase of ^ the diameter of the tube.
Strips cut from the tubes or pieces lof full section of the tubes
must have a tensile strength not less than 24 tons and not
exceeding 30 tons per square inch, with -an elongation of at
least 33 per cent in a length of 2 inches. They must idso be
capable of being bent without fracture through an angle of
180^, the internal radius of bend being not greater tiian
\ in. ; the strips to be tested either cold or after oeing heated
to a blood red and cooled in water of about 80° F, Thb tubes
for magazine coolers shall stand an hydraulic pressure of
1001b. per square inch, without leakaj?e.
The rolled material from which the flanges are made ahaU
be of the same quality as the tubes, as specified above.
Wrought Iron Forgings.
Iron forgings are to be made of the best selected scrap
iron, of approved qualitv, forged into blooms. Jump weld^
are to be entirely avoided for important welds.
Samples cut from the forginffs are to be tested a»follow8: —
The strength and du^ility shall be determined from test
pieces which are to be prepared from samplfs pieces cut
lengthwise from .the foiling from a part of not less sectional
dimensions than the body of the forging. Such teat pieces
shall be machined from the sample pieces without oeing}
forged down.
Tensile Test, — A test piece prepared in accordance with
test piece B is to have a tensile strength of not less than
22 tons per square inch of section, with an elongation of not
lees than 22 per cent on a length of 8 diameiters of the test
piece.
Bend Test, — A test piece 1 inch square must withstand
without fracture being bent cold through an angle of 180^
ADMIEALIY TESTS. 275
iih« inidrtud radius of the bend being not greater tliaii 1^ timet
the thickness of the test piece. A 8amp.!i9 is to be noiched
and bent cold to ascertain the qaality of the material.
At least one tensile and one cold bend test are to be taken
from each forging.
Gable Ibok.
The iron is to be of good welding qn&lLty^ free from
lamination. Special consideration is to be given to obtainin§p
a good fibre in the iron.
The samples -of every description of iron shall have an
ultimate tensile strength respectively :—
Of not less than 23 tons to the square inch of section, for
sizes under 2(} inches in diameter ; of not less than 22^ tons
to the square inch of section, for sizes from 2^ to 2^1)^ inches
in diameter, both sizes inclusive ; and of not less than 22 tons
to the square inch of section, for i>izds above 2^ inches ijt
diameter.
Bars less than 1 inch diameter are to be tested on the
8 diameter test piece (B), either full-sized as rolled, or turned
down, the reduced portion to be not less than } in. diameter ;
and the elongation is to be not les3 than 22 per cent on a
length of 8 diameters. Bars 1 inch diameter and over may be
tested full size as rolled, or turned down, the reduced portioa
to be not less than 1 inch diameter, and the elongation may be'
measured on a gauge length of four times the diameter of the
test piece (test piece Z), in which case the elongation is to be
not kes tlian 26 per cent ; or may be measur^ on a gauee
length of 8 diameters (test pieoe B), in which case t£e
elongation is to be not less than 22 per cent.
Forge Test, Cold. — ^Bars of 1 inch diameter and above are
tt> adi&it. of bending cold, when practicable, through an angle
of 180% thus, J j j
end bars under 1 inch shall admit of
^)
bending oold, thus, *" yv. ■ -\\ in each case to the
same radius as the end of the link for which they are generally
Qsed.
In the case of the larger sizes where this is not practicable,
the bars may be cut longitudinallv through the centre. Each
portion to admit of bending cold without fracture with the
outside original surface in tension through an angle of 180%
the inner radius of bend being not greater than 1^ times
the thicJcness of the test piece.
276 ADMiRALtY TESTS.
A flfiinple is to be notched and bent, tlias,
to show the fibre and quality of the iron.
Forge TeH, Hot. — ^Bars are to be punched with a punch
one-third the diameter of the bar, at a di«tanoe Of l\
diameters from the end of the bar. The hole Is thon to be
drifted out to 1} times the diameter of the bar, the side of the
hole split, and the ends mast then admit of turalnif bwek
without fracture, thna--
Cast Iron.
The minimum tensile streng^th to be 9 tons per squard inch
taken on a length of not less than 4 diameters (test piece Z).
The transverse breaking load for a bar of 1 inch aqoare,
loaded at the middle between supports 1 foot a|»art, is not to
be less than 2,000 lbs.
Special Malleable Iron CatiingB. — ^Each casting to be eoond,
clean &nd free from blow-holes, and to be well annealed. To
stand being dropped, without sign of injury, frOm a height
of from 9 to 15 feet on to an iron or steel filab of not lees than
1^ inches in thickness. The casting to be afterwards subjected
to such hammering tests as may be necessary to prove the
soundness and efSciency of the casting for the intended service.
Two test pieces are to be provided from each oasl^ each
having a parallel section of V x |" for a length isl «l liisi
8|" with ends to suit the mode of grip employed.
One of these test pieces is to have a tensile jstrengih of not
lees than 18 tons per square inch, with aik ^dnjraUon of not
less than 4| per cent on a length of 3 inches. The oilier test
piece is to stand binding cold in tiie direction 6f the lesser
thickness without sign of injuJry throiigh an angld of 90*, the
internal radius of bend being not greater iSttKn 1 inch.
Certain of the articles mav be tested to destruction with
the view of ascertaining the fitness of the castings for the
lervioe intended.
ADMIRALTY TESTS. 277
(hdimmtf Malleable Cast Iron, — Caating^ eliftU stand being
dropped ou a 9lab of cast iron, qp one of equivalent Hardness,
item a HelffHt of 10 to 20 feet. Strips ont from the castings
are to stand being bent cold without fracture through an ancle
of 45**, the internal radins of bend being not greater than Sie
thickness of the piece tested. Pieeea from the castings may
be broken to show the extent of annealing.
Natal Brass.
Navid braas is to be of the followincr eomposition : best
new selected copper 62 per cent^ tin at least 1 per oent, the
xemaindeF zinc. In no case will Naval brass bo accepted
having kee than 61 per cent copper. The impurities shall niot
exceed f of 1 per cent.
All Naval brass articles are to have good^ dean, and smooth
surfaces, free from black oxide, blisters, and internal spongir
aessy and are to be hard-rolled cold- Naval brass rods may be
extruded as may be approved. The bars are to be tested
withonii annealing, and i^e test pieces are not to be annealed.
JBvtrs,^*(l) All Naval brass bars are to be cleaned and
straightened. (2) They are to be capable of being hammered,
hot, to a fine point, (3) They must stand being bent»
cold, without fracture through an angle of 75^ as follows :
The test piece will be placed on two supports 10 inches
apart and forced down m the centre by a die, which has
a ntdiuB eqnal to the diameter tested. For bars over 1| inches
in dtameter or thicknes^ this test is to be carried out on
a piece of 1^ inches in diameter or thickness, selected from
the outside portion of bar, and bent so that the original
outside surface is in tension. In bara not of circular section,
the corners may be well rounded oS before the bending test.
(4) A fa£Bicieni number of bars are to be nicked at the ends
and broken so as to satisfy the Inspecting Officers as to their
general soundness. (5). They must stand the following tensile
tests, which are to be taken nromthes^me bar as bending tests:
Round and hexagonal bars fin. diameter and under are to
have an ultimate tensile strength of not less than 25 tons per
square inch. Elongation to be not less than 20 par cent in
2 inches. Round and hexagonal bars above | in. diameter are to
have an ultimate tensile- strength of not less than 22 tons
per square inch, whether tiHrned down in the middle or not.
Blongation to be not lals tiian 20 per cent in 2 inches.
Bars of any other seotion are to have a tensile strength of
not lest' tiian 22 tons per square inch, and for elongation are.
to be treated as round bars of corre^KOiding sectional size.
Bkee^.-^AU Naval braas sheets of } in. thickness or less
are, if ordered to be supplied axmealed, to be capable of being
doubled, ooldj to a eurve the inner radius of which is not
^roati»r than the thickness of sample without fraoturoi and
278 ADMIKALTY TESTS.
to stand a tensile stress of not Idss than 26 tens p» square
incli, with an elongation of not less than 30 per cent in
2 inches. Hard-rolled sheets are to stand without annealing
a tensile strain of 26 tons per square inch, with an elongation
of 25 per cent, and a bending test through an angle of 135^.
The bending test may be carried ont by bending up to a right
angle in a suitable vice, and completed by mallet on a smooth
anvil or other means.
Sheathing sheets are to be annealed and thoroughly cleaned
after rolling, and to have a clean smooth surface nree from
buckling.
Plates, Tube, and Diaphragm above }tn. in thiehnen. —
Surfaces are to be flat, smooth from the rolls and free from
scoring, laminations, pitting, or cracks. The billets are to be
machined on both sides either immediately before or after
" breaking down " ; all defects to be " dressed ont " before
the rolling is proceeded with.. Test pieces are not to be
annealed.
Plates above f in. and up to | in. thick to stand bending
tiirough an angle of 135^, uie internal radius of bend being
not greater than the thickness of the plate, the bending test
being carried out as described under brass sheets above.
They are to stand a tensile stress of not less than 26 tons per
square inch, with an elongation of not less than 20 per cent
in 2 inches.
Plates above } in. and up to } in. thick ; the bending test
angle is to be 120^. Tensile test to be 24 tons per square
indi with an elongation of 20 per cent in 2 inches.
Plates above f in. thick ; the bending test angle is to l!>e
90^. Tensile test to be 22 tons per square inch with an
elongation of 20 per cent in 2 inches.
Naval brass sheets and plates must also satisfactorily pass
a hot forging test.
Naval Brass Castings, — ^Test pieces turned, must have an
ultimate tensile strong^ of not less than 10 tons per square
inch, with an elongation of not less than 7} per cent on
a length of 4 diameters (test piece Z).
GUN-M£TAL.
The gun-metal used in the manufacture or any article,
except where specially approved, is to be of the following
composition : copper not less than 86 per cent, tin 10 to 12 per
cent, zinc 2 per cent maximum. The whole to be of godd
clean metal free from any admixture of lead.
Tensile Tests. — Pieces taken from the castings, prepared
in accordance with test piece Z, are to have an ultiniate tensilt
strength of not leas than 14 tons per sqnara inch, with arf
elongation of at least 7^ per oent on a length of 4 diameters.
admiralty tests. 279
Phosphob Bronzb.
All phosphor bronze is to have the following composition :
copper 83 per cent, tin 10 per cent, phosphide of copper
7 per cent. If preferred, the composition may be coppei*
90 per oentf phosphor tin 10, the latter to contain 6 per cent
phosphoras.
"Lsst pieees knust have an ultimate tensile strong^ of
15 tons per eqaare inch, with an elongation of at least 10 per
oent in a length of 4 diameters (test piece Z), and on analysis
to show not less than 0*3 per oent of pliosphorus.
Should an especiiUly hard material for bearings, etc.^
be desired, the composition of the phosphor bronze may b0
made, copper 85 ^er cent, phosphor tin 15 per oent, with
a (diemlcal analysis showing not less than 3 per oent of
phosphorus. In this ease the tensile strength must be 7} ton?
per square inch, with an elongation of not less than 1 por cent
on test piece Z.
OUDINART BSASS.
All brass articles of minor importance such as label plates,
buttons, hooks, etc., are to be of the ccnnposition best adapted
to the uses for which they are severally intended, but no
brass articles will be accepted which are found on analysis to
contain more than 3 J per cent of lead.
Copper.
Coppar used in the manufacture of any copper articles is
to assay not less than 99*3 per cent. The quality of the copper
may be tested by the following Huntz metal test : — 3 lb. of
the copper will be placed in the melting-pot, covered with
pieces of hard wood or charcoal, to prevent the loss of zino
when added. When the metal has melted 21b. of zinc will
be added, and the mixture stirred and run into a cake about
4 inches square in an open iron mould. When the cake has set
it will be allowed to cool gradually in air, and when cold it
will bo nicked with a cold chisel, and broken carefully to show
the fracture. If the cake be tough and break with a fine silky
fracture the quality is considered good, but if it break short
with a coarse stringy fracture, and with a yellow colour, the
quality is considered bad.
Copper Pipes. — Copper pipes are not to be made by the
electric depositing process, and copper so made is not to be
used for their manufacture unless it is remelted.
Strips cut longitudinally from pipes after annealii^ in
water are to have an ultimate tensile strength of not less than
14 tons per square inch with an elongation in a length of
2 inches of not less than 35 per cent.
280 ADMIRALTY TESTS.
Strips cut longitudinalljT and traaflversely are to staud
bending cold, double, the internal radius of bend being not
greater than the thickness of a strip, if unannealed, and until
tne two sides meet, if annealed, and in the latter ease to be
hammered to a fine edge without cracking.
Each tube is to be tested internally by water pressure,
without leakage or permanent increase of diameter.
No pipe is to be less at any part than the thioknesi ordered,
nor should its weight exceed by more than 7^ per eeat that
calculated to be due to its dimensions, taking 5551b. per
cubic foot.
Flanges of all copper pipes are to be of the following
mixture : copper, 85 per cent; zino, 15 per cent.
Copper Sheathing. — (Copper sheathing includes Bheeti of
12, 16, 18, 28, and 82 oz. per square foot. The Munis metal
test, as specified above, is to be made on selected sheets. The
sheets are to be hot rolled to about f the finished length.
To be hard rolled cold and afterwards annealed and cleimed.
The oxide scale is to be completely removed before the oold
rolling process and also after the final annealing. The
finished sheets are to have bright, clean, and smooth surfaces
perfectly free from black oxicle or discoloration. The edgeji
are to be neatiy sheared and the sheets equal in softness and
finish, and in all respects conformable to the pattern sheet.
The nail holes to be marked in the 32 and 28 oz. sheets.
Copper Sheet.' — Strips cut longitudinally from and portions
of copper sheets are to have, after annealing in water, an
ultimate tensile strength of not less than 14 tons per square
inch, with an elongation of not less than 30 per een^ in
a length of 2 inches* Strips cut lengthways and crosswise are,
if unanneaJed, to stand bending double, the internal radios
of bend being not greater than the thickness of the strip.
If annealed, tibie strips must stand bending until the two sides
meet, and hammering to a fine edge without eracking.
Copper J9ar«.-^AU ooppgr bars to be hard relied, cleaned,
and straightened. After annealing, to have an ultimate tensile
strength of not le^s than 14 tons per square inch, with an
elongation of not less than 30 per cent in a length of 2 inches.
The bars are to be capable of being bent oonipletely double
without Iractare, the internal radius of bend not being greater
than the diameter or thickness of the bar. They are also to
be capable of being hammered hot to a fine point.
Lead.
Sheet lead is to stand, without injury, cutting or bossing
up, or any other usage test that may be considered necessary.
Lead pipe is to stand, without bursting, a water pressure
test of 300 lb. per square inch up to 1^- inch diameter^ and
ADMIRALTY TESTS. 281
2001b. per square inch above 1^ inches ^nd up to 4 inches.
Tho pipe is to have sufficient ductility to admit of turning or
flanging at the ends to double the internal diameter withouti
splitting.
Zinc.
The Eino for Proteoiort, 4tc, — ^Zine must not contain more
than 1*1 per eent of lead.
Wood.
All woodworlc to be well seasoned and free from objection^
able shakes^ sap, defective knots, etc. Deck planks to be
free from heartwood. All teak is to be East Indian.
India-rubbe»
(other than that used exclusively for machinery purposes')*
(a) The vulcanize^ india-»rubber is to be of a boaofeiieQiif
character throughout, as evidenced by microscopical examina-
tion, is to be thoroughly compressed, free from air-holes^ p^nB9>
and all other imperfections, is not to roontain any oruial»
rubber, recovered rubber, or other treated or waste rubber,
or rubber substitute of any kind, and \a to stand the tests
mentioned below without its quality being imp^red.
(fi) The quality of the caoutchouc used for all vulcaiilfled
india-rubber goods described — subject to the exceptions named
below — ^must be of such a character that after it has been
made up into the vulcanized and finished article as defined
above, not more than 10 parts per cent of organic matter and
sulphur, calculated on the non-mineral matter present, oen be
extracted from the rubber by boiling it for six hours in a
finely-ground condition with a 6 per cent solution od^ alooholie
oauatic potash.
(^o) Although maximum percentages of sulphur are IMuned
in the specification the quantities used should be as locw as
possible consistent with prqper vulcanization.
(d) Where the use of pure best quality eftou^hono in
prescribed it must be of such quality that not more than
6 parts per cent of the organio matter present can be
extracted from the rubber bv boiling it for sis hours in
a finely-divided condition with a ^ per cent solution of an
alcohouo caustic potash.
Qualityi 8. Vulcanized india-rubber sheet or valves, efo.)
for purposes requiring considerable elasticity, to be made of
pure caoutchouc, of the quality specified at (5) above, and
with no other ingrediei^ts than sulphur, the proportion ot
which is not to exceed 4 per cent reckoned on the manu-
factnred rubber ; is to endure a dry heat test of 270* F
for two hours without impairing its quality.
282 ADMIRALTY TESTS.
Quality Sa, Vulcanized india-rubber sheets ^ valves,
washers, or rings, etc, for side scuttles, electrio light, and
hose fitting, etc., to be made up of pure caoutchouc, of tho
quality specified at (6) above, and with no other ingredients
than sulphur and white oxide of zinc ; the sulphur is not
to exceed 3 per cent, and the oxide of zinc is not to exceed
40 per cent, reckoned on the manufactured rubber ; to endure
dry heat test of 270** P for two hours without impairing its
quality.
Quality 8ft. Vulcanized india-rubber sheet or washers, ete.f
including armour bolt washers, to be made of the same
ingredients as specified above for 8« quality rubber, except
that the oxide of ziuo is not to exceed 50 per cent, and the
sulphur is not to exceed 2^ per cent.
Quality So, Vulcanized india-rubber sheet, valves, or
washers, etc, to be made of the same ingredients as specified
above for 8a quality rubber, except that the oxide of zinc is
not to exceed 60 and the sulphur If per cent.
Vulcanized india-rubber tubing, — ^Except for special re-
3[uirements the india-mbber for tubing is to be made of the
ollowing composition : Pure caoutchouc, of the quality
specified at (b) above, sulphur and white oxide of zinc. No
other ingredients whatever to be used in its manufacture. The
sulphur not to exceed 3 per cent and the oxide of zinc 30 per
cent ; to endure a dry heat test of 270** F. for two hours
without its quality being impaired. The canvas used in the
manufacture of all the tubing subjected to internal pressure to
be made of flax or fine hemp. The tubing is to satisfactorily
withstand the stated pressure throughout its entire length,
and is to be tested accordingly before being received.
Vulcanized india-rubber mats, perforated, to be made of
pure caoutchouc of the quality specified at (eQ above and
with no other ingredients than sulphur and the oxides of lead
and zinc ; the sulphur not to exceed 2| per cent, and tfaift
Oxides of lead and zinc 60 per cent, in equal proportions ;
to endure a dry heat test of 270' P. for two hours without
its quality' being impaired.
India-rubber Solution. — ^The solution is to consist of pure
best quality caoutchouc dissolved in good solvent mineral
naphtha, which is free from tarry matter, and which is com-
pletely volatile at or below 290® P.; 100 parts by weight of
the solution must contain not less than 13 parts by weight of
rubber. The total mineral matter in the rubber solnlion most
be under 01 per cent. Sulphur must be absent.
Canvas. -
All canvas articles are to be of good fit and well made.
Tho tensile test, weight, etc., of the several numbers of
canvas to be ai follows : —
Af^MI&ALTY TESTS.
28»
No.
1 E.N.
No. 4M
6
N.
>9
»
$f
Minimnm
Breaking -Stren.
Weft.
Warp.
lb.
lb.
480
840
460
320
400
280
350
250
390
330
240
170
210
150
193
140
WeHrhtper
Bolt of 89
Yards for l-6«
and 40 Yards
for 7.
Remarki.
lb.
46
43
36
30
27
85
80
27
To be spun
whollj from
long flax.
To be spun
wholly from
flax.
Ijenfffh of strips for testing 2 feet by 1 inch for Not. 1 to 6. and 2 feet by-
li inches for No. 7 B.N.
Aiilcles made of canvas are to be well sewn together with
best flax twine coated with a composition of five parts of.
beeswax, four parts of palm-oil, and one of resin ; if hand-:
sewn, to contain not less than 120 stitches to the yard.
Where holes and thimbles are fitted, the latter are to be of
gunmetal ; aU brass gromets to be of spur teeth pattern.
All painted articles to have three coats of best paint.
Manilla Cobdagb.
Ifantntt eordage ts iio conTomir to "the xable flheifii beioWy
and to stand the brei^king strain stated therein. -
Fathoms. Defreos.
Length and angle at which strands are to
be when formed ^ . . • 142 27
Length and angle at which strands are to
he when hardened . • • • 134 83
Length and angle at which rope is to be
when laid. ...... 118 39
284
LLOYD'S TESTS.
Size of
Rope,
DescriptioT)
of Tarn.
Totta
Nunber
of Yamn
in Rope.
Weight per Coil of
118 Fathoms.
Standard Breaking
StFftfal.
In,
i
40 thread
6
Tons
0
. Owt. Qrs
0 0
.Lb.
To
Tons Cwt.
0 3
Qrs.
3
Lb.
0
i
»
12
0
0 0
20
0
7
2
0
1
it
U
0
0 0
25
0
10
0
0
n
79
21
0
0 1
5
0
12
2
0
n
a
33
0
0 1
24
0
18
3
0
If
»
42
0
0 2
10
1
5
0
0
2
jj
54
0
0 3
1
1
13
0
0
2}
»
66
0
0 3
20
2.
2
0
0
^
»
84
0
1 0
20
2
10
a
0
2i
»
102
0
1 1
21
3
2
0
0
3
»
120
0
1 2
21
3
15
a
0
3i
30 thread
123
0
2 1
6
4
17
0
0
4
>♦
159
0
2 3
25.
6
5
0
0
^
»
201
0
3 3
1
8
1
0
0
5
»
249
0
4 2
17
9
17
0
0
6i
»
303
0
5 2
18
12
3
0
0
6
)i
351
0
6 2
23
14
7
0
0
61
2S thread
360
0
7 3
14
16
0
0
0
7
>»
408
0
9 0
17
18
10
0
0
n
}>
468
0
10 2
3
21
5.
0
0
8
»
534
Q
11 3
25
24
a
0
e
9
j>
675
0
15 0
14
30
0
0
0
10
)}
834
0
18 2
22
37
10
0
0
11
>»
1,008
1.
2 2
12
45
12
0
0
13
»
1,40?
1
11 '■ 2
6
63
15
0
0
LLOTD'ft TESTS 70B MATSBIAL8.
■ 8ieel Piat^.^^To be made by the open-hearth process, acid
or basic. To be finished fi^ee Ifom oraoks, sarfaoe flaws, and
lamination. The tensile tests shall be made on a test piece
prepared similarly to tiiat described as * A ' ( Admirali;^ tests,
p. 265). The ultimate strength shall lie between the Hmits of
28 and 32 tons per square ineh. The lover limit may be
26 tons for plates specially intended for cold flan^fing ; the
tensile tests may be dispensed with in material used where
strength is unimportant. The elongation in 8 inches shall be
at least 16 per cent below f in. thick, and 20 per cent for
thicker material.
The bend tests are the same as for the Admiralty (p. 265).
Steel Bars. — As steel plates, but the upper limit of tensile
strength may be 33 tons per square inch.
RIVlftTED JOINTS. 285
Steel Eivet9.—As for the Admiralty (p. 268), «iO*pl that
bars are to stabd 25 to 30 tons per square inch.
Steei Castingi, — The test pieces for tensile and bend tests
are to be made as described in the Board of Trade Rnles on
p. 449. The ultimate tensile strength to He between 26 and
35 tons per square inch, with an elongation of 20 per cent.
To stand bending cold tiirough 120^^ with an interna! radius
of bend of 1 inch.
St^m frames cast in olie piece to be raised through 45%
and let fall on hard ground rooe^sed as necessary. Other
important castings to hd dropped thron^k from 7 to 10 feet.
Afterwards to be slung up, and hammered all over with
a 7 lb. or heavier hammer to test the soundness of ^e casting.
Ingot Steel Forginge, — General requirements as for
Admiralty (p. 271)j but the test pieces to be as for steel
castings above. The tensile breaking strength to be between
28 and 32 tons per square inch, with an elongation of 20 per
cent for 28 ton steel, aad 25 per cent for 32 ton steel ; in
no case must the sum of the tensile breaking strength and the
percental^ elongation be less than 57.
^e bend test piece must stand bending over a radlas of
4 in. through 180*.
Use of Iron.
The rivets, keel, stetn^ rudder, pillars, etc., also the floors,
glrdcurs, and in&etf bottom in boiler space may be made of iron
withovt iacarease of size. Deck plating, floors, double bottom
struotsre lA holds, bulkheads^ eagine casings, bulwarks^ and
de<^ h<>ttsei miiy h% made of ko» 10 per oent thicker than the
steel specifled. Iron to be of good malleable quality, and
subjected to shipyard tests.
For Board of tncto testa lor maierialt for boilers, eto.>
s^ p. 449.
BiviTXD jonrxs.
DBsnm OP Bimplb Bivbted J^mTS.
A Mm^k riVateA {dint nAy fail In lev«ral wayl, aa
instaaeed by the Ik^ joint with a single rivet shown la
fig. 161.
Fig. ICI.
» \
!b 1 -^
* I »
1 ■ ■ I ■'"■■f-rfi-t
B
(1) Plate 6an tear abng AB.
(2) llivet 6ah shear.
286
STRENGTH OF RIVBT8.
(8)
Plato ofto crush in front of rivet at 0.
(4) Plate can tear along^ CD. This last ia prevented when
CD in the clear is at least equal to the rivet diameter.
^ 3=3 thickness of plate in inches.
d SB diameter of rivet hole in inches (^ in. to ^ in.
more than rivet diameter) •
5 Bsi breadth of plate in inches.
Psspull on joint in tons.
(1) P/(6-eQ^== tensile strength of plate (26* to 2S mild
steel).
(2) P/*785^2 «: shearing strength of rivet (about 22 mild
steel). '
(8) F/dt cs bearing strength of rivet (about 40 mild
steel).
Shearhto akd BEABma Values or Bivets (Wosiono).
Mild Steel.
The stresses allowable in tons/inch' are taken as :— *
Under single shear, shear 4, bearing 8.
Under double shear, shear 7*5, bearing 10.
These correspond to a factor of safety of 5 ; the corre*
spending allowable stress in plate would be 5 (Admiralty
quality) or 5*5 (Lloyd's requirements).
In the table for single shear the bearing area has beeitf
assumed increased by 6 per cent to allow for the cone caused
by punching ; if the holes are rimered or drilled, reduce the
bearing values by 6 per cent. If the rivets are countersunk
increase the bearing values under single shear by 18 per cent.
Under dead loads a factor of safety of 3 is permissible ;
multiply all shear and bearing vallies by 1}.
MUd Steel, eingle ehear, punched holee*
limed
teterof
ole.
1
^
Bearing Valae of Rivel.
Weight of Plate in lb. per foo0.
s
3|.
1
1!
an
Tons
10
Tons
121
Tons
16
Tons
Tons
20
Ttons
25
Tons
30
Tons
35
40
Inch
Inch
Incb«
Tons
Tons
i
yN
'2466
•99
ri7«
1*46
1-76
2 05
2*34 2*93
3*52
4-10
4-Gl
A
1
•8068
1-23
l-30*,l-62*
1-95
2-23
2-60 3*25
3*90
4*55
5*21
i
»
•3712
1*48
215«
2.60
2-86 3*68
4*30
6-01
5*71
H
f
•4418
1-77
1*66 1-90
2*34*
2.73«
3-12 !8'90 4-68
6*46
6-21
1
H
•518S
207
1-70 »212
2-64*2 96*
3-39*4*24 I5-08
6*93
6*73
1
H
-6903
276
1-96
2-44*2-93 13-42
3*91* 4-89* 6-86
6-84
7*81
1
1^
•887
366
222
2-78 *;8-83"jS.88
4-44 6^55 6*66*
7-77*
8-68
11
lA
1107
443
2*46
3-08 '8-69
rsj"
1*92 6*16
7*88
8^1«
9*84<
U
lA
1-363
6-41
2-74
3-42 4-11
4.79
:V47 j6*84
8-21
9*68
10*94
STRENGTH OF RIVETS,
287
Mild Steel
, double
shear, drilled holes i
•M
%-i
■
9
1
Bearing Yalae of Rivet.
ianieter o
Bivet.
/Lssaxned
lameter o
Hole.
a «
* o
Weight of Plate in lb. per fool.
1
Inch
«
»4
2*-
QQ
Tons
10
Tons
121
15
17J
Tons
20
Tons
25
Tons
30
Tons
36
Tons
40
Inch
Incha
Tons Tons
Tons
i
^
•2486
1-86
1-38*
1-72 3-07
2^41
2-76
3-46
4*14
4-82
5-52
^
a
•8068
2-30
1-63*
i-gi*
2-29T2-68
3*06
3*82
4-59
5-36
6-1J5
i
H
•3712
2-78
1-68*
2*11*
2-53*}2-96
3*37
4*21
5-06
5-90
6-74
H
3
•4418
3-31
1-84
2^30
2-76*
3-24*
3*68
4-60
5-52
6*44
7-38
I
H
-5185
3^8
1-99
2-49
2^98*
3*48*
3*98*
4^97 5-97
6-96
7-96
i
1
-6903
•887
6-18
6-66
2-30
2-60
2-87
3-26
3-45
3*91
4-02
4^56
1=6^
5-21
5-75* 6-90
8-05
9-12
9-20
10-42
6-51
7-81*
li
1107
1-353
8-30
10^15
2^91
3-22
3*64
4-02
4^37
4-83
5-10
5-63
5-82
5-44
7-29
8-15
8-74
10-19*
11-27
11-64*
12-68
HT
^ Note. — ^The bearing valaes to the right of the zigzag lines
are greater than the shearing values, those to the left are
less. To get the full value of the rivet shear, rivets must be
selected from those to the right of these lines.
The values marked* are for rivets of the sizes usually
adopted with the corresponding plates.
The table has been made complete, although certain rivet and
plate sizes included therein would not be used together in practice.
High Tensile JSteel-^Cruiser quality (.H.T.y strength 34-38
tons/inc/i^).
Add 30 per cent to aU the values given above for mild steel.
High Tensile Steel — Destroyer quality (H.H.T,^ strength
37-43 tons/inch^).
The stress allowable in tons/inch' are taken as : —
Under single shear, shear 7, bearing 9.
Under double shear, shear 13, bearing 11.
These correspond to a factor of safety of 5 ; the corre-
sponding allowable stress in the plate would be 7« It is
assumed that the rivets are also of H.H.T. steel.
All holes are assumed drilled. When countersunk increase
the bearing value by 25 per cent.
Under dead loaas multiply all shear and bearing values
by If.
2S8 gtREXGTH O^ RIVETS.
H.H.T, Steel, single shear, drilled holes.
Diameter of
Rivet.
Asstuned
Diameter of
Hole.
ft
«
o
W
■s
at
1
Shearing Value
of one Rivet.
Bearing Value of Rivet.
Weight of Plate in lb. per foot*.
6
n
10
I2i
15
17i
20
Inch
. i
h
^«
ft
H
S
Inch
1
!
Inch a
•0928
•1296
•1726
•2486
•3068
•3712
•4418
•6185
Tons
•64
•91
1-20
1-74
2^16
2^60
3-09
3'63
Tons
•38*
•45*
•62
•62
•69
•76
•83
•90
Tons
•57
■68*
•78»
•93
1-04
1^14
1-24
1*34
Tons
•76
Tons
•95
1^13
1^30
Tons
1*14
1^86
1-56
1*86
Tons
1^33
1*68
1-82
2*17
2^42
2-66
3^13
Tons
1-62
1^T9
2-07
2*48
2^76
3*04
13*31
1-04
1'24*
1^38
1-52
l-W
V79
1-55
1'73*
1^90*
2^07
£•24
2'28*
2-49
2*69
J
V^*
S.S.T. Steel,
double shear, drilled holei
r.
Diameter of
Bivet.
sss
Ui
^5
•
a
2
<
Bearing Value of tlivet.
Weight of Plate in lb. per foot^.
6
n
10
m
16
m
20
Inch
i
1
ft
H
i
Inch
ii
1
»
1
H
Inch*
•0928
•1296
•1726
•248S
•3068
•8712
•4418
•5186
Tons
1*«1
1-68
2^24
s*2d
3^99
4-83
5^75
6*73
Tons
•46*
•66*
•63
•7«
•84
•93
l-Ol
1*10
Ton<$
•70
•82*
•94*
1*14
1*26
1*40
r62
1-65
Ton3
•93
1^10
X-26
1*62»
1-68
1-86
2-02
2-20
Tons
1-16
1'3'r
1^57
1"96
2*11*
2-32*
2-63
2^76
Tons
I'Sb
Tons
1-92
Tons
im
2-19
^•53
^BT
3-37
r7i
4*05
4-39*
1-66
1^89
2^62
2^79*
3^08
3^90
2-«6
2-96
3^2«
3-65»
3-86
See notes under tablet f^r mUd steel.
Note alflo that all practicable suset of rivets in double shear
give Way by exoesSive bearing ]^reasu»e ; double batt-straps
a»e therefore of little use in this qoality steel.
MivHs in Tsfmdn,
The value of such rivets depends very much on the size of
the head and point. With hammered points the stress per
square inch of hole area is about three-quarters that ordinarily
RIVET SPACING. 289
allowable in the material^ provided that the pull is even sd
^liat there is little or no bending action on the rivet. This
gives a working stress of 4 (mild steel) or 5^ (H.H.T. steel)
tons per square inch under live loads.
If the pull is uneven take '45 the usual stress, i.e. 2^ (mild
steel) and SJ (H.H.T. steel) tons per sqffare inch. In all
cases the stresgih is uneertain, for it largely depends on the
character of the workmanship.
For areas of holes for different-sized rivets see tables above.
Spacing op Rivets.
(^Lloyd's.')
3} diameters (centre to centre) in butts of outside plating
and beam stringers (except quadruple butts).
4 diameters in edges of outside plating, quadruple riveted
butt laps and double butt straps, butts and edges of inner
bottom, butts of decks, margin and tie plates, girders and
floors.
4 J- diameters in gunwale and margin plate angles, edges
and butts of bulkheads, edges of decks, angles between webs
and side stringers.
5 diameters in angles to flat keel and connecting floors to
centre girder, bulkhead framed where caulked, butts and edges
of mast plates and floors.
6 diameters in deck plating to beams fitted to alternate
frames.
7 diameters in frames (to be 6 diameters to outer bottom
where depth is 11 inches or more, or spacing 26 inches o«r
more), reverses, floors, keelsons, beam angles, deck and hold
stringer angles, bulkhead stiffeners, various angles, and
(generally) deck plating to beams.
{Admiralty.^
Maximum : 4^ diameters In oil-tight work, 5 diameters in
W.T. work, 8 diameters otherwise.
diameter
er butts.
Clear distance of rivet from edge of plate = one di
+ } in., generally, but one diameter + J in. in destroyer
Thickness of Butt Steaps.
Lloyd* s, — Up to -30" same as plate; above '40", 1«25 plate
tbickn^s; proportionately between *d(f and •40". For double
straps total thickness less thickness of single strap is about • 10" for
i" plate and '02" for Ij;" plate and so intermediately ; if one strap
is countersunk make it about '06" thicker than the other.
^^if»»r/i/^y.— Generally same as plate. Exceptionally two
straps are used with combined thickness 1| times plats
thickness.
For VY.T. work one strap should generally be used.
U
290
SIZE OF RIVETS.
00
i
I
cB
IB
cB
§
OS
I
bo
u
o
eiaiiis
fl O m g9^.S OB
??W<N I1?3JS? llfS'^'5?
•AU
eiqnoa
i ^f^«^ I '&'^(^ I tS'<D^S^
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eiqejJi
eec^io I ^<o«? I ^?^g^
ca
i
•AU
•ATJ
eiqnoQ
1r^«o^ I t^So* I ^a^jg
•AU
oiqwji
fl « 9
*8d«I 9Sp9
•AU oiSuig
ua to
. ? S 9
'SWWw m^llfeaco I
*ed«i espa
'AI2 aiqnoQ
I 1 I I I i ^ ! '9 1 TS'.o^S^
'fld'BX 44Qq
•AU 9I8U|S
^^Tf^*^ i 'd^ 1 CO I MM
■Sdvi ^^nq
^r^«'S 1 ^9« I .o«« I I i
■Bdvi |9nq
■Apxeiqojj^
M M M 1 I'^l^^sf
•sdvifs 9<^iiq
*A|2 eiqnod
•Aft eiqaij,
H4 W^ '4c*
M I M M iHlS^SJSS
M M M I M I I I TT
31383 SSI £S$^s:3
'%9A-TiI JO lo^mvfcx
♦ * ♦ » ♦_ ^. jH. .
CTtQ
«
rts
;s2
«
2 00-^
fl _ •
o
S aO
I SS
i^S
9 ^
** "C *•
iw *
J^ ■*> H
S 18
S •
■t 4
••• -^
5^
PROPORTIONS OF RIVETS. 291
Countersink of Rivets.
Lloyd' % (for outside plating).— The proportions are as
shown in fig. 162. The depth of ooontersink is ^ plato
thickness when '6Q" or more ; if less, to be conntersank for
full depth.
Fio. 162.
i*RiVET f Rivet , iRtVET
For yaehts, the holes of rivets smaller than I" are as follows : —
in. in. in. in.
Diameter of rivet • • ik I iV \
hole . A A A A
countersink A iJ « M
Admiralty practice* — ^Angle of ooantersunk point is equal
to or slightly greater than that of countersunk neck (see
below). Depth for thick plates about A" ^^^ ^k^n that
of plate.
Lengths and Proportions of Bivets.
Admiralty dimensions of 1 inch rivets of the several
descriptions are specified in the schedule below.
u — li"— -J
03
la*
292
RIVETED JOINTS.
S<HBDrLE OF » 'ImENSlUNS.
>
I
Inch
p
With Pan or
Snap Head
and Straight
Necks, to Draw-
ing D or F
With Counter-
sunk Head,
to Drawing A
With Pan or Snap Heads and
Conical Necks, to Ihuwing b or o
11
Inch
H
5
Inch
1
I
M.
I
^1
Inch
Inch
tV
t
1
Inch
3
Inch
o
A S
Inch
* -^v. lo. for snap heads. ** i In. for pan heads
t ^ in. for pan heads. ^ l-^ in. to* pan beads.
Zloi/d's. — ^The rivets specified for use in outside plating
have pan heads and conical necks with the following
dimensions : Diameter of rivet d, diameter of t6p of head d,
diameter of bottom of head I'Qdy depth of head '7d, diameter
of top of neck l'l2d, depth of neck to suit ihiokneas of
plating.
Lenqtqs foe Ordebino.
The lengths under heads of various descriptions of rivets
for two thicknesses of plates may be determine as follows :--
Countersunk points about f diameter greater than length
of hole ; snap nydraulio points about 1^ diameters greater
than length of hole ; rough hammered points about } of
the diameter greater than lengtih of hole.
For three thicknesses allow ^" more ; and for any nnnsual
length proportionately.
Design of Riveted Joints in Gexebal.
Treble riveted butt connections may bo broken in five
different ways, see fig. 162 A.
(1) The plate may break through the line a a,
(2) The strap may break through the line b b.
(3) The plate may break through the line oo and shear
rivets r,r.
BIV£T£D JOINTS.
293
(4) The butt, strap may break through line ee and ahear
rivets 8, «.
(5) All rivets may shear on one side of batt strap,
tlsually (3) and (4) need not be considered.
Fig. 163a.
The strength en the basis of (1), (2), and (5) should be
about the same ; frequently the strength with (5) is made
rather greater than the other two. To obtain deduction for
rivet holes from plate area, increase diameter of hole by
6 per cent if punched, and by 25 per cent if countersunk.
The least rivet spacing desirable is about SJ diameters
between centres.
Example, — A 301b. stringer plate 4 feet wide is to have
a double butt strap. Design the joint, given that the plate
is already pierced with beam rivets spaced 8 diameters.
Take 1" rivets or 1^" holes. Assvime holes ion beam rivets to
be countersunk ; there are 6 rivets. Working strength of plate
at 6 tons/inch^ = 6 x i x (48 - J} x IJ x 6) = 1«0 tons. Taking
mild steel double shear, shear ralue (from table, p. 287) per rivet
is 6*65, and bearing value is 7*81. Take the smaller. Least
number of rivets is 150/6 '65 = about 23.
A convenient arrangement will be found with 28 rivets
in rows of 6, 11, and 11. Thei outer cow having only 6 rivets
makes plate when breaking as in (1) as strong as through
beam holes.
To get combined thickness (t) of straps, allow punched
holes in one strap and countersunk holes in the other. Mean
percentage increase of hole diameter is ^ (6 -|^ 25) or 15.
Hence <x (48 - 1-16 x iJ x 11) = ifA, or i = -87. Straps
should each be •43" or approximately 17^ lb<
Note, — ^If the plate is unweakened, the strongest possible
strap is diamond-shaped.
In tiie joints (figs. 162b and o) the 8treng<h of the plate is
only weakened by the extent of :o»e rivet hole. Double butt straps
should be fitted if this standard of strength is to be maintained.
Zigzag Riveting,
The clear area of the plate between zigzag rivets should
be 35 per cent greater than that required directly across the
294
BRACED STRX7CTURES.
plate. In high tenBile steeli destroyer quality, thiA per-
centage should be 60.
Thia apparent weakening along the diagonal shonld be
allowed for when determining the rivet spacing of ordinary
joints, if for any reason the rows are placed unusually dose
together.
Fig. 163b.
Fig. 162c.
Effeot of punching, driiling, and annealing.
In mild steel punching lowers the ultimate strensfth while
drilling raises it. Since the latter effect can only be due to
circumstances which do not influence the elastic or working
stress, it may be inferred that the real loss of strenja^th due to
punching is about 20 or 25 per cent. This is partly restored
by hot riveting, and wholly restored by annealing or rimerin^.
In high tensile steel punohing'is even more deleterious.
BRACED STRUCTURES.
Certain types of structures or girders' consist of a number
of pieces jointed together by pins. If the joints can, without
large error, be regarded as frictionlqss, and each piece or
member has no more than two pin joints, the stresses are
wholly those of tension or compression along the lines joining
the pin centres, provided that the loads are assumed con-
centrated at the joints.
If number of members =» (2 X number of pins) — 3, the
structure is termed a perfect frame. Its stoesses eau be
determined by elementary statics.
If the number of members is leas than that g^ven by the
above formula, the structure is termed an incomplete frame.
It is free to move, and the stresses depend on the position
taken up, and conversely ; e.g. chain, suspension bridge.
If the number of members is greater than that eiven by
the formula, the structure is termed an overbracea frame.
It may be self -stressed, and unless some members be
PERFECT FRAMES.
295
disregrc^d^ >B redundant, the detennination of the stresses
involves a knowledge of the elastio properties of the material
of the members ; e.g'. many types of boats' slings*
Perfect Frames.
If the loads are not placed at the joints, divide that on each
member into two equivalent portions at the two joints.
To find the stresses by a reciproocd diagram, — Determine
the external reactions (if unknown) and represent them by
lines on the diagram. Let the spaces on the diagram be
lettered by large letters a, b, o, d . . . separated either by
members of the frame or by external reactions. Then ab
represents the reaction on the member separated by the
Fio. 168 a).
Fio. 168 <2).
b
8]|f aces A and B, and so on. Commence a " reciprocal "
diagram, making a known force ab represent to scale the
force AB^ and parallel to it Tfig. 163). Then draw ad in
the reciprooal diagram parallel to ad, and bd parallel to bd.
Let ad, bd intersect at d. Then these will represent the
stresses in AD and bd respectively. Continue thus with the
next letter, and so on until the reciprocal diagram is finished}
and all the stresses are known. A simple examination will
show which members are in tension, and which are in
compression.
To find the stresses by the method of sections. — ^To deter-
mine tiie stresses in any one member (as, for example, in
certain cases when the above method is inapplicable), divide
the structure in two by a line cutting through not more than
three members^ including the one whose stress is desired.
296
SHIPS DERRICK.
Find ther point of intersection of tlie other two menoibera
(produced if necessary) and take moments about it for all
the external forces on one side of the section, including the
unknown stress in the member cut. Then the stress in the
member is determined hj the fact that the algebraical sum
of the moments is eero. If the other two members are
parallel, resoWe the external forces perpendicular to their
direction.
Example, — In fig. 163, to find the stress in de^ draw
-^ section xy cutting through it, and through ae^ bd in
addition. These two latter intersect at r. By taJcing moments
about r of tlie external forces on one side, i.e. either HQ, ac
to the left 0/* AB to the I'lght, and of the unknown force D%
and equating to zero, the unknown force is obtained. Here
force DE X its perpendicular distance from r is equal and
opposite to the moment of ab about p. Similarly, ae is found
by taking moments about q, and bd by taking moments^
about p. If AB and bd were parallel, db would be "f ound by '
resolving the forces on one side perpendicular to ae.
Note, — ^If any member of tne structure have joints at
more than two points, it must be treated as a beam. In that
case deal with the foroes on each side by the method above,
and put their total resultants and moments equal to zero.
Ship's Derrick.
To find stresses in parts,
BuLE. — Find the tension in all parts of purchase. TIiIb
is deduced from the load lifted -, it is usually assumed that the
FiQ. ]<M.
rsmoie
:i LINES TO
vTOPpme LIFT
DOUBLE
;rSiNOL&
SIN6U
SHIP S DERRICK.
297
friotioiL of the blocks inoieases the tenmon by 10 per cent * at
each tnrn, and that the load is being lifted. Draw OQ
(fig. 16$) to represent ii| a difurraoi the direction of the
purchase (shown in fig. 164). Draw OB to represent the
Fio. 16S.
mean direction of the topping lift for the particular position
OP of the derrick. Without great error the purchase and
topping lift lines may be taken to intersect at the end of the
derrick o.
Choose a scale such that PQ = total vertical load lif ted^ on
this scale set off Qs ^ tension in purohase OQ. Join ts, and
draw ST parallel to ob. The polygon PQST is the reciprocal
diagram for the forces on the end of derrick ; and on tl\|e
scale above^ ST=: total pull in topping lift and pt = thrust
in derrick. From these data, the sizes of derrick and of steel
wire rope for purchase and topping lift may be determined.
The maximum B.M. on mast, if not stayed = vertical load at
o X horizontal distance of o from mast.
Example, — ^To proportion the parts of a battleship's derrick
shown in fig. 164. The load consists of —
Tons.
. Aetual load (maximum) ...«=: 160
Proportion of weight of derrick » g^ x 7 tons =a 3'8
3 blocks >s '9
Wire in portions of purchase and topping lift = *3
Total weight to be lifted , . . = 21*0
Total to be lifted by purchase = 16*0 -|- 3 (one block)
= 163 tons. The latter load is borne by three vertical lines,
* This is an extreme estimate ; the actual loss of power is generally
}unch less.
298 SHEER LEGS.
mean pull ^ X 16*3 = 5'4 ; so that their actual tennons
are (allowing 10 per cent friotion for each turn) 4*9, 5'4,
and 6*0 tons ; tension in portion OQ of purchase == 6 -}- 10 per
cent = 6*6 tons.
In fig. 165 put R at about J the distance up between the
two highest blocks in the mast. Find the scale that will make
PQ = 21 tons ; this will be the scale of the force diagram.
Make QS = 6*6 tons. Draw ST parallel to OB and measure
ST and PT.
Bepeat this process for several positions of the derrick.
It will be found that the thrust in derrick is nearly constant
— say 35 tons ; and that the total pull in topping lift if
a maximum when derrick is as low as possible (here 8^ to
horizontal) ; take it as 30 tons. The maximum tension in
topping lift is (allowing for friction) ^-}- about 15 per
cent = 8*6 tons.
A factor of safety of 6 or more is usual in steel wire
rope ; here 4'^ rope breaking at 58 tons could be used.
The section of derrick at middle should be sufficient in the
basis of the formula for pillars (p. 328), omitting the area
of stiff eners, but omitting also the term jup/t since the stiffeners
serve the purpose of preventing local collapse.
To find greatest height of purchase block for stabilitff of
derrick when topped.
EuLE.— The height of block Q (fig. 165) above the heel of
the derrick must not be greater than po— Qs. Since QS/qp u
the ratio of teiLBion in purchase to load, and » equal to n
(say), where n is about *3 in ordinary cases, PQ must not
exceed the length of derrick divided by (l -|- ii).
In the example, n = 6*6/21 = *31. Hence PQ should be
less than 60/1*31 or 45 fccf. Actually pq = SO feet.
Sheer Legs.
Let w be the load lifted, including half the weight of the
sheer legs.
In fig. 166, DC is the back leg, on the perpendicular from
0 to the line joining the feet of the front legs OA, OB. cm Is
vertical. The forces at o in the plane of the figure may
be graphically determined by means of a triangle of foroes or
reciprocal diagram.
Alternatively if a, j8, 7 represent the three angles mod, MOO,
DOB, then
WT &1 f\ at
Pull in back leg =* . /^ — r
® sin(i8-o)
Thrust in each front leg = » . /o — x sec 7.
BOATS SLTKGS.
299
Horizontal force required at 0, the foot of the back leg, is
equal to ^ta^ x{Bm$ + /ieo8fi}; where /i is the coefficient
of friction at o, say • 15. Take plus sign if sheer legs are being
topped, and conversely.
Fio. 168.
SHEER LEGS.
C 0
SlOe ELEVATION
A D B
ELEVATION IN PLANE
OF FRONT LEGS. .
Boats' Slings.
In heavv pulling boats not lowered from davits^ the sling
shown in fig. 167 ia frequently adopted. The stresses are
readily obtained, as shown (for one side)- in the figure, bq
represents weight of boat ; qp is parcdlel to as and gives tha
stress in that member ; bp gives that in ab. Similarly bb «■
BP, B9 Is pacallQl to bo : and bs^ bs give the tensions in bo
and bd.
Fig. 167.
Each portion of the sling is proof-tested to double its
working load.
In steam- and motor-boats, the sling shown in fig. 168 is
sometimes used. The stresses are here indeterminate, for they
depend on the adjustment of the length of AO. This should
be arranged so that the boat is deflected as little as possible
when slung. The middle leg ao should be capable of supporting
(say) f the Whole weight ; the other two legs ab, ad, should
zoo
BEAMS.
be cftpabk ol supporfeing } the weight, for the diBtribation
of the tensions is ancertaln.
Example. — ^A boat weighing 15 tons when fully equipped
is hoisted by means of a sling of the type shown in fig. 168 ;
the angles BAC and dac are each 45**. Determine the sizes of
wire rope, etc.
Working tension m ac is 10 tons, and in ab or ad is
5V2 or 7 tons. Allowing a factor of aafoty of 4, it appears
that the wire rope ao should be about 4}'^ circumference ; ab
and AD would probably be made the same size for oonTMiience,
but thedr si^e could lie safely reduced to 3|^'.
The ring A should be proof-tested to 80 tons, and the sling
fittings at B and d to 15 tons, and at o to 20 tons. Using
the formula on page 336, it appears that if the mean diameter
of the ring A be 12'^, the diameter of the iron should be 2|*.
SHEADING FOKCES AKB BENDING MOMENTS OF BEANS.
Shearing force, — ^At any section this is the total force
tending to break the beam in the manner shown in fig. 169.
It is equal to the algebraic sum of all the forces acting
u>[
Fio. 169.
Fia. 170.
. <'»I
(b)£
between that section and either end of the beam. For dis-
tributed loads of weight w per unit length, the shearicg
BEAMS.
301
force F is given by the formula — F « \ «? dx, when (2a; is an
element of length.
Bending moment* — ^At Any aection this is the total oouple
tending to break the beam in the manner shown in fig. 170.
It is equal to the algebraio sum of the moments of all ihfi
forces acting between that section and either end of the beam.
It is also equal to the area of the shearing force up to that
section. For distributed toads the bending moment H b g^ren
by the formulas-^
M= ^wxdzoxu^ {Fdx.
Note. — ^Since loads act nsoally downwards, downward
forces are regarded as positive, and are set off on the positive
side, i.e. belovf> the beam. Similarly + shearing forces (repre-
sented by (fi) in fig. 169) and H- bending moments (repre-
aented by (6) in ^» 170) are set off below the beam.
Gbaphical Method of deterhinimo the Bendino Mobibnts
and shearina foeges in a beam.
Concentrated Loads acting at various points.
Fig. 171.
In fig. 171 AB represents the beam supported at points A and
B. Set off continuously along a line EF, the forces w^, w*, w', w*
= C* C^, the resultant of the forces. Take any point o, and join
302
GRAPHICAL BENDING MOMENTS.
0C«, oc', .... oc*, &c. Draw the parallel lines ad®, wi>',
. . . BD" through the lines of action of the forces. Take
any point, D" in AD", and draw D®, d* parallel to oc^, D', i^
parallel to oc*, . . . dS d»» parallel to oc*. Join d^, d«*, com-
pleting the funicular polygon D^, d', d**. Draw a line oc parallel
to D^, d®*, cutting BP in c ; then c*c equals the supporting force
at A, and cc^ equals the supporting force at b. Also through o
draw OH perpendicular to ef.
Through the points c®, c', . . c* drop perpendiculars on to
ad®, w'd*, . . bd", and form the hatched figure S*KS*QS*'.
Then the vertical ordinate of this diagram, measured at any
point in the length of the beam, gives the shearing force at that
point, measured on the same scale as used for setting off w\"w\ &c
To obtain the bending moment at any point x make a scale as
follows : —
If beam is drawn to a scale 1 foot =i at inches,* and loads are
drawn to a scale 1 pound —^ inches, then the ordinate 'y* of
the funicular polygon is the bending moment at the point x on
a scale such that 1 foot-pound is g=^^^, in inches always ; OH
OH
is measured on an inch scale.
Forces acting in different directions.
Fig. 172.
These diagrams are constructed in a similar manner to (fig.
171), the lengths of the forces being also set ofiE in the direction
GRAPHICAL BENDING MOMENTS
803
of tbeir line of action. c*o is the supporting force at A, and OC*
the holding: down force at B as it lies to the left of c — ^that is, it
is measured in the opposite direction to 0*0^; the bending
monient at any point o' w^ere the sides of the funicular polygon
cross is zero, and the bending moments to the left of o* are in the
opposite direction to those on the right of o'.
Tablb of Graphical Bending Moments and Shbabino
Forces of Bbam&
w » load. L « length of beam.
w » uniform intensity of load.
M — maximum bending moment.
m — bending moment at any section.
Sj = maximum shearing force.
B s shearing force at any section.
The diagrams are constructed by setting off to scale M, and s„
or s ; then the ordinate measured at any point represents the
moment or shearing force at that section.
Bending Moment
Shearing Foroe
Fixed at one end, loaded at the other.
Fig. 178.
Set down M a WL, and join
BO.
Pro 174.
Set down s »= w parallel to
AC.
QBAPHICAL BBHDINO U0UEKT8.
Table of G84PHICai. Bkndinq Moments, &o. — coi
Banding Homenta I SbeulDg Fmett
Fitted at one etui and leaded imifermly.
parabola bc, wJtose vertei Is at b
Set off 8, » wL, and j<
Set oft it = WL, Mid H,-
..„(,.-|).
to a point D at middle of load.
Draw BB, a semiparabola, aa for
a beam unifoTiiily loaded of a
Table of Qkaphical Bendins Udubkts, &c.—
gbmrlne Foiwi
Fixed at one cad mth several eoncesirated Uadt.
Set off U-WL, H,^W,L„
», = W,Lt. Join B to C. P to D.
and B to H. The beading mo-
ment is eqnal to the snm of the
bending moments at the section
prodacid by each loctd separately.
u "9® ^■
M J '
. i
i" !■
Set off 8 = W, B, = W|, 8, = W,
ABDE, KFOH, and HKCJ.
Set off M = — T— 1 and conatrnct
Set off 9, above and s
, and join DC.
)6
GRAPHICAL BENDING M0UEKT6.
Table of Graphical Bending Moments, &c. — cont.
Bending Moments
Shearing Forces
Stiffported at both eTids, load out of centre.
Fia. 188.
D
Fi(J. 187.
Set off M = ^^, and join AC
L
md BG.
B
— o -v < — h sj
Set off 8
— , ana. b, = — •*
L L
and construct rectangries ACDB
and EBGF.
Stipj?orted at both ends, v/aeqiiaUy distributed loads^
Fig. 190.
M
T«i
.a^ j h H
The bending moment at any
point is equal to the sum of the
bending moments produced at
that point by each of the weights
separately. Set off
L * L ^ L
Then set up jh, kg, and nf,
making the length for whole
ordinates equal to the sum of the
three ordinates at those points
due to the several bending mo-
ments.
Setup
E '
L
02» W^j"""li
s
Si
w.
s
i
2 _ »»«— "It
L
4
ORAPHIOAI. BEMDlNti H0ME8I8.
Tablb ov Gbapbical Besdino Moments, Uc—ema.
Supported at iotk endi, and pajiial na^arm had not extetuHttg U
ttAer mpjtort.
ria m.
r-'-X"-
ends of t erect perpendicnlats
iHF and £G, cutting AO and bc
in H and E Join d luid b, then
on : draw the parabola 7aK,
whose middle ordinate eqnftb ~'
808
CURVES OF BENDING MOMENTS, ETC.
Beam supported at both ends and loaded continuously, hit
unevenly distributed, (Fig. 194.)
Curve of l/oad.— Set up weight of load per unit of length, say
in tons per foot, at Pj^; jg^^
suitahle points in the jinvt
length of the beam; a
curve ACB then drawn
through the points thus
found will form the
curve of loads whose
area will equal the
total load on the beam
in tons.
Supporting Pressures.
— Find the distance
the centre of gravity G
of the area of curve of loads from either support. Then if
w »•- total load, p aod p, » supporting pressures at A and B
respectively, d and <^, =* distances of a- from A and B re-
spectively.
sures. r
ice of I
Then PL = w<J|, and p «
p,L = w<J, and p,
17dy
L
Shea/ring Stresses, — Set up ad at A » p, and set down bf at
B = P,. To find the shearing stress at any point K, calculate the
area of the curve of loads from A to the point K » area of amk,
and deduct this from the supporting force p, and set this up as
an ordinate, kl, of the shearing curve. At B a point will be
reached where the difference between the curve of loads and the
supporting force P will be zero ; this spot is termed the point
of reverse racking. The differences from this point on will be
negative, and are to be set down below the line AB.
Bending Moment. — This is found in a similar way to the
shearing curve, only the area of the shearing curve between the
end of the beam and the section, say at K, is equal to the
bending moment at that point. That is, the area adkl is
equal to the bending moment at the point K, and is set up as an
ordinate, KN, of the curve.
The maximum bending moment occurs at E, where the
shearing force is zero, and is equal to the area ade ; th®
shearing stresses from this point on, being negative, have to be
deduct^. The part area of the load curve ace is equal to tiie
supporting force ad.
The part area of the load curve BCE Is equal to the sap-
porting force bf.
STRENGTH OF MATSBIAIB. 809
8TBSir0TH 07 XATEBIALS AHD 8TBS88S8 IH SHIP'S
STEUGTUBE AND 7ITTIN08.
Definitions.
1. Btreat is the matnal action between two parts of a body
which preserves them in nearly the same relative position
when acting upon by forces.
2. Normal or direct stress across a plane is the component
force per anit area perpendionlar to the plane. If the
external forces tend to press the two portions together, the
stress is termed compressive ; if the two portions tend to
separate, the stress is termed tensile,
3. Shear or transverse stress across a plane is the com-
ponent force per nnit area in a direction parallel to the
plane.
4. Strain is the deformation of a body produced by stress.
5. Longitudinal strain is the extension produced in unit
length. If negative it is a compression.
6. Shear strain \b the relative rotation of particles produced
stress. It is measured by the change of inclination of two lines
in the plane of strain which were originally perpendicular.
7. Strength is the amount of stress a body can stand under
certain assumed conditions.
8. Ultimate strength is the maximum stress that can be
applied without rupture.
9. Proof stress is the maximum stress that a body oap
bear without injury, i.e. without a permanent change of
properties.
10. Proof loeid is the load or total force producing the
proof stress.
11. Set is permanent strain after stress is removed.
12. Elastic limit is the utmost stress or strain thai can
be given without inducing set (or, in practice, much set,
since all stresses are found to cause a minute amount of set).
13. Wif^king load is the maximum load or total force
obtained under working conditions.
14. Working strength is the corresponding stress under
working load.
15. Stiffness is a general term denoting the load or stress
required to produce a certain strain.
16. Factor of safety generally denotes the ratio in which
the breaking load exceecU the working load.
Elastic Coefficients.
Within the elastic limit it is assumed that stress is pro-
portional to strain. This is approximately correct for ductile
materials, but brittle materials show a considerable variation
from the law.
810 cthength of materials.
The Uodulua of Bla$iioify, or Young*8 Modulus, denoted
by E, denotes the qnotient of the stress divided by the straio
ia a bar nnder longitudinal tension or compression. Thns
in a bar of length L inches, which \a stretched a inches by r
tons per sqnare inch, the strain is a/L» and B = TL/a. b nuiy
also be denned as the stress required to double the length of
a bar, assuming the law of elasticity to hold good.
Fig. 193. ,
The coefficient of rigidity " 0 " is the
shear stress divided by the shear strain. In
Fig. 195 if the square abcd is distorted by
a shear stress B acting along the sides as
shown into a rhombus dbcd^ the shear strain
is equal to the amount by which the angle
ahc is less than a right angle, expressed in
circular measure — eivy ^. Equally it =
^ B B . AG B . BD
Hence C = r =
^ 2(AC-ac) 2(6d-BD)
Foisscyii's ratio ((t). When a bar having free sides is extended
under longitudinal stress, its sides contract so that tiie lateral
strain is times the longitudinal strain. The contraction of
" 2
area per cent is then — times the longitudinal 'strain per cent.
<r
The ratio tr is termed Poisson^s ratio ; it generally yaries in vftloe
between 8 and 4. — is '86 for brass, •84 for copper, •28 for iron,
•38 for lead, -dO for eteel, •20 lor zinc, '5 for india-rubber,
and 0 for cork.
Bulk Modulus (k) is the volumetric strain divided into the
pressure producing it, assumed uniform. Thus under a
uniform pressure p tons per square inch a bodv of volume v
originally is reduced in volume to V»v, The volumetric
strain is v/vj and the bulk modulus K s pyfv.
Note, — The linear contraction is one-third the volumetric
strain ; if the latter be 3 per cent, the former is 1 per cent.
These coefficients are interconnected by the following
formulae, so that when two of them are known, the othen
can be determined.
•• =
0 + 8K
6K + 2C
8K-2C
Note that b/O generally lies between 2*5 and 2}; and
E/K between 1} and 1. For values of E see p. 260. Average
values for E and 0 are reproduced here : —
FACTOE OF SAFETY.
311
Material.
Tons per square inch.
Mild or high tensile steel (tr =: 3*5)
Biveted stractural material as a
whole . . . . .
Wrought iron . . . .
Cast iron
Copper
Gon-metal . . . . .
13,500
10,000
12,500
7,500
6,500
5,500
5,500
5,000
8,000
2,500
2,100
Npte,—X for steel is 10,500 ; for water it is 130 or ^
as maoh.
Factob op Safjett.
The factor of safety, or the ratio of the nltimate load to
the working load depends on the material and on the nature
of the load. In certain materials, e.g. castings and timber,
wide fluctuations in strength, caused by flaws or other defects,
are observed ; the factor for them is correspondingly higher
than for rolled metals, which are more uniform in strength.
With regard to the nature of the load, the factor is least ip
a ^dead' load, i.e. one which is either constant or varies
slowly. For ' live ^ non-alternating loads it is nearly double
that for a dead load, since the momentary stress, due to
a load suddenly applied, may be nearly double that due
to a steady load. For live alternating loads it is further
increased, since the experiments of Wohler and others show
that stresses continually reversed even within the elastic limit
lead to ' fatigue ' and lower the elastic limit. Finally, under
shock or impact very large stresses may be momentarily
produced) though it appears that these do not generally
damage the material to the same extent as do similar stresses
acting during some appreciable time. The factor of safety
due to such stresses is indeterminate generally, as it depends
on the d^ree and nature of the impact.
Table op Faotobs op Safety. (Unwin.)
Material.
Dead
Load.
liive And Varying
Load.
Structure
subject to
Shook.
Stress of
one kind
only.
Bevetsed
Stresses.
Cast iron .
Wrought iron and steel
Timber . •
4
3
7
6
5
10
10
8
15
15
12
20
S12 EFFECT OF BENDING.
The following working stresses are deduced therefrom*
Table op Wobkino Stresses.
Material.
Workingr Stress in Tons per SQuare
inch, under -
Dead Load.
Live Load
without
Reversed
Stresses.
Live Load
with
Reversed
Stresses.
Wrought iron .
Mild steel .
High tensile rH.T.) steel
„ (H.H.T.) steel
Forged steel .
Rolled Naval brass
Cast steel
*Cast iron
„ special malleable
Gun-metal
fOak or Canada Elmi
fTeak or Pitch Pine .
fFlr or Mahogany •
(Board of T.6)
7
(Board of
T. 6'6)
8
10
12
9
n
2i
4l
3|
•6
•4
•3
4
5
6
7
H
4
IJ
8
H
•36
•25
2
2*
8
il
8
?
2
1*
•25
•2
•15
* For tension only. Under compressiTe stresB multiply by two.
f Spars or timber subjected to bending may safely receive workinf
stresses up to H times those given In the table.
Stress due to Bending.
Let u =3: bending moment in inch tons. ^
I S3 moment of inertia of section about neutral axis
in (inches)^.
p =s tensile stress in tons per square inch at any point
p in section,
y^ distance of p from the neutral axis in inches,
p^ radius of curvature in inches to which beam is
bent along the neutral surface.
B =s Young's Modulus (modulus of elasticity) in tons
per square inch.
Then ^ =» — = ~ i' beam is originally straight, and
£.=. — a=E( ) , approximately, if beam has originally
a radius of curvature po*
If the beam be very broad in relation to its depth, as, for
instance, a thin flat plate, change E to o^^lio^-l), or about
1 • 1 E for steel, <r being Poisson's ratio.
Ill M lU UXI X U MX XX« J9XV X XA (
Note. — ^The neutral sarface is that surface which is n
strained when the beam is bent. It intersects eaoh seotii
of the beam at its neutral (ucia ; this is a straight line passii
throngh the centre of gravity of the section, and perpendicul
to the plane of bending.
When y is the greatest distance of any part of the sectii
from the neutral axis, p, which is equal to— ^^ is the maximv
stress at that section. Also — p \r equal to' the bendii
moment v., so that it is the moment of resistance of tl
beam. — is then the moment of resistance corresponding
unit stress, and is termed the modulus of the section.
Then, Stress = Bending Moment -r- Modulus of Section.
Moment or Inebtia.
For methods of calculation the moment of inertia
various sections, etc., see pp. 69-75.
The moments of inertia and resistance of various sectio
are given in the tables below. (Moment of resistance =
X modulus of section = p\[y^
Tablb op Moments of Tnbetia and Rbsistanob op
VARIOUS SBCTIOUS.
Form of Section
REOTANQLe.
REOTANQLe.
Ai
4
wAmx
TRIANflLS;
Moment of Inertia
through Centre
of Gravity
5£
12
Mf-"')
36
Moment of Besistance
^6
'{&('-■)}
I>
24
814
MOMENTS OF INERTIA.
Tablb of Moments op Inertia, &c. — oont.
Vorm of Seotion
Square.
Hr-d-i^
Square.
Hexaqoh
HEXAQOtft
^*^^
OCTAQON.
m0
w//mwM
Begular polygon with
n sides.
^— side.
rsiadias of circum-
Bcribed circle.
_ w* . 2ir
a = area*-—- sin. - .
2 n
CiROLE.
Moment of Inertia
through Centre
of Gravity
£1
12
4!
12
>V5
«-64l3ft*
»-5413&«
i*
1+2V?
6
= -638 J*
i.e-D
Moment of Bedatance
if*
•ll^Nf
|l?5»«-626pa^
jgP*«V3»-641%?>»
«690^i^
4
64
•049W*
H7'*
4
=-7854y*
MOMENTS OF INERTU.
816
Table of Moments of Inertia, &o. — eofU»
Form of Section
Houow OmoLc.
SEMroiROLE.
Hollow
SEMIOinCLE.
Hollow Square.
Hollow Square.
Moment of Inertia
through Centre
of Oravity
•11<W*
•110(r*-r,*)
"" r + 7*,
64
»0491W»
12
12
Moment of Resistance
19^
y
^^^«.0982j»J^
\/d*- dy\
Ad)
Table op Mohbnts of Inbbtia, &.c.—oont.
T^ijJbj
of the whole sectioc taken i
solid about that axia and then
deductings the momeDt of tbe
hollow ^t from it about the
,(5^A-)
MOMENTS OP INERTIA.
811
Table of Moments op Inertia, Ac. — eont.
Momeni of Inerii*
Uirough Centre
of Gravity.
12 1 bidi + kd)
Moment of ResfstaDoe
" y
d
yi^-2
d
2 '
where
hd
y
hidi + kd
2 Ml + M
bidi+kd
where A — area « biCi + b^+ kd
di^d(2biei-\'kd)l2K
d%^d('Xb^+kd)l2jL
The formuIsQ for the last two figures are approximate
the thickness of flanges and web being assumed com
paratlvely small.
Note. — ^From the aboye figures tibe moments of inertia an*
resistance can be reduced very approximately for many sections
materiale, including angle bars, T bars, I bars, Z's, and ohannele
See also pp. 241-254.
Example. — A cast-iron bar whose section is of the form givei
last in the above list, and where d = 12", ci = Ca = 1", k = 1", 6i = 3"
hi—S(', is placed on supports 25 feet apart. Find the limitin
distributed live load that it can support.
From the formula i = 495 (inches)*, di «= 6-57", da = 6-48*
Since the upper flange is in compression, and therefore ampl
strong for cast-iron, y should be taken as 5-57" for the lowe
flange in tension. Also ^=1-5. Hence moment of resistanc
s= — = — ----; — = 133 inch- tons.
y 5*67
Bending moment = -5- =
vrl w X 25 X 12.
8
inch-tons when w i
133 X 8
the load. Hence load in tons = ^ — r^ = 3 • 55.
as
MOMENTS OF INERTIA.
Table of Momrnts of
Inertia
AND Moduli of Sections
FOB
CiRouiiAB Sections
Diameter in
Inches
Moment of
Inertia
Modnlas of
Section or
Diameter in
Inches
Moment of
Inertia
Modnlas of
Section or
i/v.
Diameter in
Inches
Moment of
Inertia
Is
m
88861
1
0-0491
0-0982
85
78662
4209
69
1112660
2
0-7864
0-7864
86
82448
4580
70
1178588
83674
3
8-976
2-661
87
91998
4973
71
1247393
86138
4
12-67
6-288
88
102364
6387
73
1819167
36644
5
80-68
12-27
89
113661
6824
73
1398996
38192
6
63-62
21-21
40
126664
6283
74
1471963
89783
7
117-9
33-67
41
138709
6766
75
1563156
41417
8
2011
60-27
42
162746
7274
76
1637668
43096
9
8284
71-67
43
167820
7806
77
1786671
44890
10
490-9
9817
44
183984
8863
78
1816978
46689
11
718-7
130-7
45
201289
8946
79
1911967
48404
12
1018
169-6
46
219787
9666
80
8010619
60866
18
1408
816-7
47
839631
10193
81
3113051
63174
14
1886
869-4
48
260676
10867
82
3319347
64130
U
8486
381-8
49
888979
11660
88
8839606
66136
16
8217
403-1
60
806796
18278
84
8443920
68189
17
4100
482-3
61
888086
18088
85
2568898
60398
18
6163
672-6
52
858908
18804
86
8686180
63446
19
6397
678-4
68
S87888
14616
87
2812305
64648
90
7864
786-4
64
417893
16469
88
2943748
6C903
SI
9647
909-8
56
449180
16334
89
3079863
69310
83
11499
1046
66
488760
17841
90
3320623
71569
83
18737
1194
67
618166
18181
91
8366166
7398S
84
16286
1367
68
666497
19166
98
3516686
76443
26
19176
1634
69
694810
80163
98
8671998
78968
26
22432
1736
60
686178
21806
94
8888498
8154S
87
26087
1938
61
679661
88384
96
8998198
84173
28
80172
2165
62
726388
33S98
96
4169320
86859
29
84719
2394
63
773878
84648
97
4346671
89601
80
39761
2661
01
838660
86786
98
4537664
92401
81
46333
2926
65
876240
86961
99
4716315
95869
32
61472
32ir
66
931480
88886
100
4908788
96176
33
58214
3628
67
989166
89687
84
66697
8869
68
1049566
80869
IJbto. -For ghafti subjected to torsion, the torsional inertia and modnlui
ire fonnd by doabllnc their raloes in the above table.
MOMENTS OF INERTIA.
81S
Table op Moments of Inertia and Moduli at SscnoifS
(i-s-y) FOB Hollow Tubulab Sections.
!
s
ThicknesB in InohM
1-a
1-5 1-8
2D
2-2
1
2*6
a-8 .
81)
8-5
10
8a7-i
66-42
878-0
74-60
408-«
81*70
4S7-S
86-46
11
406-8
81-85
5171)
9411
571'»
108-9
600-8
109-2
026D
118-8
la
601-0
loo-a
696-8
116-0
7785
128-9
816-8
186*1
654*1
142*4
9000
UOD
18
782-3
iao-8
911-1
140-2
1019
166-8
1080
166-1
1184
174-4
1201
184-8
14
996-9
142-4
1167
166-7
1811
187-4
1806
190-8
1460
209*9
1664
288-4
15
1248
166-4
1467
196-«
1666
280-8
1766
886-5
1866
248-8
1904
266-9
2101
9B0-2
16
1688
192-3
1815
aas-9
9066
867-1
2199
274-9
8829
281-1
2498
812-8
9648
880-8
17
1869
S19-9
2214
260-5
2617
296-1
2608
817-4
886-8
8062
882-6
8271
864-8
881
897-8
IB
2246
9ttr6
996-4
8042
886D
8907
868D
8475
886*1
8761
416-8
8099
448-6
4186
489-6
19
8180
884-8
8686
888-8
8912
4U'8
4168
488*7
46U
474-8
4814
806-8
4996
698*8
90
8764
875-4
4808
480-8
4687
468-7
4948
494*8
6889
586-9
6748
674-8
6008
696-8
6452
646-i
tt-6
6819
561-7
6881
607*2
78U
660D
7977
709-0
8676
762-8
8942
794-9
9747
666-4
86
8860
710-4
wnss
770*2
10884
827-0
11820
906*7
12^
977-7
12778
1022
14022
1122
87-6
18602
962-9
14096
1026
16498
1127
16782
1221
17686
1278
19897
1411
80
17827
1166
18678
1246
90686
1872
22859
1491
28472
1505
26021
1786
82-5
82877
1877
24167
1487
2SDDOD
1648
29066
1788
80654
1680
84005
2098
85
28826
1619
80619
1760
88896
1987
86968
2114
88888
-8226
48484
2486
S7'6
86246
1880
88146
9085
42801
9966
4mt
9*06
4S786
2599
64688
2912
40
48210
3im
46818
2841
61996
2600
66917
2B46
60068
8006
67440
8872
Ifote.— The first flgures
second fbe modall of sectit
1 giren are fhe
ons(I-^v).
momen
ts of in
ertia. ai
id the
Note.— ¥or shafts sabjected to torsion, the torsional inertia and modul
^re foon^ by doubling their values in the above table.
820
BEAMS of: equal STRENGTH, ETC.
CoKTiNuous Beams.
Distribution of load on each equidistant level support of a
continuous beam uniformly loaded.
Divide the load on each span by the dividing factor in the
table below ; then multiply by the corresponding figure under
'Reactions at props'.
Number of
Spans.
Dividing
Factors.
Reactions at Props.
2
8
3
10
8
8
10
4
11
11
4
4
28
11
32
26
82 11
5
88
15
43
37
87 43 15
6
104
41
118
108
106 108 etc.
7
142
56
161
137
143 143 etc.
8
888
152
440
874
892 886 etc.
9
530
209
601
611
535 529 etc.
Any large
number.
1
'894
1134
•964
1*01 about 1 etc.
TABiiE OP Beams op Equal Strength throtjghoitt
THEIR Length.
Ncte.—Th^ Eections ore in all cases sapposed to be rectangnlar.
Pig. 1S6.
Depth equal throughout.
Breadth proportional to dis-
tance from loaded end.
Pio. 197.
Depth equal throughout.
Breadth proportional to square
of distance from unsup-
ported end.
BEAMS OF EQUAL STRENGTH.
82
Table of Beams of E<iual STRSNexH THBoueHOur
THBTR IiBN6TH (concluded).
Breadth eqtial thrtnighmit.
Depth proportioDal to square
root of distance from loaded
end.
FialM.
Breadth eqna^l thnnighowt*
Depth proportional to dis-
tance from unsupported end.
Depth equal throvghotit.
Breadth proportional to dis-
tance from nearest point of
support.
Depth eqiial throughout.
Breadth proportional to pro-
duct of distance from both
points of support.
Breadth equal throughotU,
Depth proportional to the
square root of the distance
from the nearest point of
support.
Breadth eqiial throughout.
Depth proportional to the
square root of the producti
of distance from both points
of support.
Fiu. aoo.
Flo. aoi.
FXU. 2Ui2.
1 IG. 20B.
m99Q9
322 BENDING MOMENTS AND DEFLECTIONS OF BEAMS.
JHitribution of 2 or S supports for mimmum Barldinff
Moment,— l^i w be total load asnimed aniformly distributed
and I the total length of beam.
For 2 supports, place each *207 / from end ; b.m.
(maximum) at supports and at middle is '0215 wl.
For 3 Gupporto at the same level place one at middle, and
the others '145/ from end ; bm (maximum) at supports is
'0102 wl. If the level of the central pi^p is iidjnsted so that
it supports w/8, and if the other supports are the^ '13 I friovn
end;, BM is only :0085 wl,
Bensikq Moments and Deflections of Beams.
L s= length or span of beam in inches.
I = moment of inertia in (inches)*, of greatest cross
section if not uniform,
w = total load on beam in tons.
£ = Young's modulus (pp. 260, 311) in tons per square
inch.
D =3 maximum deflection in inches.
Mo = bending moment at end.
M = maximum bending moment at or near the middle.
Then D = ; Mo = K2WL ; M = KaWL where Ki, K9, Ks
EI
have the values in the table on p. 323.
Strength of Bulkheads under Water Pressure.
It is assumed that the bulkhead is suffioientiy wide for its
central portions to be uninfluenced by the connexions at the
sides (otherwise a narrow deep bulkhead, horizontally stiffened,
may be treated as a series of horizontal girders uniformly
loaded). For stresses on the plating between the stiffeners
see ' Stress in flat plating ', p. 825.
Consider the stiffener tog*ether with a strip of the adjacent
plating as forming a girder ; calculate the position of C.G.
of its section, the moment of inertia I about an axis through
G parallel to the bulkhead, and the modulus l/y. The width
of the strip of plating should not exceed the stiffener spacing ;
neither should plati^ extending more than fifteen times it9
thickness beyond the line (or outside lines) of stiffener rivets
be included.
Let as=head of water in feet above top of bulkhead (if
negative, take a = o).
A = height of bulkhead in feet.
b = stiffener spacing in inches.
I =s moment of inertia of section, including ons
stiffener and plating of width b, in inch units.
£ = modulus of cla;iticity in tons per square inck.
i
1
III!
i"
Ml *
—
■" =|3| "i"-'?"*' a "^
1
1
- — -
1 1 1 1 1 1
ii'^^^
.-«.
. ^
Hi •« ■«
■5 -fihl-Hs
4h '
1
j ;:
i s
E 1
- 1- ?
■ -:■ 1
i -s' 1
1 !
\ 1
1 {
111
pi:
If
il
1 . • III-
1 a 1 nil!
1 H f Hill
III illili
Hi Ijlil!
^
• "' -
"" s
aa a
3 a S E2 S
324
BULKHEADS UNDER WATER PRESSURE.
Then the pressure on top and bottom boundary angles,
the bending moments, and the deflections are obtained from the
following table : —
For width ' h \
Total pressare on plating in tons . «
Support at top boundary in tons .
Support at bottom boundary in tons .
Bending moment at top in inch-tons .
Bending moment at mid-depth in inch-tons
Bending moment at bottom in inch-tons
Deflection at mid-depth in inches .
Deflection for steel (e = 18.500 tons/in.^)
Depth of. position for max. b.m. below top \ when
M fi *• deflection ., / a*=o
Stiffeners
unbracketed
and assumed
' free ' at top
and bottom.
840
840 V 8/
660
{2a+h)
lfl2
hh*(2a+h)
BI
604,0001
•58 7i
•52 7t
Stiffeners :
well bracketed
and assumed
fixed in
direction at
ends.
hh
840
bh
(aa+h)
840 \ 10 J
840V 10/
1680
700 V 8 /
^^ — il
2.520.0001
•55 h
•52 7i
Note. — From the last two lines it is evident that the
bending moment and deflection at mid-depth may be assumed
to be the maximum without appreciable error. The stiffener
stresses may be deduced from these bending moments ; the
boundary bar riveting from the top and bottom supports ; and
the top and bottom fitiffener brackets may be roughly con-
structed to take the corresponding bending moments. The
deflection and b.m. at mid-depth are the same as if the total
pressure was uniformly distributed. Calculated stiffened
stresses up to 18 or '20 tons per square inch are permissible.
Example. — A bulkhead is stiffened with channel bars 12f x4*
X 4" X '60", spaced 2' 6" apart, the mean thickness of plating being
•40". Find the stresses, etc., assuming stiffeners unbracketed;
the depth of the bulkhead is 20 feet, and the head of water 5 feet
at the top.
Consider a strip consisting of stiffener and plating. The
width of the latter is limited to 30" (stiffener spacing) and
also to 12" (fifteen times the thickness each side) ; the
smaller figure must be taken. Find i and y, taking trial
neutral axis through centre of stiffenof .
BULKHEADS Xti^DxiB. WATJ3R FRESSuj»,E.
825
Item.
£3
60
vi
9
■
i
1
Moment of
Inertia.
Is
13
i
72
Web of channel .
^^^
^m^im
Flanges of channel
1-75
1-76
+ 6-75
-6-75
+ 10-1
-lO-l
68
68
^im^m
„_
Bulkhead plating .
4-8
6-2
+ 29-7
184
300
72
14-3
29-7
72
1
872 «ic.i. about
assumed
axis.
29*7
Beal neufaral axis = t7~7 — 2-08" from assmned axis.
14 '3
M.I. about real axis = 372 - 14-3 x (2-08)* = 310. y = 6
+ 2-08 = 8-08. Modulus l/y = 372/8-08 = 46-1. Maximum
B.M. = ^ (2a+h) = ^^g^ (20 + 10) « 643 ittoh-tons. Stress
in stiffener = My/l = 643/46*1 = 14 tons per square inch.
\f • :i a L' 30x20^x80 „ . ,
Maxmium deflection = g04 000x372 = *^^ ''''^^''
Stress in Flat Plating under Water Pressure.
h = he&d of salt water in feet.
t = thickness of plate in inches.
a = one-half the breadth between supports in foot.
/= maximum stress in material in tons per square
inch.
8 = central deflection in inches (Young's modulus
assumed to be 13^500 tons per square inch).
In long plates whose edges are free to angle put
*=807^'''(f)'! then /= jig BM»/t= anl i^ ^^chaVt\
when B and C are read off opposite A in the table on p. 326/
Similarly, if the edges are fixed or continuous, put
D =
h^ (f) ; then/= --~ E7^7^2and5= T^nF/ta*/^',
7-82x10®' V^/ ' -^ 19-2""^"'" 7,200
when E and F are read off opposite D in the table on p. 326.*
• Extracted from Trans. I.N.A., 1902 ; paper by Mens. Boobnoff.
826
STRENOTH OF DAVXT8.
A.
B.
C.
D. E.
F.
•804
•901
•908
•908
1^012
•960
1-974
•889
•711
•988
•997
•926
10 -la
•683
•496
1-as
•976
•877
87-4
•512
•880
2-88
•954
•884
57-4
•4B2
•808
412
•988
•794
108 -2
•aP76
•259
6^M
•912
•758
167-8
•884
•228
7^21
•893
•725
355
•802
•196
909
•874
•606
510
•265
•168
11-19
•867
•668
884
•222
•182
88^T
•701
•401
1658
•191
•108
160
•464
•262
If the plate is stiff, so that the deflection is slight, do not
calculate A or D, bat take B = 0 = B==Fsl.
If the plate is so flexible that the resistance to flexure is
negligible compared with the hoop fusion, /being uniform,
/'
h^a^l^\ a»=
. ha^lL
78^5 '*-'-' 4,880
If however the edges are allowed to approach, as in the bottom
of a box, or if the plate is initially curved, S being the ilnal
deflection, / = /ta^/70 U ; if initially flat the sides approi^ch by
»V9a in.
If the length of the plate be 21 feet, and its ends are supported
like the sides, substitute alj^a* + l^ for a in the above lormule.
A high nominal stress, e.g,t 18 tons per square inch, is
frequently adopted in bulkhead and even shell plating.
DiAMETEE OP A DavIT.
Let d = diameter in inches.
H S3 height of davit above uppermost support in feet.
s == overhang in feet.
w = maximum load in tons on each davit.
d^=:ZO w (s + Jh) for wrought iron, allowing
4 tons/inch^.
(f8= 22 w (8 + J h) for forged steel, allowing
6i tons/inch^.
Lloyd's Sule. — (a) For boats and davits of ordinary
proportions,
d= i length of boat in feet,
(ft) Otherwise, eZ» = L.B.D. (Jh + s)/40,
where h, B, and D are the principal dimensions of the boat
in feet.
Board of Trade Mule. — rf3_-.L.B.D. (jH + 8)/c,
where o = 21'5 for iron and 26 for steed.
Unstmmetrical Beasis.
If the beam or loading is unsymmetrieal resolve the b.m.
into components in the planes of the principal axes of
inertia of the section. Treat each component separately and
UNSYMMETRICAL BEAMS. 327
find the stress and deflection due to it. Tiie final stress is
the sum (or difference^ if of opposite signs) of the two Qpm-
ponent stresses at the same point. The final deflection is
compounded of the two components.
In pp. 2il to 254, particmars of the principal moments of
inertia, etc., are given for the British Standard Sections.
If lateral deflection is prevented, as in a beam connected
to deck plating, the section can be treated as if symmetrical,
and the stresses and deflection found by the ordinary method,
using the moment of inertia about an axis perpendicular to
the plane of bending.
EoMmvple, — Find the greatest stress and the deflection of
a standard angle bulb beam 4" x %^' x *2i\ resting on supports
10 feet apart, oanying one ton unilormly distributed (a) if free
laterally, (&) if lateral movement is prevented.
Maximum bending moment is 15 inch-tons. Angle a » 14}°.
(d) Least moment of inertia » 2-17 x (-548)' » -652.
Greatest moment of inertia = 4*461 4- '915 - '652 — 4*724
(see p. 241).
In plane of minor axis, m/i = 15 cos a/4* 724 = 3*07.
In plane of major axis, m/i » 15 sin a/ •652 = 5-86.
The greatest compressive stress is evidently at the comer of
the bulb, which is 1*46" above the greater axis and '98" from the
least axis.
Stress due to bending in plane of minor axis = 307 X
1*^6 = 4*48 tons/inch^
Stress due to bending in plane of major axis := 5*86 X
•98 = 5-76 tons/inoh*.
Maximum compressive stress is 4*48 -\- 5*75 =» 10*23 tons
per square inch. ' Similarly the maximum tension is found
to be at a point near the bulb; and it is equal to 10*86 tons
per square inob.
^ - . . . , , . . 5 «;Z< 5 mP
Deflection in plane of mmor axis = r^ ~^ — T^ ZiT
Oo4 EI 4o EI
5 X 3*07 X 120 X 120 ,, . ,
" 7Z — io gAA — '34 men.
48 X 13^500
Deflection in plane of major axis is similarly *65 inch.
Net deflection is V(-34)* + (*65)^ or •73 inch, in a direction
tau'^ 65/34 or 59^ with the minor axis, that is 44° with the
vertical* towards the right as drawn in fig. 156.
(b) Section modulus (i/y) about horizontal axis is 1*907.
iGrreatest stress = 16/1*907 = 7*87 tons per square inch.
This is at the bottom where y is greatest.
At the top y = 1*66. I - 4*46.
Maximum compressive stress ,=> 15 X 1*66/4*46 = 5*59
tons per squaxe inch.
^ - . / ..IV 5 X 16 X 120 X 120 o^ * 1
Deflection (vertical) = ^g ^ 13,500 x 4-46 '^ '^^ ''''^'
8^28 PILLARS.
Strength of Fly-wheels and Pulleys.
Stress = density X G^^*^' velocity of rim)^/^.
For cast-iron wheels, stress in tons per square inch =
(speed in thousands of feet per minate)2~T-83. In pulleys
the maximum working stress is about *45 ton/inch^, the corre-
sponding velocity being about 6,000 feet per minute. In
solid fly-wheels, working stress is '28 ton/inch*, velocity 4^600
feet/min., although the higher stress given for pullers is
sometimes admissible. In built-up fly-wheels, stress is *1
ton/in ch^, velocity 3,000 feet/min. A cast-iron fly-wheel
bursts at about 25,000 feet per minute.
Pillars.*
Let w = breaking or crippling load in tons on pillar.
A =3 sectional area of material in square inches.
/ 8= length of pillar in inches, if round-ended f ; b=
^ length of pillar if square-ended or fixed at
ends ; ^--Jn^engih if fixed at one end.
p =s least radius of gyration of cross section in inches,
or kp^ = least moment of inertia of cross section in
(inches)*.
ft a^^ coefficients depending on the material.
t =s minimum thickness of material, if hollow^ in
inches.
N =3 coefficient for a hollow pillar depending on shape
of section = 500 for a circle, 600 for a square,
1,200 for a cross, 800 for an equilateral triangle,
700 for an I bar. In a solid pillar put N = o.
For very long pillars where lip is greater than 150 (i.e, if Z is
more than about 40 diameters, if circular) T ***e/(-~ + -5Ji
(Me Z'\
-T + -gj for
mild steel.
For long circular solid pillars of length L in feet^ diameter d in
incJies the collapsing load is i5d*li? for iron or steel, 2i*/i** for
oak, 2}dVl'* 'or pine. If hollow the load is A;iV(l+6-5^)»
where k = 2,800 for steel, 100 for oak, 180 for pine, the thielmess
t being supposed relatively small, d Is then the mean diameter.
For pillars of ordinary lengths —
* Based on data recorded by Professor W. E. Lilley In Amer. Boo. C.E.,
1912.
f Unless pillar is very securely fastened at ends, it should be regarded
as ronnd-ended.
PILLARS. 829
Material f a
Nickel steel .... 54 2,800
Bessemer steel ... 49 2,600
Mild steel .... 36 4,000
Hild steel, annealed . 27 5,000
Wrought iron . . . 24*5 4,600
Cast iron .... 49 1,300
For dry timber, Rankine gives / = 3*2, a = 3,000.
A factor of safety- of 5 or 6 is generally allowed.
Eccentrically loaded or initiallff bent pillare.
Ziet e = eccentricity of loading In inches ; = ^ of distance
from centre of section at middle from line of
action of load (if initially bent).
p = distance of outer fibre from neutral axis in inches.
w f
Then— "
(^ + rP(^ + ^0+'S
For solid pillars put—; = o.
Example. — A hollow circular pillar, 12" external and 10"
internal diameter, 30 feet long is l(Maded centrally. If the pillar
is initially bent 3>75" out of the straight, find the collapsing load.
Assume pillar of steel, having fixed ends.
f^^ts Wi" + ^^) = A (144 + 100) = 15-25 ; p = 3-9.
N=500; a = 5,000; t = l; e = 4 x 3-75/5 = 3 ; 2/ = 6^
Z « 30 X 12/2 = 180.
A = -785 (144-100) = 34-6. / =» 27.
w 27 •
Hence g^.g - ^^ ^ g.^ /, . 3 x 6\ . 180 x 180
/ 5UU X 3.;^w gj^x
V 6,000 X lA^ ^ 1625/ ^
6,000 X 1/V 15-25/ 6,000 x 15-26
or W = 450 tons = collapsing load.
Working load = w -i- factor of safety, say 6, = 90 tons.
Note. — Values of p for various standard sections are given on
p. 76 ; and values of a/>^ for other sections on pp. 313-317.
CaipPLiNO Loads op Solid oe Thick Mild Steel Pillaes,
(In accordance with the formula above, using the constants
for mild «t«el annealed, ^ = length in inches if ends are
rounded, = half length if ends are fixed ; p = least radius of
gyration in inches. Load is in tons per square inch.)
mo
PLATING UNDER COMPRESSION.
l/p
0
27-0
10
20
250
100
30
40
20-5
120
50
60
70
Crippling load
26*5
22*9
18-0
15-7
13-4
IIP
80
11-8
90
10-3
110
7-9
130
140
160
Crippling load
90
6-96
6-17
5*49
4-91
Maximum Compressive Stress in Thin Steel Bectanoular
Plates. •
Let thickneas be t ins., leng^ ^ ins., and maximum com-
presBive etress in the direction of the length fi tons per
square inch, then, if the breadth of the plate is very large oom-
pared with its length, fi = 12,000 t^lJ^,
If the breadth of the plate be b ins., multiply /^ by the
factor k in the table below : —
bll CO B 2 1-5 1-25 10 -75 -5 -33 -25
k 1 1*24 1-56 208 2-69 4 7-7 16 36 64
It is iabove assumed that the edges of the plate are simply
supported. If they are rigidly fixed mahe I and b one-half
the length and breadth respectively ; if partly fixed, as at
beams and stiffeners, reduce I and & by an intermediate ratio
(say *75), using judgment.
If the compressive stress /i also acts across the plate so as to
produce uniform compression /i in all directions,
A = 12,000^2(1 + 1).
If the compressive stress across the plate be fg (different &om
where m and n are integers chosen so that /i and fy are as small
as possible. If one of the stresses be tensile, change the sign of
/i or/s. For a material other than steel replace 12,0QO by 8B/9.
Strength of Thin Hollow Cylinders and Spherical Shells
UNDER Internal Pressure.
p = internal pressure in tons per square inch.
T = tensile stress in material in tons per square inch.
i = thiojcness of material in inches.
r = radius in inches*
For Thin Hollow Cylinders,
Tt ^ P^ P*'
r T t
The longitudinal tension is one-half the oircumfierential
tension T. For Board of Trade rules re cylindrical boilers,
see p. 462. > ;
♦ These resnlta are based on a paper by Professor G. H. Bryan, Londoa
Math. Boo. Proo.. 1891.
CYLINDERS. 881
For Thin Spherical ShelU.
Strength of Thick Hollott Ctundbrs and Sphebioal
Shells under Pressure.
Let Bx) B} a» internal and exteraitl radii in indies.
Vv Pt '^ iateraal and exterxial pressares ia tons per
eqoare inch.
Ta« tensile eircumfereaiial stress at any point
in tons per sqoare inch.
p C8 compressive radial stress at any point in tons
per square inch.
For ihick Cylinders,
When «i => o, greatest valoe ol T ia at internal oiioamfeience,
E,«+Bi2
^or tAtoft Spheres.
At ra4ius «. T = bA^I* "''"'11^ '^^ +i>iBx' -|..B,»}
^_ 1 fBi'B,'(Pi-l>«) „R.4.^i.j\
When p2 — 0, greatest value of T is at internal circumference,
CoLLAPSiNQ Pressure of Thin Hollow Cylinders.
Let 7 ss length in iaohes.
r S3 mean radius in inches.
<=s thickness in inches.
p Es collapsing pressure in tons per square inch.
For short wrought-iron or steel tabes — at least |" thick,
2,150^
"^ iT-
3 700 i'
For long tubes P = -—3 — If ribbed, calculate the moment
of inertia pf the perimeter, assumed unrolled flat, and divide it
by that of the same plate unribbed. Multiply P by this ratio.
338 SHEAR..
Shear and Kesultant Ste esses
1. Each shear stress is accompanied by another equal
shear stress in a perpendicular plane.
2. These two shear stresses are also equivalent to equal
tensile and compressive stresses across planes inclined at 45^.,
e.g. in fig. 195, on p. 310, a shear stress along ab is necessarily
accompanied by one along AO ; these produce an equal tension
across AC and an equal compression across bd.
3. Two tensile or compressive stresses pi and P2 Cp^^ o^^
of them negative if they are of opposite signs) together with
stresses q in perpendicular planes are equivalent to a maximum
direct stress of Hpi+pjj + Vjipi -JP^f + g^ and a maTimnm
shear stress of Vi {pi -pij^+^t these being in planes inclined at
45° to one another.
4. A bending moment M combined with a twisting moment
T produce a maximum direct stress such as would be caused
by an e<juivalent bending moment of J (m -(- a/(m^ -\- T^) ;
the maximum shear stress is equal to that caused by aju
equivalent twisting moment a/(m2 + t2).
.5. The shear stress permissible varies from one-half to the
full tensile stress permissible ; about 80 per cent is fre-
quently taken. (See riveted Joints, shafts, plate web girders.)
Sheab Stress in Beams.
To find the shear stress at a point p in the section (fig. 204),
let
>' = shearing force at section in tons.
I ss moment of inertia of section about neutral axis in
(inches)*.
a =s area (shaded) of portion of section lying beyond p
in square inches.
p=^ distance of centre of gravity of area A from neutral
axis in inches.
:; = thickness of web or breadth of section at P in inches.
g = shear stress in tons per square inch.
g = FAf^/lZ,
--.JlNlillBXL
-AXiSL.
Note, — ^The shear stress is very small in the flanges or
horizontal portions of a beam. In an i beam of ordinarv
proportions it is practically constant over the web, and equal
to F-=-web area.
PLATE WEB GIRDERS. 883
Plate Web Giudebs.
F s=s shearing force in tons.
D =s depth of web in inches.
t=s thickness of web in inches.
9 =3 spacing of stiffeners in inches.
/= spacing of rivets connecting web to flanges or in
web seams in inches.
7 =s shearing or bearing yalue (whichever is less) of one
rivet in tons, taJcen in single or doable shear as
the case may be (pp. 286-8).
nssnnmber of rows of rivets.
t = F/2*5 D (minimum).
«2= 1500<«
(^-)-
8 == IIVD/P.
Note. — (1) It is assumed above that the shearing stress
allowable is one-half the tensile strength, and that the latter is
5 tons per square inch. For any other tensile strength f, change
F to 5f//. (2) It is frequently advisable to make t thicker
than would be given by the first formula in order to avoid
too close spacings of stiffeners or rivets. (3) The size of
the stiffeners can only be determined by experience, e.g. make
total weight of stiffeners the same proportion of weight of
web as in a similar suwessful design.
Excmtple.^-DeBign a solid plate transverse frame of a large
ship ; spacing 4 feet, depth 8 ft. 6 in., weight of ship 70 ions
per foot run, /=: 8 tons per square inch.
• A rough estimate ojf F is found by assuming half the
weight of the ship to be taken on centre when in dook^
the remainder being taken on the side docking blocks. Theo
F near keel is i the weight over a length of one frame spaoe^
or 70 tons.
Change F to 5f// or to 45 approximatoly.
Then t = 45/2-5 X 42 = *43 inch minimum.
On trial this would be found rather small ; take t = i'^.
, 1500/5x42 x»5 ,\ ^^ .„ ., . .,
tr = "~T~\ 45 1 ) or « = 22-4 ; so that the
stiffeners should be spaced about 2 feet apart.
To obtain v, assume rivets }^ diameter. Shear value from
table is 2*76 tons. Take « = 1 ; then a' = 276 x 42/45 =
about 2}.
Shafts.
H = horse-power transmitted.
T = torque or twisting moment in inch-tons.
d^, d^ = external and internal diameters of shaft in inches.
^ = maximum shear stress in tons per square inch.
0 S3 coefficient of rigidity in tons per square inch ;
= 5,500 for steel.
e = angle of twist of shaft over length I feet in degrees.
N == number of revolutions per minute.
884 SHAFTS.
_ _T g __ irc0
T = -196^ (<^i*-<^2*)/<^i = q X torsional modalos of section,
fl (steel shafts) = 1-27 Tl/(d^* --d^^) ^ '25 qlfd^.
TN = 282 H.
H es ^ (di^-d2^)/U7d^ = -426 qix X torsional modoliiB of
section.
The torsional modulus of section and polar moment of
inertion for shafts of various signs are tabulated on pp. 318^
319 ; see note at foot of tables.
Working Values of Shear Stress q (Jtons per sqwsr§ ihcK).
Wrought iron about 3^
Forged steel (from scrap) . . . 4 to 4|
Forged steel (ingot) . • . • 5
Cast iron 3
Gunmetal 2
Copper IJ
Note. — ^The stress allowed depends on the fluctuation of
load. Where load varies greatly, as in factory shaftingy talce
about } the above values. In ordinary engines divide q by
the ratio in which the maximum torque exceeds the mean,
i.e. by 1*6 with single oylindersi by 11 with two, and by 1*05
with three or more oylinders.
When the shaft is subjected also to a bending moment M,
substitute for t the equivalent twisting moment M +Vm^ + t*.*
Frequently the size of the shaft i& determined from considera-
tions of stiffness ; the twist in long shafts should not exceed
1^ in 20 diameters' length, which makes q about 2}.
For Board of Trade Eules, see pp. 464 and 465.
Square Shafts.
See note at foot of table on p. 336.
TWISTINQ KOMEXT OF A CSANK.
Fig. 205.
V T J
Let AB = centre line through cylinder and shaft in inches.
AG = line ^rouffh centre of crank.
AD s= line at right angles to AB.
* fPhis irires th« maximnm direct stress, which is asnally taken as
a criterion of a shaft's strength. To get the maximnm shear stress, take
the moment Vm*+t*.
COUPLINGS, BEARINGS. 835
BO s= connecting rod.
CD = line BO produced.
AE = line perpendicular to bd.
P = load on the piston.
B = thrust on connecting rod.
0 = angle ABC as angle bad.
Twisting moment :^BXae = bXad cos9==px ad.
Couplings.
Flanged Couplingt,
End of shaft enlarged to 1*12 diameter to take keyway.
If (2 = shaft diameter in inches,
Number (n) of 1x>lt8=}i2+3 for shafts over IJ, diameter.
Diameter of body of coupling =r: 2(1? -f-. ^n
Diameter 0) of bolt = .6d/ -/n.
Diameter over flange =» 2(2 -{- 1^ -{- 6| 9.
Tfaic&ness of each flange =» *5<? + *25.
Total length of coupling » 2^(1 4- 1-25..
Diameter of recoas for bolts =r 2*5S.
Box Coupling,
Fig. 206.
y_ ZAi.-.. L = 3d + 11"
HI ]""! 1 * = -45D+r.
Beabings.
DUtanee between bearings in line shaft loaded with pulleys.
Diameter of shaft in inches 1} 2 2} 3 3} 4 4}
Distance in feet . • .7 8 9 10 11 12 13
Distance between bearings for high speed unloaded shaft,
*/d^~+~d^
Distance in feet = 175 V a • (Symbols as on p. 338.)
For tk ship's shaft with thrust, change 175 to about 160.
Working Pressure on Bearings,
Pressure in lb. per sanare inch of
BTain crank-shaft bearings
» t> »
n » f>
19 it »
Line shafting on gun-meta
,y ,, cast iron
Pivots on gun-metal
„ lignum vitse .
projected area (I x d),
200 (slow).
300 (fast passenger).
300 (warships).
500 (T.B. destroyers).
200
50
200 to 500.
1,200
Also pressure in lb. per square inch multiplied by rubbing
speed in feet per minute should not exceed 50,000.
386
HELICAL SPRINGS.
a " A
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VIBRATION.
387
Helical Springs (see table opposite).
D &= mean diameter of coil in inches (from Zd to Sd in
general).
b, d = 8izQ of section in inches (5 greater than d, ace
I table).
ft SB number of complete turns of coil,
w = load in lb.
8 = deflection in inches under load w.
/ = shear stress due to w in tona per square i^oh^
about 25 for ordinary working loads in tempered
springs (9*1 in circular and 11*3 in square safety
valve springs under Board ol Trade Bules).
c .= coefficient of rigidity in tons per square inch =
5,500 for ordinary steel, up to 8,000 in special
steels.
Frequency dp Vibration.
N = number of complete (to and fro) vibrations per minute.
The material is assumed to be steel, having Young's
modulus = 13,500 tons per square inch, and coefficient of
rigidity = 5,500 tons per square inch. For any other material
the frequency ia proportional to the square root of the elastic
» coefficient.
Conditions.
N.
1. Mass hung from spring which deflects 9 in. .
2. Mass w lb. hung from spring with n coils
diam. Din., diam. of iron d in. (w includes
} mass of spring).
3. Mass as above, radius of gyration E in.,
torsional vibration.
4. Mass as above, suspended by wire diam. c2 in.,
length I in.
5. Taut wire rope, circumference c in. , length I ft. ,
tension t lb.
6. Do., but weight w lb. given in lieu of
circumference.
7. Weight wlb. on the end of rectangular canti-
lever I ft. long, moment of inertia of section i
in inch nnits, about axia perpendicular to
plane of vibration.
8. As above, but weight w at ipiddle of beam
I ft. long, ends free.
9. As abov0, bat ends of beaxQ fiz^d in direction
10. As 7, but weight uniformly distributed . . .
11. As 8, ,,
IB. Uniform bar unsupported
it
>>
I ■ p ■■■ 1 1
188/ V«
223,000<ig
V'wnD*
129,000d8
KV^WnD
206.000 (f'
170\/—
V
I
X 43,000
X 172,000
X 86,000
X 88,000
X 250,000
X 550,000
X 550,000
888
HOOKS.
BiNQS.
D = inean diameter of ring in inches..
d = diameter of iron in inches.
Proof load in tons == X^d^jn.
Working load = one-half proof load.
The Admiralty proof test for standard rings and ring bolts^
where D = ^d, is 4<^.
Hooks.
The following proportions and working loads of crane
hooks are given by Mr. Towne (fig. 207). If the top be
formed into an eye, make external diameter of eye 1*8,
fnterniU diameter '1,
Pio. 307.
Diameter i
in inches. |
Safe load in
Diameter
in inches.
Safe load in
11
5a
IS
2
2i
2i
21
Safe load in
lb.
tons.
lb.
tons.
lb.
tons.
1
1
690
830
1,200
1,480
1,870
•26
•37
•63
•66
•83
u
18
14
li
li
2,420
2,820
3,450
3,910
4,720
1-08
1-26
1.64
175
2*10
5,370
6,080
7,700
9,460
11,400
2-4
27
8.4
4-2
61
For heavier loads take square of diameter to vary as the load.
KEYS, BOLTS. 339
British Standard Keys and Key ways.*
(See table on following pago.)
The keys to be cut from standard key-bars whose width and
thickness are '002 inch greater than nominal size of key.
Keys whose length is not more than 1^ diameters of shaft
to have a taper in thickness of 1 in 100. Nominal thickness is
that at large end. The depth of keyway at centre line to be
j- thickness of key.
Screwed Bolts.
The strength of a screwed bolt under shear, or under
tension if not screwed too tightly, is about three-quarters that
of a bar or rivet whose .area is that of the bolt to the bottom
of the thread.
Under tension an unknown factor is usually introduced
by the stress caused by screwing up. In the following table
allowance is made for this.
Working street (/) in lb, per square inch of screwed holts.
(H. J. Spooner, Esq.)
/ for steel.
/ for iron.
Largest sizes of bolts and studs
Under 5" diameter .
Ordinary marine practice .
Cylinder under 10" diameter .
6,000
4,500 to 3,000
(least stress in
5,000
2,500
4,800
3,600 to 2,400
smallest sizes)
4,000
2,000
For rougher joints with packing which must be compressed
to make the joints tight, halve the above values.
For areas of screwed bolts to the bottom of thread, see
p. 535.
Paessubb op Water on Dock Gates.
D B= depth of water in feet.
L ss length of one gate in feet.
T«B mutual pressure between gates at middle in lb.
N x=s normal water pressure on one gate in lb.
£^ KB distance from point where gates meet to a right liae
joining their hinges.
d
* Reprodaced by permission of the Bngineerinjr Standaras Committee
from fhelr Report No. 46, British Standard Specification for Keys and
Keyways. published by Messrs. Orosby Lockwood & Son. Price £«. M.
840
BRITISH STANDARD KEYS AND KEYWAY8.
(See p. 339.)
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8TBEN6TH OF WHEEL GEARINO. PAl
TOOTHED-WHEEt GEARING.
Ihsy Met?wd of Setting Out the ShetJi. (Fig. 208).
Let AB be the pitch circle. From the same centre draw
circles CD, kf, so that their distances from the pitch circle are
Fm. 208.
respectively -35 P and -45 p (p being the pitch). Tiie points of
the teeth will end on CD, and the roots on ef.
Round the pitch circle AB, set oif the pitch and the edtfes of
the teeth, making the thickness of each tooth 45 p.
Through the edge Q of onte of the teeth draw ttqs inclined at
an aagle of 76° to the radius through q. Make rq equal to -95 p
and QS -55 P ; and through R and s draw ch-cles GH, bp. From
R as centre strike in the curve db for the lower half of the
tooth, and keeping this radius constant and the centres always
Fig. 209.
on the curcle GH, dnw the lower half of fettch too*h in turn
iben from centre s strike m the curve ed for the uppw half!
and keepmg the centres on KL, and the same radius, draw in
842
STRENGTH OF WHEEL GEARING.
the upper half of each tooth. The complete shape of eacli
tooth is now drawn in.
In order to draw the line BQS, the instrnment shown in
!fig. 209 is often used.
The parts BO and ad are inclined at 75**, and from A two
scales are set off along ab and ac, so that QB and Q8 may be
at once measured when the pitch is given in inches.
This method gives a tooth approximating to involute form ;
it is sufficiently accurate for gears wording sluice valves,
w.T. doors, etc., where evenness of running is of minor
importance.
Usual ProportianB of Teeth.
*
Common
pattern
Moulded
Wheels.
Maohlne
Moulded
Wheels.
Worm
Wheels.
Pitch line to tip . . .
Pitch line to root . . .
Totaldopth
Thickness at pitch line .
Width of teeth— large .
„ „ small .
ZZp
•42 p
•75;?
•45 2?
2 p
3 p
•3 p
•4 p
•7 p
'ASp
2 p
3 P
•37 p
'61 p
'48 p
1-5 p
2 p
ft. per min«
1,800
2,400
2,400
2,600
3,000
3,000
Length of worm, ^p ; p = pitch.
Limitinff Speeds of Toothed Gears,
(Mr. A. Towler.)
Ordinary cast-iron wheels
Helical „ „
Mortise „ „
Ordinary cast-steel wheels
Helical „ „
Special cast-iron machine-cut wheels
Strength of Cast-iron Teeth.
P = pressure in pounds transmitted (assumed concentrated
on one tooth).
/ = stress allowable in lb. per square inch.
b = breadth of tooth in inches.
j9 = pitch in inches.
Then / = 10,000/\/ speed at pitch circle in feet per second;
this is given approximately by the following table :—
Speed-ft. per min. 100 or less 200 800 600 900 1200 1800 2400 9000
/ . . . . 8000 5500 4500 8200 2500 2200 1800 1606 1580
P = '05/pb,
8TBEK6TH OF WHEEL OEARINO.
3i
Relation of Horsb-powrr Transmitted and Velocity
AT the Pitch Cibcle to Pbbssube of Tkkth.
's|
Si
Velocity in Feet per Minute.
Nnmber
Power T]
60
180
800
490
540
600
TOO
900
laoo
1500
H.P.
Lbs.
Lbs.
Lbs.
Lbe.
Lbs.
Lbs.
Lbs.
Lbs.
Lbs.
Lbs.
1
660
183
110
79
61
60
42
37
28
22
S
1,100
367
220
157
1«2
100
86
73
65
4A
3
1,650
660
830
236
183
160
127
110
88
6C
4
S;300
733
440
814
224
200
169
146
110
8£
6
8,7M
917
660
393
306
360
212
183
188
lie
10
6,800
1,833
1,100
786
611
600
423
867
276
22C
15
8,260
2,750
1,660
1,179
917
760
636
650
413
830
20
11,000
3,667
2,200
1,671
1,22a
1,000
846
733
550
440
25
13,750
4,583
2,760
1,964
1,527
1,260
1,058
917
688
660
90
16,500
6,600
3,300
2,367
1,833
1,600
1,269
1,100
825
660
40
22,000
7,333
4,400
3,143
2,444
2,000
1,693
1,467
1,100
880
60
27,600
9,167
5,600
3,928
3,066
2,500
2,116
1,833
1,376
1,100
60
33,000
11,000
6,600
4,714
3,667
3,000
2,638
2,200
1,660
1,320
70
38,600
12,833
7,700
6,500
4,278
3,600
2,962
2,567
1,925
1,640
80
44^000
14,667
8,800
6,285
4,889
4,000
3,386
2,933
2,200
1,760
90
49,600
16,600
9,900
7,071
6,500
4,^00
3,808
3,308
2,476
1,980
100
66,000
18,333
11,000
7,857
6,111
6,000
4,231
3,667
2,760
2,200
110
60,600
20,167
12,100
8,643
6,722
6,500
4,654
4,033
3,025
2,420
120
66,000
22,000
13,200
9,423
7,333
6,000
6,077
4,400
3,300
2,640
130
n,600
23,833
14,300
10,214
7,944
6,600
6,500
4,767
3,576
2,860
140
—
25,667
15,400
11,000
8,566
7,000
6,923
6,133
3,860
3,080
150
—
27,600
16,500
11,786
9,167
7,600
6,846
6,500
4,126
3,300
160
—
29,333
17,600
12,671
9,778
8,000
6,769
5,867
4,400
3,520
170
—
31,167
18,700
13,357
10,389
8,600
7,192
6,233
4,676
3,74C
180
—
19,800
14,143
11,000
9,000
7,616
6,600
4,950
3,96C
190
—
20,900
14;929
11,611
9,500
8,038
6,967
6,225
4,18C
200
—
—
22,000
16,714
12,222
10,000
8,462
7,333
6,500
4,40C
300
—
—
33,000
23,671
18,333
15,000
12,692
7,700
8,250
6,60(
400
—
—
44,000
31,428
24,444
20,000
16,923
8,067
11,000
8,80(
600
—
55,000
39,285
30,555 25,000
21,154
8,433
13,760
11,00(
844 PROPELLER STRUTS.
N&^.^l. Pot roti^Iy-cttt wheels where the Whole pre&sare
may come on one corner of the tooth, make b in the formuls
Up whatever its actual value. ^
2. Mr. W. Lewis has shown thai the coefficient in the above
formula ia '067 for well-shi^ed teeth ; and that it increases
according to the number of teeth.
Number of teeth ... 20 30 50 rack.
Coefficient— radial flanks . . -06 r065 '069 'OTiJ
,, well-shaped teeth. -09 -102 -112 •124
3. For teeth liable to sadden shock? / should be halved.
4. The pressure P is related to the horse-power M, the
pitch diameter d inohes and the revointionB per minute i^
by the formula PN£? = 126,000. The relation between power,
pressure, and velocity is expressed in the table on p. 343.
5. For teeth of materials atber than, oast iron, multiply /
by the multipliers below.; or alternatively multiply the pitch
deduced from the formulee by the relative pitches below,
keeping proportions the same.
i\Ta^<ii.iai ttrA»;i Oxin- €a3t Wi^oDgrht Phosphor Kickel Vailadiam
Maxenai. vvooa. ^^tal. Stedl. Iron, Bronze. SteeL Steel.
Multiplierfor/— 1-5 .2.1 2-3 2-3 3-7 4 to 6
Belative pitch 1 -82 -69 -65 -65 -52 -5 to -45
Wood teeth, although weaker than iron, are differently
proportioned so that their relative pitch is about the same.
•e. The least number ef teeth (Unwin) == 791S/>3x for iron
and 961h/^3n for mortise.
7. To secure quiet running of wheel gears the diameter
should be kept lowf ^ as to reduce the peripheral velocity;
to secure adequate strength the teeth should be as short and
wide as possible.
Dimensions of Propeller Struts (' A ' BBaoKEXs).
(A. W. Johns, Esq., M.t.N.A.)
The size of the struts may be determined from the stresses
caused by the loss of one propeller blade. The strength should
be the same as that of the end of the shaft under the same
conditions. This gives the following formula for a cast-steei
bracket : —
Rr = -62- • — - — cos— , where
a B 2
R, r »tiie greatest and least dimensions of the arms (R is
conunonly about Z'5 r).
D, d ~ the external and internal shaft diameters.
ni = longitudinal overhang from oentre of propeller to
centre of bracket.
a s= longitudinal clear overhang from centre of propeller
to after end of bearing on bracket.
0 « angle between arms of bracket.
All ia inch units*
LONGITUDINAL STRESSES IN SHIPS.
845
I^ea^ik ol i>earii]g ia commonly abeni 6} d ; it is dtoter-
nuned from the bearing ptoasnre, Which should not exceed
about 20 lb. per square inch projected a^a. Thickness i^onnd
bearing (ex bush and gland) about |f .
LOKOrrtrBIHAL SiritESSES 11^ SHIPS.
The vessel is assumed to float in a wave of its own length
(h.p,^ whose height is ^ the length.^ The maximum B.M. is
calculated when the middle of length lies over (a) the crest
of the wave, (5) the trough ; th^ moments being tearmed
hogging and saggin? respectively. Begardiiu^ the ship as a
bean^, zhe moment of inertia is calculated amicUhips, or at any
weakened section near amidships ; the stresses in keel and
upper works are then calculated by the ordinary beam
formulce. These stresses are nsually limited to certain amounts
appropriate to the conventional conditions assumed.
Peactical Consteuction op Bending Moment Cueves.
1. At sections spaced about one-twentieth t of the length
draw curves of areas or ' bonjean ' curves. These curves give
the immersed area of each section at any draught. They are
conveniently constructed with the help of an integrapn ; or
otherwise the information- on the displacement sheet should
be utilized.
2. Set off the wave on a contracted longitudinal scale ; the
vertical scale should be the same as that adopted in the above
curves. The wave is assumed trochbidal ; it may be con-
structed as on p. 34, or by means of the following table which
gives the proportions of a standard trochoidal wave.
X
h
0
1
•034
2
•128
3
•266
4
•421
5
•677
6
•720
7
•839
8
•927
9
982
10
1
g = distance from crest divided by ^ length. * '
h = depth of wave surface below crest divided by height
of wave (or by ^ length)*
3. Deteifmine the Weight and centre of gravity (longitudinal)
of tbe ahip. It is nsval tO assume that condition which provides
the greatest 3B.M. ; e.g., for hogging take out all midahip
weignts such as coal, reserve fieed" water, etc.. and for sagging
take bitnkers and midship tanks full.
* A ibore lojgrical asstuni»ti6n is to limit tUls i-atio to Ahips less than
470 feet In len^tti. In longer ships the Iielfflit shcHild vfttv as the square
raotqf the length, tbos U^2' ior 500', 96-4' fob 650', 26'6' for 600', S8-6' for
TtXr , 80*6' for fiX)', and so on. ^is leads to calculated stresses of reasonablo
magnitnide in evcm the ilatgrest ships. The proportions given above for
a standard ttoodhoidal wave ^en n<o longer hdld.
t One-tentb spacing is sdmetiiUes sufficient. The number of divisions
depends on the regularity of the form of Jthc ship ; in most cases the
* 0B6«twentieth ' need only be used at the ends.
/
846
LONGITUDINAL STRESSES IN SHIPS.
4. Determine the position of the ship relative to the wave
60 that the displacement is equal to the known weight, and
the longitudinal position of centre of buoyancy is below tho
known position of the O.Q-. This is done by trial and error ;
the midship section of the ship is maintained at the crest or
hollow of the wave. Draw on the diagram of wave a line
representing any fixed line in the ship — say, the load water-
line. The position of this line is guessed in the first place ;
from it the draught at each section is measured, and thence
the immersed area is obtained by the bonjean curves. From
these areas the displacement and C.B. are calculated by
Simpson's Bule. If these are not correct take a second line
on the wave diaffram. and so on until agreement is finally
obtained.
Fig. 210.
BJH
5. On a base of length draw) a onrve of buoyancy in tons
per foot of length, fiBB in ^^. 210. The ordinate of this curve
is ^ the immersed area of section—both sides.
6. Draw a eunro 6f weight per foot ran. This entails
a rather laborious calculation whiob may be performed
as follows : Divide the ship into oompartmeate separated*
by the sections used in the buoyancy caloulations. Tak^
each item of weight from the weight oalcnlation of
tlie ship (p. 102) and distribute it, finding what amount
lies within each compartment. Thus, the skin plating,
if uniform, can be distributed by drawing a curve of
girths,; (finding the area interceptea within each compart-
ment, and multiplying this by the tetel weight divided by
the total area. Framing is distributed by calculating weights
of a few typical frames, and setting them off in a curve.
LONOITUDINAL STRESSES IN SHIPS. 34^
Concentrated items — engines, boilers^ etc. — are easy to dis-
tribnte, foif tiiey; usnally lie entirely in one or two
compartments.
These component weights are placed in a table hating
a column for each compartment ; by addition the total weight
is found for each compartment. Set off in the middle of
each compartment (fig. 210} the mean weight per foot run
within, i.e. total weight -r- length, and draw a line parallel to
the corresponding portion of the buoyancy curve. This givef
a stepped curre of weights www . . . The scale should be
the same as for bbb . . .
7. Keasure the intercept between the weight and buoyancy
curves ; set them off above the line for excess buoyancy and
below the line for excess i^ight. This gives a series of
rectangles forming a stepped curve of loads llll ....
8. Determine the area of each reetangle, counting negative
below the line. Commencing from the left, set off the area
of the first rectangle on the ordinate tb the right ; then the
area of the first two rectangles on the ordinate to the righ^
of the second ; and so on, each ordinate representing to scale
the area of the portion of the loads curve lying to the left.
This gives the curve of shearing force s.f. (In fig. 210
it is shown reversed.) The curve should close ai the extreme
right-hand side.
9. In the same manner find the areas of portions of the
S.F. curve, setting them off to scale on the right-hand
ordinate. This gives the B.M. curve which should also close.
10. The maximum ordinate of the B.M, curve gives the
bending moment required.
Alternative Method.
In order to avoid the laborious process described above, the
weights of each half of the ship are often calculated separately,
as shown for the battleship On p. 109. If this is done, proceed
as indicated in paragraphs 1 to 5 above, and then as follows : —
6. Treat the buoyancy curve as directed above in par. 8
for the curve of loads, but commence from each end, and
continue only as far as amidships. We thus get two curves
whose ordinates represent the area of the buoyancy curve
outside it-^on the left for the left-hand portion, and on tl\e
right for the right-hand portion.
7. Treat these two curves in the same manner, again com-
mencing from either end. As a result we have two curves
which may be termed the curves of bending moment due to
buoyancy.
8. Find the arithmetic mean of the two bending moments
amidships as obtained from the curves just drawn. Call
thi5 M^^
/
648
LONGITUDINAL 8TSBSSES IK SHIPS.
9. Find also tiie atithmetio mean of tho nomenld of fotwaitf
and after weights about amidships. Thus, in the batttesliip,
p. 109, this is } (1,694,800 + 1,052^250) or 1,373,500 foot tons.
Call tliis M^,
10. ^he difference betv^een Mr and it^^ is the required
maximum B.M.
APPBOXIMAT& Values of B.M. in vabioub CLAsaes.
w s=: normal displacement of ship in tons.
L = length of ship between perpendicular*.
yATiTmtm B.M. = wl/k ; where K is about 25 for modern,
battlefdiips, steam yachts and scouts, about 20 for desttoyers
and very fine vesBels, 25 te SO for liners, 30 to 35 for carg^o
ships and older batileshi|>s.
The hogging moments are %istially great<9r than the sagghig,
except in very fine vessels. Warships nsing ^il fuel are
liable to have large hogging moments, unless the oil tanks
iwe confined to the machinery ispaoe.
Shost Method of coNdTRCCtiNa Wskibt Cn&?«.
(Sir J. Av BUea, LL.D.)
The laboni^ xnTolved in drawing the curve of weights id
due principally to the hull items. In this method the curve Is
assumed to txmsuit of three straight lines (fig. 211), the
ordinates being — » — > and — > where Hs« weight of hull
L L L
in tons, h =s length of ship in feet, and a, b, and o are co«
efficients depending on the type of ship. Note that a -\- ib^
-{-.0 = 6 in ail cases.
In an ordinary passenger or cargo vessel a = *566^ b =
1*195, 6»-65a.
The weights of equipment, eargo, machinery, and fuel are
readily distributed and added to the above distribution of
hull weight.
Sta. ill.
AFT
.aj 4
U— ^ — 4-
-^-*h-i--i
F^O
Weight CuEvfe for Warships.
In large warships a very close apprexlmatioa to the dis-
tribution of hull weight is found by taking it f aa tin
buoyancy in still water and i as a trapezoid proportioned
so that its CO-, lies over that of the hull.
LONGITUDINAL STRESSES IN SHIPS. 849
MoMByT OF Inebtia of Section.
1. Bouglil^ predict the position of the neutral taxia, and
draw in a trial axis near it.
2. Oftletthtte for each portion of the straotnre which hae
longitudinal continnity the area ▲ in aqnare inohes, the dis-
tance y feet from neutral axis assumed, and the depth d feet
(for side plating and other vertical portions).
3. Arrange these in two tables (pp. 850 and 351), putting
in one the portions above and in the other the portions below
the K.A. Insert also the products Ay, Ay', and Ad^.
4. Find the soma of a. Ay, Ay', and Ad^- ; subtract one-
twelfth of ihe last fvom the sum of Ay' ; call the difference i.
5. Taking the two portions find the sums of A and i, and
the difference of Ay. Call these a^, x^, A^iA. The position
of the real neutral axis from that assumed is equal to A^y^
divkled bj A^ ; while tiie moment of inertia about the real
neutral axis is l^ — (A*y*)«/A.
Thus, in the exaviple —
A Ay I
Above NA. 2271 84,200 654,000
Below N.A. 1873 21,700 382,000
3644 12,500 1,066,000 in.' x ft.'
Excess above.
Beal neutral axis above assumed = 12,500/8,644 = 8*4 ft.
M.I. about real N.A. = 1,036,000 - (12,500)'/8644 «
993,000 in.' x ft.' ; or 1,986,000 for both eidea of the ship.
6. The new distances y of N.A. from upper deck and keel
ace now fonnd. They are 20*6^ and 23*5' in the example.
7. The stress in tona per square inch is tlien My/i. That
in the portion in tension is Increased owing to the rivet holes ;
usuallv it J3 conaldared that ^ ol the area is thus loai^ so
that toftatreia must be iBcreased by V« ^^ ^^ example^
Hogging,-^}!^^ 437,000 feet tons.
rr -1 *«. •' J 1, 487,000 X 20-6 11 ^ _
TenaUe stress in opptr deck » " i 995 000 — ^ "o ~ ^'^^
tons per squiuw inch.
. ^ . , , 487.000 X 23.6 ^ ,^,
Compressive stress m keel = — 1 flee OQQ — "* ^'^^ *<»>* P*'
square inch.
3^n^--^M ^ 386,000 feet tons.
n • *«^ • J 1. 336,000x20-6 ^'^
Qf^fi^^9$i^ stress m upper deck = ' 1 986 000 ** ^'^^
tons pex square inch.
rr -1 ^ • 1, 1 836,000x23-6 11 ^ ^„ ^
Tensile stress m keel = — , ^^r^ ,^r^ — ^ -s* = 4'87 tons per
l,V9O,0Ul7 If
square inch.
350
Calculation for
Battleship 590' x 90' x 27' x 25,000 tons.
POBTION ABOTE ASSUMED NEUTBAIi AXTH.
-
from
N.A.
tt.
o .*;
*
o
Item.
Scantlings.
1
t
Area
sq. in. *
Distance
assumed
Se<
S.S
6
Upper deck
(
' 39'x70 lb.
819
24-0
19656
471740
Gunwale bar ....
1 (rxe"x87-6 „
11
23*2
265
5920
Plating behind 6" armour .
s'xao „
78
19-4
1897
27100
4610
Deck angle to main deck .
1 6"x6"x28 „
8
15-6
125
1950
Main deck
! 38' e"xl4 „
102
15-8
2560
40450
Angle underside main deck
6"x6"x28 „
8
16-1
121
1830
Plating behind 12" armour
12-2' X80 „
110
91
1001
9110
16400
Angle at heel of armour .
r'xr'xas „
10
80
80
90
Top girder behind armour
8"xe"x6"x34„
10
11-8
113
1280
2nd „ „ tt
8"x(Jf'x6r'x34„
10
7-6
76
560
Top angle Bhd. mid. to main deol
i 3j;;x^;xio „
3
16-8
47
740
Bottom „ „ „
3j"x8l"xi0 „
8
9-4
28
2'/0
Topstrake.. .. ..
8-7' X14 „
16
14 0
224
8140
220
Bottom ti II 11
8-4' X20 „
20
10-9
218
2380
280
Middle deck (flat)
26-5' X 40 „
818
9-2
2926
26930
„ sloping top thickness
13-6' X 60 „
248
6-5
1580
10280
9620
M t» bottom M
14-8' X 40 „
178
6-4
1189
7290
8720
Inner girdle, top angle
3i"x8j"xl0 „
8
90
27
240
„ bulkhead, top plate .
1 5-8' X12 „
21
6-8
182
880
700
Wing „ „ angle .
; S§"x3i"xio „
8
9-0
27
240
If 1* II strake
1 5-8' X 12 „
21
6-4
184
860
TOO
1, II 2nd 1.
4'xl2 .,.
14
2-0
28
60
220
Inner bottom angle to mid. deck
1 6"xe''x28 „
8
8-9
81
120
„ t. top strake .
8-9' X 14 „
16
2-0
82
60
250
Outer „ angle to mid. deck .
e"x6r'x87-6 „
11
20
22
40
„ „ P strake .
2-rx85 „
23
10
22
90
100
Cover plate to armour
1-6' X 14 „
6
2-0
12
20
Strap to 6^^ backing .
1'X35 „
11
19-8
212
4100
.,1^
1'X85 „
11
11-6
127
1460
•I 13f »» ...
1'X85 „
11
6-8
75
610
Angle, middle deck to casing .
3j"x8i"xl2 „
4
9-8
87
840
„ main „ .,
3*"x3rxi2 „
4
16-8
65
110
„ upper „ If ■ .
3j"x85"xl2 „
4
24-8
97
2860
Casing ooaming, middle deck .
1B^'X25 „
11
9-0
99
890
I, I, mam „
1B^'X26 „
11
16-0
176
2820
upper „
IST'xaS ..
11
240
264
. 6880
„ middle to main
6-4' X20 „
88
12-6
479
6040
1660
M main to upper „
rxl4 ..
29
20-0
680
11600
1420
2271
84200
660U0
44780
Ax
44780
Total
8780
668840
Moment of Inertia.
Trial neutral axis assumed to lie 20 feet above baae.
POBTION BEIOW ABBDMBD NbUTB*!, ASM,
362 LONGITUDINAL STRESSES IN SHIPS.
Alternative correction for rivet holes.
Alternatively the correction for rivet holes iai made in the
moment of inertia by reducing a. Ay, and I for the half in
tension by ^ Thus for hogging, in the example —
A Ay I
AboveN.A. XA 1»858 28,00a 635,00a
Below N.A. . . 1,373 21,700 382,000
3,^31 6,300 917,000
N.A. above that assumed = 6,300/3,231 = 1*95 ft.
Moment of inertia = 917,000 - (6,300)2/3,231 = 904,000.
The new y ie found and the stresses calculat.ed as before
except that the factor y* is omitted.
The resulting stresses are usually within 5 per cent of
those found by the other method.
Stresses allowable.
The conventional method of determining the stvesses greatly
exaggerates them, particularly in large ships.
In ships about 400' long, or less^ allow 6 tons per square
inch ; 8 is permissible, for portions in tensions only, wher«k
high tensUe steel is used. In larger ships 8 tons (compression)
and 10 tons (tensile with h^h tensile steel) have been taken,
Stbess dub to Shearing.
The shearing force on the hall is greatest at about one-
quarter of the length from either end, and is 'approximately
one-seventh to one-eighth of the displacement. It can be
determined exactly from the curves of shearing force (fig. 211).
The shearing stress on the side plating and edge rivets k
found by the method described on p. 332.
Effbct of Continuous Superstruotube.
Let A == area of original section of ship in square inohei.
a =s area added in the form of superstructuie in square
inches.
h ^ height above original neutral axis at whi^eh a may
be supposed concentrated, in feet.
I = original moment of inertia of section in (inchefi
X feet)«.
y sa orif inal height in feet of top deck about neutiaj
axis.
New moment of inertia = i -i -, —
A + a
New distance 'y' = distance of superstructure above
new liT-A. ^ aA/(a— a). In order that the stress on the upper
part of structure ^hall be reduced, the area added a most be
greater than a , ^ • ^^ ^^ This quantity is greatest wkem
^ =» y + V y* + i/A ; the minimum effective added area is then
equal to i/2y(y + y/y^ -[- i/a).
MECHANICAL POWERS.
353
MECHANICAL POWERS.
The power applied and the weight lifted are directly propor-
tional to the distances moved through by each body in a given
time.
w = weight to be raised.
p =s power applied.
D = distance of power from fulcrum.
d » distance of weight from fulcrum.
n » number of movable pulleys.
L = length of inclined plane and wedge.
H B height of inclined plane.
0 s= circumference described by p.
t B thickness of wedge.
s s distance moved through by P.
* « distance moved through by W.
E -resistance to wedge.
p =■ pitch of screw.
Genbbal Formula for all the Powebr.
w«
«r sp
s
p =
B
8-H
P
BP
w
The Lever and Wheel and Axle.
w =
PD
d'
wd
wd
W
Fia. aiS,
I
®
Fig. 2ia,
"rit-'i
©
PlO. 214.
♦ P
K ■
"•^^
Fig. 215.
Aa
354
INCLINED PLANE, WEDGE, SCREW.
Thk Pulley.
w = 2p» p^* —
2)1
Fio. 216. Fig. 217.
ONE MOVABLE PULLKY TWO MOVABIJ: rUT.LEYS,
"^ ^
Wi
Note, — For revolutions ot wheels see p. 362.
The IsrcLiNED Plane.
H
p^:^s
L
W
, WH
P
The Wedge.
t
L
B
The Screw.
2»
c
PC
-* w
p
FlQ. 218.
I IQ. 219.
Fig. 220.
Note. — One-third more power thaa is obtained by the fore-
going fonnulae is generally allowed, in order to overcome the
resistance due to friction, &c., weight and power being in
equilibrium.
MECHANICAL POWERS.
o ," .■•
Fia. 221.
r-IPPEBENTIAL PuLLBY.
A Dlfffr&niial Pulley consists of two blocks (see
fig. 221). The upper block contains two sheitves
of sllgbtly different diameters, secnxied so as to
revolve together. A chain is wound on the blocks
93 ahowD, the blocks having projections on their
rims to fit the chain and prevent slipping.
Sappose the upper block makes one revolntfon :
Then the length of loop abod is shortened by a
length = circumf ereyoe large sheave, and the loop
is lengthened -circumference of small sheave.
Circumference of large sheave « 2ir<^,
n „ small sheave » 2vdf| ;
/. Difference in length of loops2ir(<f — <^i),
and the weight will be raised ir(rf-<f,).
If p =s force acting on chain, friction n^lected,
for one revolution of wheel p moves 2wd ;
/. px2x<f=ir(/i-<f,)w,and p=^Z^»w.
• Pulleys with Friction.
where ^' and k are constants which can be detennined for every
system of pulleys by two experiments.
If the weight to be lifted is very large, p^ can be neglected ;
if very small, k can be neglected. *
SFFICIEMCT of SCREW&
J^ffleienejf tf Sorewi, — Let ab be one turn of the screw
developed, then BC = pitch, and ac = circumference of screw,
w « weight lifted, r = reaction of screw thread. When R makes
an angle ^ with the normals angle of repose» coefficient of
* A rough rule is to Bsanme the tension of the rope to be diminished by
6 or 10 per cent, after each turn round an ordinary sheave. See example
on p. 298.
856 SCREWS.
friction =/i. Now h is caused by the power applied to turn the
screw ; .', its vertical component = w, and its horizontal com-
ponent s is such that if P = moment of force used, p = s x - ;
.'. p = _ sin (fl + <^), and w « R cos (0 -f ^).
2*
Work done by power in one revolution - p + 2ir « -Rwd sin(9 + ^).
Work done on weight = wp = Kjp cos (^ + ^).
Efficiency = r^— — r.
* tan (e+<^)
This is a maximum when 6 = 45°— J<^.
Then taking Tan 4) = 2 Tan %, we get
Maximum efficiency = / - — |-** )
Conversely, if action be reversed,
Efficiency- ^^
For an irreversible screw 0 must be less thad ^. In screw
steering gear and W.T. door screws, which should be jast irre-
versible, 0 is made '08 or 4\ degrees, giving a pitch equal to about
4 mean (pitch) diameters.
The thickness of a square thread is tisaally one-half the
pitch; the depth is about if pitch. In the "Acme** s(»ew
thread the longitudinal section is trapezoidal, the depth being
J pitch + •01", the tip thickness •3707 pitch, and the angle
subtended between the two dicing sides 29^. The bearing
pressure on the threads in the direction of the axis when
transmitting motion varies from 200 to 1,000 Ifo. per sq. in.,
depending on the lubrication.
The proportions of standard (Whitworth) V-screw threads are
given in pp. 533-5 ; in the formuln above for efficiency change
a
tan ^ to tan ^ cos 0 sec ^- for V -treads of angle a ; for standard
threads of small pitch this virtually increases ^ by aboat
IE per cent.
The Sellers' thread (U.S.A.) consists of equilateral triangles
of depth d = pitch x V3/2 or '866 pitch. The tips and roots
are flattened \d from the vertices, so that the actual depth of
thread is f d.
BELT GBARjyG- 357
BsLT Gearing.
m
Length of CroMed Belts. — If two pulleys of diameters Dand d
distant o apart from centre to centre, be connected by a crossed
belt, the total length of the belt =
This length is constant provided that the distance between the
centres and also the sam of the diameters are constant. In
designing speed cones for a lathe, the same belt will drive
equ^y well on all if the sum of the diameters of each pair of
pulleys be the same.
Length of Open Belts. — No simple exact rule can be given, but
the following, though approximate, is generally accurate enough
for practical purposes. Let one pair of pulleys have diameters
D, and dy It is required to find the diameters of another pair of
pulleys of different ratio, but driven by the same b^t. Treat
them first as if the belt were 'crossed, and find the diameters
D, and d^ of a second pair, so that d^ + <3?s = ^i + ^i- Then
calculate (p, + <g^) + (^i ~ ^'^ " ^^« ^ ^»^', and taking this ex-
pression as the sum of the two required pulleys, and D^— ^^ ^^
the difference, recalculate D, and d^ which will be the diameters
required.
MeHstanee to Slipping —A «nd 8 (fig. 223) are the points
where belt leaves pulley T,^ and T2 are the tensions of
belt at A« B when on the point of slipping.
Fig. 2aa.
If 0 » the angle AOB in radians,
/i » coeffieient (rf friction between belt and pulley.
e^ 2*119.
or logio(Ts. Ti) = • 434 M^.
•> Too ^ angle aob in degrees.
858
BELT GEARING-'
Gbratest Value of the Ratio of Tensions on
Light
AND Slack Sides of Belting fbom
Equation I.
Angle embraoed by Beltatf
Batio of Tensions si 1
IB
In
In Fraction
Circular
of Circum-
ft =0-2
ft=0-3
fi=0-4
M=0-5
Degrees
Measure
ferenoe
30
•524
•083
1-110
1170
1-233
1-299
45
•785
•126
1-170
1-266
1-369
1-481
60
1047
•167
1-233
1-369
V621
1-689
75
1-309
•208
1-299
1-481
1-689
1924
90
1-571
•250
1-369
1-602
1-874
2193
105
1-833
•319
1-443
1-733
2-082
2-500
120
2-094
•334
1-521
1-875
2-312
2-851
135
2-356
•376
1-602
2-027
2-566
3247
160
2-618
•417
1-689
2-194
2-849
3-702
165
2-880
-468
1-778
2-372
3163
4-219
180
3142
•600
1-875
2-566
3-514
4-808
196
3403
•641
1975
2-776
3-901
5-483
210
3665
•683
2082
3003
4-333
6-252
240
4-188
•666
2311
3-514
5-340
8119
270
4-712
•750
2'666
4112
6*689
10-56
300
6-d36
•833
2-849
4-806
8-117
13-70
Let P s resistance at circumference of driven pulley, then
p « Tg - T, ; H = horse-power transmitted, and v = velocity of belt
in feet per minute ; then PV = 33000 H, and /. T, - T, = 2^^22L^.
If N = number of revolutions of pulley per minute, <? = dia-
meter of pulley in inches ; then velocity of pulley =
12
= v
Ta-Ti =
33000 H X 12 896000 H 126000 R
irdN vds dH
The ooeiBeientff of friction between belt and poUej are
about '48 with leather belt on wood palley» *28 (dry) or
•88 (wet) with leather belt on iron pulley, and '6 with hemp
rope on wooden pulley. Take •S in general.
The speed of main belting should vary between 3,000 and
4,000 feet per minute. At high speeds both tensions are
increased by centrifugal force ; this increase is 86 lb. per
square inch at 3,000 velocity, and it varies as the square of
BELT GEAHING. 359
the Speed-. This provides a limit to the efficient speed at
which beltings may be driTen ; for both T3 and t^ in the
aboTO equations are correspondingly reducod.
The thiohness of a single belt is about «^" ; of a double belt
I" to i".
The width varies from about 32 (small) to 70 (large) times
the thickness.
The. weight of 1 foot of belting, 1 square inch in sectioQ^
is about '45 lb.
The maximum working stress is 320 lb. per square inch far
single belts and 2401b. per square inch for double belts.
G^e conve^^ty of the pulley should be ^" up to 0" width/
^" up to 12", and i" beyond.
Approximate creep of belts is 2} to 3 per cent maximum ;
i.e. this is the excess speed, of the driving over that of the
driven pnlley.
Approximate fninimum pulley diameters for durable running.
(Mr. H. J. Spooner.)
Thickness of belt in
82n|isioch . .4 5 6 7 8 9 10 11 13 18 U 16 16
Di&meter of pulley
in inches . . 3-9 61 8-8 12 l$-7 20 24-5 29-5 3^ 41 48 65 98
Size Op Belt required.
A; =3 ratio of tensions (see table on p. 353).
V = linear speed of belt in, feet per minute = ird^l\2,
H = horse-power transmitted.
/=..= working stress of belt in lb. per square inch (see
above).
A = sectional area of belt in square inches.
Ex. — Find the area of belting required to transmit 25 H. P.
at 4,000 feet per minute. 6 = 165° ; /* = •$. Take / = -320.
From table h = 2-37.
1 1 , r>rf
^^ ''^ 3T0OO ^^ 2^ ^ ^'^^^ ^ ^^^^ " ^^^ • Whence A = 6-5
square inches.
360
WORK DONE BY MEN AND ANIMALS.
Table op Work Done by Men and Animals. {Ik-oni
T/visde?V8 'PracUcal Meohmics.')
Nature op Labour
Daily Duration
of Work
in Honrs
No. of Units
of Work
per Day
No. of Units
of Work
per Ifinute
i Weight Raised,
or Mean
Pressure, in Lbs.
Velocity
in JPeet
per Mlnnte
1. Raising Weightt VerHcaUy.
A man mounting a gentle in-
cline or ladder without biir-
den — i.e. raising his own
8-0
203,200
4,230
145
29
weight ^ '
Labourer raisiog weights with
rope and pulley, the rope re-
turning without load
60
563,000
1,560
40
39
Labourer lifting weights by
hand
60
531,000
1,480
44
34
Labourer carrying weights on
his back up a gentle incline
or up a ladder, and returning '
60
406,000
1,130
145
8
unladen '
Labourer wheeling materials
in a barrow up an incline of
1 in 12, and returning ^vith
100
313,000
620
130
4
empty barrow f
Labourer lifting earth with a
spade to a mean height of 5^
feet f
100
281,000
470
6
78
2. Action on Machines.
Labourer walking and pushing
or pulling horizontally
8-0
150,000
8,180
27
116
Labourer turning a winch .
8-0
1,250,000
2,600
18
144
Labourer pushing and pulling \
alternately in a vertical di-
rection
80
1,146,000
2,390
11
216
Horse yoked to a cart and)
walkinff /
10-0
15,688,000
26,150
150
175
Horse yoked to a whim gin
8-0
8,440,000
17,600
100
175
Do. do., trotting
4-5
7,036,000
26,060
66|
391
One man can lift with both hands 236 lbs.
„ „ „ support on his shoulders 880 lbs.
A man*s strength is greatest in raising a weight when his weight is
to that of his load as 4 is to 3.
Note, — In the above table the unit of work is taken at a
pressure of 1 lb. exerted through 1 foot.
UNIVSBSAL JOINT.
36i
Tablb givino the Useful Effect of Agents employed
IN the Horizontal Tbanspobt of Bubdens. (From
Twisden's *PraeHeal Mechanics.')
AaxNT
Man walking on a horizontal
road without burden — that is,
transporting bis own weight
Labourer transporting material
in a truck on two wheels,
returning with it empty for a
new load
Do. do^ with a wheel-barrow
Labourer walking with a
weight on bis back
Labourer transporting mate-
rials on his back, and return-
ing unburdened for a new
load
Do. do., on a hand-barrow .
Horse transporting material in
a cart, walking, always laden
Do. do., trotting
Do. do., transporting materials
in a cart, returning with the
cart empty for a new load
Horse walkmg with a weight
on his back
Do. do., trotting •
0
10-0 25,398,000 42,330
100 13,026,000
100
7-0
7,816,000
5,470,000
21,710
18,030
18,080
60 6,087,000 14,100
I
100| 4,298,000 7,160
100200,582,000,334,800
4-6 90,262,000;334,300
10-0
10-0
70
10,940,800182,850
34,386,000
32,072,000
67,310
76,410
145
220
180
90
145
110
1,600
750
1,600
270
180
... ^
292
99
160
145
97
65
223
44
121
212
424
Noic—Uhe useful effect in the above table is the product of
the weight in lbs. and the distance in feet.
Universal (Hooee'b) Joint.
For sketch of joints see p. 549.
0 = angle between tne axes of shafts,
s ss ratio of the angular velocities.
B varies between sec B and cos 0\ attaining each value twicf
daring a complete revolution.
In practice 9 does not usually exceed 85^ ; R then varies
between 1*22 and '819.
362 CRANES, PULLEYS, ETC.
Hand Cranes.
p = power applied to handle in lbs.
D = diameter of circle described by handle in inches.
w = weight to be lifted in lbs.
N = number of revolutions of handle.
71 = number of revolutions of barrel.
d = diameter of barrel in inches.
I =: length of handle in inches.
^ DPN N_W^ ^ _ -Wdfl ^y - PP^
nw n 1>P PN dn
__ wdu , W(fn /i=2pn2
~ DN ~ 2pn wd
Mte.—The ordinary height of handle above ground is 36
inches. Diameter of circle described by handle, 32 inches
Power imparted by one man, from 15 to 20 lbs.
Steam Cean^.
s = speed of piston In feet per minute.
D = diameter of main drum in feet.
w = load to be lifted.
N = number of revolutions of main drum per minute.
p a: pressure on one pist-on.
8 » speed of n^ain drum in feet.
n = number of revolutions of cran^ sh^jb p^i; minute.
I a length of stdToke la feet.
d = diameter of piston in inches.
P s pressure of ste^ in lbs. per square inch.
S = 2/i^ <«=aUl,6ND T==-7S64^d^
ND
Velocity of Pulleys.
V - velocity of driving pulley.
D = diameter of driving pulley.
V - velocity of driven pulley.
d = diameter of driven pulley
D,= ~ d = — v = ~ v^~
V V T> d
The final velocity of any number of pulleys
V X D X D' X D" &C., r ^ ^t ^11 p xl J3» I e
" — ^ — V. — ;;r— « where D, D, D , &c., are the diameters of
dxd' x,d' &c.,
the driving wheels or pulleys, and <Z, d'^ d'\ &:c., the dianieters
of the driven pulleys.
36^
Coefficients op
Fbiction.
1
Covffioient of Fxiotian.
MateriaU (dry).
IVom
To
]\CetaI on juefcal . . . «
•10
•30
Wood on metal ....
•10
•60
Wood OD wood ....
•10
•70
Leather oa metal
•25
•60
Leather on wood
•25
•70
Metal on stone . • • .
•26
•50
Stone on stone ....
•40
76
Ice on ice ....
•018
•028
Steel on ice ....
•014
•027
Hemp on oak ....
about *
5a
Materiah (lubritaUd^ .
Ifetal on metal ....
•009
•10
Wood on metal ....
•02
•10
Wood on wood (see also p. S77)
-oaa
•10
Leather on metal
•12
•26
Hemp on wet oak
about '
93
Ball-bearing races (Goodman).
Cylindrical race
•0012
•0018
Thrust— flat race
•0018
•0012
One flat race, one V-raoe
•0018
Two V-races . . .
•0055
i\'o<&.— Under forced lubrication the friction is farther
diminished, the coefficient being found to vary with speed
and temperature.
Friction of journals and pivots.
D == diameter in inches (larger diameter for pivots).
d =: smaller diameter for pivots in inches.
w = load on journal or thrust on pivot in lb.
M =: frictional moment in inch-lb.
H = horse-power lost.
B =: British thermal units generated per minute.
N = revolutions per minute.
/i = coefficient of friction.
H = MN/63,000 ; B =s= 425 H.
Loose journal M = j^ M wi>.
Tight new journal M = • 78 ^ WD.
Worn journal M = • 64 ju WD.
New conical pivot, angle 2a, m = J /i W cosec a (d' - <?')/ (d- - d").
Worn ,, ,, M =« J ^ W cosec a (d + d).
Fiat pivot (new) M = J ai wd.
Flat pivot (old) M =» J /* wd.
864
STEERING.
NOTES OS STEEBIKO.
to the tiller .•—
Means that rudder is a-pori,
or inolined towards the left.
Means that rndder is a-star-
board, or inclined towards
the right.
Means that rndder is a-
weather, or inclined to
windward.
Means that rndder is a-le^, or
inclined to leeward.
Terms used with reference
Helm a^atarboard, or inclined
towards the right.
Helm a-port, or inclined to-
wards the left.
Helm a-lee, or inclined to lee-
ward.
Helm a-weather, or inclined
windward.
Steering Indicator. — ^Tiller and indicator should mo?e the
same way ; rudder, wheel, and ship's head should more the
same way, and opposite, of course, to tiller and indicator.
Four features chiefly affect the readiness of a ship tc
answer helm : (1) time occupied in putting helm hard over ;
(2) pressure on the rudder when hard OTor ; (3)
moment of inertia of ship about vertical axis passing throng'h
the centre of gravity ; (4) moment of resistance to rotation-
For good steering it is also necessary that there are no
eddies at the stern, and that the water flows steadily past thn
ship, so that the fairness and fineness essential for speed are
also necessary for good steering.
Path when Tubninq.
Fig. 224 shows the path described by a ship whose rndder
is put oyer ; bbb ... is the path of the centre of gravity
Fig. 224.
STEERING.
365
of ship, CO ... is that of the stem, and aa ... is a line
drawn to tonoh the middle line of the ship in its successive
positions. The point of contact o of the ship with aa . . .
is termed the pivoting point ; it is nsually situated very
slightly abaft (in quick turning ships at) the bow. The ship
thus appears to point inwards across its path except just at
or near the bow.
The path of the ship differs very slightly from a circle ; the
taoiieal diameter is the maximum distance travelled perpen-
dicular to the original direction ; the advance is the total
distance travelled in the original direction from the momen#
of putting over the helm.
Steebino data fob Warships (Full Speed, 85*» Helbi). 1
Ship.
490
Area of
immersed
lonflritudinal
plane divided
by area of
radder.
Advance in
yards.
Tactical
diameter
in yards.
Tactical
diameter
divided by
lensrth.
Battleship* .
37-6
490
440
27
ti
410
40*5
400
370
2-7
»f
400
45-2
440
500
37
Cmiser
490
48*4
480
600
37
»» •
440
44*4
590
790
5*4
„ t . .
435
44-5
650
920
6-3
„ t .
500
50*3
800 1120
6-7
If •
350
48-3
540
770
6*6
»i • <
320
83-5
350
380
3*6
T.B. Destroyer f27 knots)
}270
4A.n
(390
560
6-1
„ (12 knots)
^ " 1280
800
3-3
* Two rudders.
f Stern not cnt away.
Influence op Vabious Features on Steeeinq.
Zength. — ^In the above table, tactical diameter is expressed
in terms of the length, though in very long ships this ratio
tends to increase.
Rudder Area, — ^This is usually a proportion of that of the
immersed middle line plane of the ship. The ratio is given
above for warshipe ; . it is about 60 in many passenger
and cargo ships, 20 in steamboats, 15 in yachts and sailing,
boats. In long narrow ships the area should be increased
relatively to the size in order to maintain facility of turning.
Form of Rudder. — ^A narrow deep rudder develops eet. par*
more pressure and requires a smaller force to handle than
a wide shallow radder. The rudder should always, therefore.
366 STEERING.
he as deep as poesi'b'le ; but the depth is limited in warships by
the necessity of keeping the top well immersed. A mdder may-
be balanced to redaoe the power required to control it ;
not more than 30 per cent of its area should lie before the
axis, or there may. be difficulty in bringing it back to the
middle.
Form of Ship, — ^The resistance to turning is greatly
diminished by cutting away the after deadwood (see table
above). With unbalanced rudders this may reduce the dead-
wood pressure (which assists steering at small angles) to
such an extent as to render ship rather unmanageable. In all
cases it reduces the space required to turn, but in excess it may
make vessel rather slow te answer helm.
Position of Mttdder, — ^This should be as nearly as possible
directly behind the screws so as to have the benefit of their
race. Twin rudders utilize this and enable vessel to be steered
from rest ; the resistance of the ship, however^ is at the
same time slightly increased (see p. 162).
Speed, — This affects, in general, only the time of turning ;
the path is nearly the same at low as at high speeds. An
exception is found in destroyers and similar ships whose
rudders exhibit ^ cavitation ' at tiieir highest speeds, thus in-
creasing the space required for turning. In quiok-turnlng
vessels the spe&d after turning through 180® may sink to hau
or even one-third of its original amount.
J)raught» — Inprease of draught aft enlarges the circle, since
the resistance to turning is augmented.
Sorew Propellers, — ^By reversinff the inner screw at the
same time as when helm is applied in ships having more
fjian one shaft, the tactical diameter may be reduced to about
two-thirds its usual amount, but the time required for iuriuji^
is bxereased. By stopping way on the ship, it is possible
by manipulating the propellers to turn a vessel without helm
about her own centre. When stopping, with both pyepelleis
reversed, the effect of helm is uncertain.
With a lEiingle right-handed screw, well immersed, the ship's
head usually tends to turn to starboard, but the contrary
may result if the sorew breaks the surface of the water.
Melm Angle, — ^At reduced hebn angles the space required
for turning is increased. Approximately if 1 represents the
^ace <taeUcal diameter or advance) with 35® of helm, then
1*4 is that for 20® and 2 for lO"". Usually nothing is gsinedi
by inereasing heim beyond 85® or 40°.
Method of careying out Turning Trials.
Throw out two buoys that are easily visible about 2 milef
apa :. The circles are turned round each buoy alternately ;
this enables the vessel to pick up her full speed before turning
each time.
STEERING.
367
At two points A and B (fig. 225), usuaily on the upper
deck at the middle line, sights are erecbed with quadrants so
that the angles oab, cba, made by the bnoy (c) can be
measured. These sights may be of the form shown in fig. 226,
where C is a batten hinged at A, carrying two upright wiro
sights 8,8. At intervals a signal is given, and observers measure
simultaneously the time, the angles at the quadrants, and
Fig. 225.
♦
e
the angle (e) made bj^ the ship's head with a fixed bearing.
Since the distance ab is known, this information enables the
triangle abo to be constructed and laid in its correct position
for each reading. If this is done after turning through every
feur points (45°) from the original ooune, or oftener if
desired, until 82 points are turned through and the original
course regained, the path of any point of the ship (usually
the C.G., assumed at mid-length, is selected) majr be drawn in.
The speed at any point may be roughly determined from the
observed times.
On the completion of a warship these trials are carried out
usually (a) at full speed, (5) at 12 knots, (c) with revolutions
corresponding to 12 knots, but with the inner screw or screws
reversed at the moment of putting over the rudder. In each
case two circles are made, one to port and one to starboard.
Alternative method. — Instead of the buoy, a boat is used,
from which the distance of the ship is determined either by
measuring the masthead angle or by means of a range-finder.
The time tq take the observation is signalled from the ship.
A second observer on the boat simultaneously measures the
compass bearing of any fixed point (e.g. a mast) in the ship.
pBESstJEE ON Rudder.
If a rudder is held in a stream of water or, equally, is
moved through still water, the normal pressure on it varies
with the angle of inclination in the manner indicated in
fig. 227.
It follows no simple mathematical law, but increases up to
a 'htimp' at A^ then suddenly drops to B as the angle is
dightly increased ; finally, it increases sliffhtly, attaining a
868
STEERING.
▼alae at o (90**)^ wiiioh is nsnallj less, but someiimes greater,
than that at B. The simplest formtda by which it may be
approximately expressed in terms of the angle is that given by
two straight lines od, df, of which df is horizontal.
If a be the angle OE^ and B the value of the ordinate DE^ then
normal pressure p = B0/a when 9 is less than a.
~ B when 0 is greater than a.
The value of B in salt water is given by— r
B = KAV*
where B is in lb., A is the area in square feet, and v th.^
velocity in feet per second. The constant K varies in salt
water from 1*1 to 1*2^ say 1*15 average, though in plates
of extreme proportions it may be slightly greater. See also
pp. 409 and 431.
The angle a where the pressure first attains a maximnm
varies with the shape of the plate. It U approximately ai
follows :—
Shape of Plate.
Angle a.
greatest
pressure.
Garr*>
npoffMliiv
angle.
Circle or square
Ellipse or rectangle, horizontal
side twice vertical .
EUipse or rectangle, horizontal
side half vertical .
Ellipse or rectangle, horizontal
side one-quarter vertical
25*
28°
23*
16*
15
17
37*
40*
Note, — ^The greatest pressure given in the right-hand
columns is that at the ' hump ' ; this is unstable, and the
pressure there is liable to fluctuate considerably. In the last
two results the hump pressure is less than at 90°.
From this it is evident that the rudder pressure for the
greatest angle of helm, usually 35*, may have a value of K
as great as 1*2 or even more. On the other hand, this is
considerably modified by the immediate diminution in the
ship's speed on taming and by the reduction in the effective
STEERITCO, 869
helm angle caused by the lateral movement of iho stern.
From experiments made it appears that this redacas the co-
efficient to about one-faalf its Talne, rather less (40 per cent)
at high speeds and rather more at low speeds, agreeing)
fairly welt at 35° with the usual formula^
p = 1-12 Av3 sin e.
The speed v is greater than the speed of the ship by about
200/0 in twin-florew ships, and 30o/o in single or quadruple*
screw ships where rudder is directly behind propeller.
Hence, finally, if p is in tons, A in square feet, and x^
the speed of the ship in knots, at 35^ helm,
jf f
P = AV^/900 for twin- or quadruple-sciew ships with single
rudder,
p = AV^^750 for single-, triple-, or qnadraple-screw ships
(rudder directly behind screw).
P s XvJStOOO for ships going astern.
In the last formula Vg is the speed ahead, that astern being
assumed % Vg. Where associated with ' live-load ' working
stresses, &ese figures may be regarded as on the safe side.
Position of Centbb of Psessuke.
The distance of the centre of pressure from the leading
edge of a rectangular rudder is *2 X breadth at small angles
of inclination, about *3 X breadth at 15% and about *4 X
breadth at 35<», except for wide short rudders, where the
proportion becomes '33. It is usually assumed to be f breadth
at 35^. See also p. 415.
To obtain the.C.P. of a rudder with a curved outline,
divide the surface horizontally into strips of equal depth.
Find the C.P. of each strip, taking it to be at § the mean
breadth from the front edge. By adding the areas of the
strips and their moments about a fixed vertical axis, the total
area and moment are obtained ; the distance of the C.P.
abaft this axis is the quotient when the total moment is
divided by the whole area.
When going astern substitute the after edge for the leading
edge.
Strength op Rudder Head and Pintles.
Unbalanced rudders (fig. 228!).— Assume the rudder dis-
continuous at the pintles. Find the reactions at a and b
due to die pressure on the portion AB ; that on b will^ generally
be rather more than half the total. Treat similarly the
pressure from B to o* Force cto pintle B is the scoji of the
reactions due to the tw'o portlotis $ forc!e an pintle c is usually
about i the pressure on the lower portion.
With several pintles proceed similarly. The lowest pintle
takes about one-half the pressure taken by each of the others
Bb
370 STEERING.
The bending moment on the head A is generally smalL
The twisting moment is equal to the rudder pressure multiplied
by the distance of the centre of pressure abaft the axis.
Balanced rudders supported at the bottom. — ^The lower
bearing usually tt^s about | of tiie total pressure. Fin^
twisting moment on the head as before ; it is usually greatest
when going astern. The bending momejit oo the head' is
uncertain, but it cannot exceed ^ of the total pressure x depth.
Fig. 239.
C
Balanced rudders (tig. 229) iupporiing midway. — The pres-
SFure on the pintle with average shapes of rudder is § the
whole pressure. The bending moment on the head is equal
to the rudder pireflBare multiplied by ^ the whole depth of
the rudder.
Balanced rudders supfported wholly inboard, — The bending
moment on the head is equal to the rudder pressure multiplied
by the depth of the centre of gravity below the lower edge
of bearing.
Size of pintles, — ^If D = diameter of pintle in inches.
L = bearing depth of pintle in inches.
K = a constant varying from 2 to 2).
Pressure on pintle in tons = kld.
Diameter of rudder head.
T s=^ twisting moment in foot-tons.
/ = stress allowable in material, expressed in tons per
square inch.
= 5 for cast steel, 4 for forged iron, 3 for phosphor
bronze.
D =B diameter of rudder in inches.
d3= 61 T//.
= 12 T for steel.
= 15 T for iron.
= 20 T for phosphor bronze.
If there Is a bending moment M in addition to the twisting
moment, replace T by the equivalent moment M + ^M? + t".
In all balanced rudders calculate when going both ahead and
astern and take the greater combined moment.
•' Note, — ^For. low-speed ships or sailing ships, take an
'equivalent speed* which will represent the action of thei
waves. See British Corporation Rule below.
STEERING. 371
£xample.-A twin rectangular rudder, hung wholly out-
board, is 18 feet broad and 14 feet deep. Determine the size
of the Bteel rudder head, if the rudder axis lies 6 feet abaft the
leading edge. Speed of ship 21 knots ; 4 screws.
(1) When going ahead, centre of pressure is f X 18 feet
or 6f feet abaft leading edge ; that is f feet abaft axis. It?
distance below bearing would be about 8 feet.
Eudder pressure = avJ/760= 18 x 14 x 21 x 21/750= 66 tons.
Twisting moment T = 65 x -75 = 49 foot-tons.
Bending moment M= 65 x 8 = 520 foot-tons.
Equivalent twisting moment =M+ v^ilF+T^= 1040 foot-tons.
(2) When going astern, centre of pressure is 6J feet abaft
the after edge ; that is 5i feet abaft the axis.
Rudder pressure = AvJ/sOOO =* 16 tons.
Twisting moment = 16 x 5 • 25 = 84 foot-tons.
Bending moment « 16 x 8 = 128 foot- tons.
Eqnivalent twisting moment = 280 foot- tons, less than when
going ahead.
Hence d' = 12 x 1040 or D = 23 J inches.
Bjutish Coeporation Eule fob Size op Euddeb Heads, etc.
B s diameter of head in inches.
E = distance in feet of centre of gravity of immersed area
of rudder from centre line of pintles.
A = area of rudder up to l.w.l. in square feet, .
V =: maximum sea speed in knots.
D = • 26 Vl^ ^ V- for steamers.
J) = l'2B\/BiLior sailing vessels.
In the above formula take v at least 11 in vessels of
250 feet length and over, and at least 8 in vessels of 100 feet
length, proportionately for intermediate lengths.
Thickness op Eudder Plate. 1
Leogth of Vessel.
Thickness of Single Plate
Thickness of Doable Plates
(fortieths of an inch).
(fortieths of an inch).
100
25
12
200
30
14
300
35
16
400
40
18
500
45
20
600
50
20
Gudgeom and Pintles. — Space of gudgeons not more than
4 feet in vessels 10 feet deep amidenips, and 5 ft. 6 in. for
vessels 40 feet deep ; interpolate for other depths.
Depth of gudgeons to be not less than 75o/o diameter of
stock (d) ; thickness '275 0 if unbushed, and 25 D if bushed.
Diameter of pintles '5 D.
Steering Chains, — See p. 509.
872
LAUNCHING.
LAUNCHIire.
700
BOW REACHES
END OF WAYS
STIRN LIFTS. CLOVER TRAVaOF SHIP
END OF WAYS. -
AA = momeni; o7 buoyancy about fore poppet (foot-
tons).
BB == moment of weight about fore poppet (foot-tons).
CO = buoyancy (tons).
DD = weight (tons).
EB^ moment of baoyancy about after end of ways
(foot-tons).
FF = moment of weight about after end of ways
(foot-tons).
00 8=s length of ways in oontaot (feet).
iiH = mean pressure on ground ways (tons per f(y>t
length).
RK = position of resultant ground way pressure (dis-
tance from fore poppet in feet).
LL, 3IM =3 lines having ordinates equal to j ar^ i those
of GO.
TfN a Depth of lowest point of cradle below w^tor,
speed of launch being asstnned very slow.
prs=r depth of lowest point, including allowance for
speed of launch.
LAUNCHING.
Fia.fl8l.
878
MOMENT
ANGLE OF INCLINATION TO
HORIZONTAL.
Laukchino Calcuiatioks.
The calculations usually made include the determination of
the position* where the stern lifts, the maximum pressure on
fore poppets, and the liability to tip over the end of the
urays. These are found from the curves aa, bb, go, dd,
CE, and FF (fig. 230).
In special cases it may be desirable also to find jthe
amount of dredging necessary (from the draught aft), and
the amount and distribution of the pressure on the way? up
to the point of lift. These are found from the remaining
curves in fig. 230.
Method of Procedure, — (1) Construct ' bonjean ' curves,
or curves of areas of sections up to the highest water-line
likely to be found. With twenty-one ordinates it is sufficient
to do this at alternate sections only.
(2) Estitmate the draught at every section after running
various distances, ' the greatest being slightly beyond tihe
probable point of lift. Tida is done by calcmating the draught
at any two sections — say the fore and after poppets ; on
plotting them with a contracted longitudinal scale, the
draughts at all other sections can be measured off.
L
.874 LAUNCHING.
Let D = depth of water over end of ways
A = length of ground ways up to fore poppet.
B = length of cradle (fore to after poppet).
c := height of camber on the length of ground ways
(A).
H = height of keel at fore poppet above surface of
ways.
0? = distance run.
All in feet and decimals of a foot. ^
a = inclination of ship's keel at start.
fi = mean inclination of ground ways expressed as a fraction
(thus r to 1' = ^5).
Then, radius (r) of ways = a'/8o.
Starting declivity = fi - -^ = i8 - 7? (a - b).
^ "*" "72"/*
Note. — ^Negative draughts denote height of keel above
water.
(3) Using these draughts, read od the areas of the sections
from the bonjean curves, and put them in a table. Then,
using Simpson's rules, find t)ie total dLn>lacement and the
longitudinal centre of buoyancy. Thence plot the three curves
(a) buoyancy (GO), ' (6) moment of buoyancy about fore
poppet (aa), (c) moment of buoyancy about ftfter end of
ways (ee).
(4) Estimate approximatdy the weight and longitudinal
C.Ot. of ship. This can be fairly readily done when the
design information is available ; in other cases judgment
shoiHd be exercised as to what proportion of the final dis-
placement will be launched. For battleships 40 0/0 is nsual.
In Trans. Inst. Nav. Arch., 1913, Mr. A. Hiley gives the
following average launching displacements on a length of
400 feet : light craft and T.B.D., 1,000; sea-going T.B.D.,
1,600 ; light cruisers, channel boats, 2,000 ; passenger boats
liners, cruisers, 2,500 ; cargo boats, liners, cruisers, 3,000
cargo boats, battleships, 4,000 ; heavy craft, barges, 5,000 ,
ice-breakers and submarines, 6,000 tons. For other lengths
vary displacement as the cube of length ; for liners 600 feet
take 10,000 tons, 700 feet 14,500 tons, 800 feet 20,000 tons,
900 feet 28,000 tons.
For the distance of tiie OtG. of ship abaft fore poppet,
'42 length is given as an average value.
LADI^CHINO. 876
(5) Pliot ibe straight lines representing weight (dd,
fig. 280), uuDlnent of weight about fore poppet (bb), moment
of weight about after end of ways (ff).
(6) The interseetion of eurres aa and bb gives the point
of lift, B ; the difference between the corresponding ordinates
of the carves cc and dd gives the weight on the fore poppets
at that instant. The clearance between the curves be and ff
should be stufficient to obviate the possibUity of tippling about
the after end of ground ways.
Additional Investigations. -^(7) The draught at the stern
if ter lift is found as follows : Determine the draught at
the fore poppet for various distances run, all beyond B (see
formulsB above). For each run considered keep this draught
and calculate the draughts at all sections when ifrimmed to
various angles to the horizontal, e.g. J°, 1% IJi**, and 2°.
Thence determine the buoyancy and its moment about t^e
fore poppet at each .angle, and plot them (^g. 231) on^ an
angle base. The intersection of the latter curve wit|h that
of the moim'ent of weight determines the balancing angle and
the buoyancy xy. On plotting tiie latter in Sg. 230 tiie
left-hand portion of the curve <x; -may be completed ; the
difference between it and i>D gives the w^ght on the poppets.
(8) The draught at the stern is easily calculated from the
balancing angle, and is plotted in the curve nn. In a quick
launch l^is drauglit might be exceeded owing to the inertia of
the ship. It most cases it would be sufiSeient to allow
2-3 feet, depending on the size of the ship ; pp can be set;
this distance beyond nn, the right hand beii^ terminated by
a line tangential to itn at the point of lift.
(9) The line gg represents the length of ways in contact
before lifting ; the intercept between dd and oo gives the
total pressure on the ways. By division the mean pressure
per foot length is obtained and plotted in eurve hh.
The distribution of this -pressure is calculated by dividing
the difference between by the moment of buoyancy and weight
about fore poppet (intercept between bb and aa) hy the
total pressure on the ways. This gives the distanoe of t^e
resultant pressure on the ways from the poppet ; it is plotted
as EE. ^e straight lines ll, mm have ordinates respectively
equal to § and i those of GG — ^the length of ways in contact.
When SK crosses ll tiie maximum . pressure is at the after
end of ways, and is double the mean. Where ee is above ll
the fnaYiTTrmni pressure is more than double the mean, and
the pressure is concentrated near the end of ways ; the
portion of the ship in wake of the after end of wajys over
this range may require extra shoring in order to withstand
the concentrated pressure. Conversdfy where ee lies below
MM, the pressure is concentrated near the fore poppet.
(10) The shearing force is usually greatest at the fore
876 LAUNCHING.
poppet at the moment of lifting^. Conalderablo shear forces
also oeear before and after this position ; in large ships it
may be advisable to calculate them for several amounts of
travel. The force and stress are calculated by the usual
methods (pp. 332, 346) ; if the factor of safety is less than Z,
additional shoring^ should be provided.
(11) The bencUng moment is large (sagging) at the moment
of Uft ; it is frequently ako large (hogging) at a smaller
travel. The curve of bending moment and the resulting
stress are obtained by methods similar to those used for a
ship on a wave (p. 346) ; the factor of safety should be
at least 5. The decks should be riveted up sufficiently to
withstand these stresses.
Launching Paeticulars foe the 'Lusitaxia'.
(W. J. Luke, Esq., Trans. Inst. Nav. Arch., 1907.)
Diaplaoement, 16,000 tons ; declivity of ship, i^" to 1' ; over-
hang at bow, 58' ; overhang at stem (to aft perpendicular), 48' 4" ;
length of cradle, 653' Bl' ; breadth of cradle, 6' ; pressure per
square foot, 2*04 tons ; length of ground ways, 795' 6"; breadth
of ground ways, 6' ; mean declivity, '51" to 1' ; camber, 1' 4" ;
distance of centres of ways apart, 25' ; length of slip, 760' ;
breadth, 87' 6" ; depth to shelter deck, 60' 4^" ; load displacement
at 82' 6", 36,840 tons.
In the Mauretania^ a similar ship, the coefficient of friction
was '0232 ; maximum velocity, 14 knots ; lubricant per
100 square feet—tallow 1901b., train oil 81b., soft
soap 14^ lb.
The following data are given for a battleship launched at
7,300 tons, about 35<Vb of the load displacement : Declivity
of ways, §$ to a foot ; camber, 1" in 420 feet ; length of
ground ways, 510 feet ; length of cradle, about
400 feet « '8 X length of ship ; weight per square foot,
2*47 tons ; coefficient of friction, '05 ; maximum speed shortly
before lifting, 12 knots ; temperature, 58'' F.
General Notes.
The distance apart of ways between centres is equal to
the breadth of ship divided by from 3 to 31*
877
•
G
1
O
00
1
O
00
1
1
p
g
QQ
•
CO
o»
•
1
•
c
•
i
H
•
Coefficient of
Sliding
Friction.
In
82nd
inches
per ft.
r-r-oOf-it.<oQooo
• ••..•...
r-'<^cocic40»oooooo
f-4 f-l f-l 1-H fH f-l
4!
)OG0«O^i-l00V»e0f-ltH
<^cocoeococic«c9cie<i
oooooooooo
Decrease of Friction
per increase of
1 ton per sq. ft.
pressure.
In 82nd
inches
per foot.
ua »o wa ^^ ••^ ••^ *^
• ■ ■
1-1 i-H 1-H
:«.
CP Q0<O
IOtOkQ^<^^OI04f-IW
OOOOOOOOOO
oooooooooo
Starting
Declivity
in 82nd
inches
per foot.
UMOOOOOUdUdCOOUd
.........
r-COrHO»0Ot»CDUdUd^
C4C«C4rHiHi-1rHf-1f-li-l
Extra
Starting
Slope in
a2nd
inches
per foot.
«a 04
ioeoeoo40404e4C404e4
Coefficient of
Sticking Friction.
In83nd
inches
per foot.
udceooudMdto lo
...... .
e40QO»««OIO^COC004
04C4iHi-li-<lHiHf-lfHrH
4.
OOe40»<O^iHOO<0'^CO
»ft»ft"*'^'*"*coeoeoco
OOOOOOOOOO
•
Preesore
per sq.ft.
on
Sliding
Ways.
OeO'^>OCO0df-l<^So>
,:HihA|iHihA4O4O4 04O4
Breadth of ^
' Liannohing
Ways.
9 IB. to 1 ft.
1 ft. 2 in.
1 ft. to 2 ft.
2 ft. to 3 ft.
5 ft. to 6 ft.
6 ft.
6 ft. to 7 ft.
7 ft.
7 ft. to 8 ft.
Approxi-
mate
Length
of Ship
in feet.
OOOOOOOOOO
»ooococoeo«)uac4ooeo
1 1 1 1 1 1 1 1 1 1
OOOOOOOOOO
00004rHf-H»000^'^0
»HfH04eO"^»o«or*c300
Latinch
Displace-
ment
in tons.
SSSSSSSSS8
r-IIAOUdOOOOOO
00
CO
o
a
o
SQ
I
C4
i
P4
t
00
M
O
«H
08
-a
o
«H
I
878 ARMOUR.
The effect of camber on the ways is to reduce the distajice
run, to inorease the foro P9ppet pressure, and to diminiah
the possibility of tipping.
^e pressure on dogshores or holding arrangements is equal
to the launching weight multiplied by the difference betweeoi
the starting declivity and the coefScient of sliding friction.
Adoptingt the data in the table above gives a force equal to
about ^ the launching weight in large ships.
The force required to draw a vessel up the ways is e<|aal
to the weight multiplied by the sum of the declivity and the
greatest possible coe£Bcient of friction.
The declivity of the ship is fixed in relation to tiiat of
the ground (height of keel blocks riiould be about 5 feet) ;
it is conveniently ■^" to a foot less thui i^e starting declivity.
With small declivities the -stresses and pressures on poppets
are diminished ; but the distance run and the length of Hie
ways are increased. ,
The informatI(m in the table on p. 377 applies to 65** F.
temperature. At 80° F. decrease /t by IQo/o; at 40" F. add 5
or lOo/o more to the friction. In cold dry weather fi oan.be
reduced by lO^/o by adding ^ gallon of train oil per 100 square
feet of ways. Grease assumed i soft soap and | tallow; for 100
square feet allow 1001b. for 1,000-ton ship, 1201b. from
2,000 to 10,000 tons, 1401b. for 15,000 tons, 180 to 2201b.
for larger eiiips.
. ABKOVB AVB OBDVAKCS.
Fesfobation of Abmouh.
w = weight of projectile in lb.
V = velocity »t impaot in feet per second.
D = diameter of projectile in inches.
t = thickness of wrought iron perforated in Inches.
'' = D log-^ 8-841 (Ti«sidaer) ;
-,, . . thickness of wrought iron perforated , . .
The ratio ., . . 5 — *=- — . , • is termed
thickness of armour perforated
the figure of merit,- For cemented armour plates the figure
of merit lies between l'7d (thick plates) to about 2'5 (thin
plates) when attacked by a capped armour-piercing shell.
Against an uncapped shell add about '6 to these figures.
The formula does not apply to plates more f^an 12^ in
thickness ; the resistance to perforation for thicker plaites
is proportionately small.
Oblique Perforation, — If 0 be the angle between the axis
of projectile and the normal to armour surface, tlio
perforation is roughly equal to < cos 9 when 0 is fairly
small (i.e. up to 80°) ; for larger obliquities the perforation
is smaller than that indicated by the fbrmula.
rHOjECTiLES. 879
Sorizontal Armour, — ^It is usual to make thickness \ or
^ of that which would be required in vertical armour to
obtain the same degree of protection.
HOIION OF P£OJ£CXILES.
For muzele velocities of various types see pp. 380 eitc.
VelocUy. — lAit any range the striking velocity may be
roughly determined by taking the velocity at the end of each
3,000 yards to .be a certain percentage of that at the com-
mencement of that distance. T)iis percentage is about 00
for tiie lai^^ guns, 85 for 12 and 0*2" guns, and 75 for
•6" gVBm, Thus, in 6" guns, if the muzzle velocity be
2,500 feet (per second, after 3,000 yards it is 2,500 X '75 or
1,870 feet per second, after 6,000 yards it is 2,500 X (•76)3
or 1,400 feet per second, after 9,000 yards it is 2,500 X C75)*
or 1,060 feet per second. The rule does not apply when the
velocity is reduced to or below that of sound (14*00 or 1,200
feet per second), since the law pf resistance is then con-
siderably modified.
Arigle^of 4^scent.
Let V = velocity in feet per second.
B s= nuige in yards.
9 = angle of descent.
When resistance is neglected
_,. .. 3(7 R 100 R ,,
Sin 2fl = -^ = --r roughly.
Actually, the resistance causes v to vary ; the above
formula still roughly holds if V be taken as the 0am of a
quarter the initial velocity and three-quarters the final
velocity.
{The angle of elevation of gun is, for short ranges, very
sligjitly Iftss than that of cLescenit ; i^heu final velocity is
one^half initial velocity, this angle is from J to ^. that
of descent.
{The ^«<7*r*«er, ;Septe^ber 26, 1913.)
Pahtjculars op 15 in. 40-caubre Gun.
Weight in tons.
Two guns ...... 165
Turret .complete wHh armour . • 546
16iO rounds of ammunition . • . 1^3
Toiial . . . . . b^4t
Weight of steam pumping plant and hydraulic piping
ifor 4 turrets (8 guns), 102 tons ; weight of projectile,
1,950 lb. ; weight of charge, 610 lb. ; muzzle velocity,
2,300 feet /second.
((.' on tinned on j>. 388.)
380
Ballistics, Weights, etc
., OF
Sir W. G. Armstrwig, Whitworth dt Co.,
Ltd,,
Semi-Auto-
Semi-
Naval
Land-
matio.
Auto
ing.
Diameter of Bore . in.
1'86
1-86
1-85
2-24
2-24
2-953
3
>» t» nun.
47
47
47
67
67
76
76
Length of Bore calibres
40
46
60
40
60
1413
18-8
owt.
cwt.
owt.
owt.
owt.
owt.
' owt.
Weight of Gun . . .
4-51
5*0
7-5
7-6
10*6
1-876
4
t* t» kilos.
290
254
381
881
633
95
203
„ Projectile lb.
3*3
33
3-3
6
6
11-76
12-6
» »* kilos.
1-6
1-5
1-5
2-722
2-722
6*33
6-67
oz.
oz.
lb. oz.
oz.
ib. oz
oz.
OS.
J, Charge M.D.Corditn
8-25
10
1 0
10
1 2-6
7-76
16
*» f> kilos.
0-234
0-283
0-453
0*283
0-525
0-22
0-369
Muzzle Velocity . F.S
2132
2300
2680
1968
2400
IICO
1685
>» >f • Jl»K>-
650
701
817
600
731
383
483
Muzzle Energy. . F.T.
104
121
164
161
240
98
213
t> (> • • Al. 1 .
32-2
37-6
50-8
49-8
74-3
803
66
Penetration at Muzzle in.
615
6-8
7-3
6*4
7-3
^
_
(Txeddder wrooght-iron plate).
«» I* ^^•
130-8
147-3
185-4
137-2
186*4
..
_
Rounds per Minute . .
26 25
25
25
25
20
20
Diameter of Bore
Length of Bore
Weight of Gun
• in
calibres
ft
»*
kilos.
Projectflc lb.
kilos.
»»
„ Charge M.D.Corditr
„ „ kilos.
Muzzle Velocity . F.S.
ti t» • M.S.
Muzzle Energy. . F.T.
Penetration at Muzzle in.
(Xraddder ^m>aght-iron plate)
_ »» , » nim.
Rounds per Minute . .
6
7-6
7-6
8
8
9-2
9-2
152
190
190
203
203
234
234
50
45
60
45
60
46
60
tons
tons
tons
tons
tons
tons
tons
8-75
13-8
16-75
18-0
20-4
26-76
28-82
8 90
14021
16'^03
18289
20727
27179
29289
100
200
200
260
260
880
880
45-26
90-73
90-72
113-4
113-4
172-37
172-37
lb.
lb.
lb.
lb.
lb.
lb.
lb.
830
74
76
80
90
122
186
160
33-566
34-473
36-29
40-82
66-.^4
61-7
3000
29~0
3000
2846
SOO.)
2760
8000
914
8^4
914
867
914
838
914
6240
ir6'?
12481
14031
166C0
19926
23n4
1932
36119
3865-2
4345-2
4831-2
6171
7340
26 6
306
32-3
32-2
34-9
36-2
40-1
647-7
777-2
820-4
817-9
886-4
8941
1018-6
9
6
6
6
6
4
4
381
EiiSWicK B.L. AND Q.F. Guns.
Elswick Works, Newcastle-tm-Tyne.
Semi-
Joinfd
Auto.
Gun.
3
3
3
3
4
4
4-7
4-7
4-7
6
6
70
76
76
76
102
102
120
120
120
152
152
40
60
60
19-2
40
60
40
45
60
' 40
46
cwrt.
cvrt.
cwt.
cwt.
cwt.
cwt.
cwt.
cwt.
owt.
tons
tons
12
17-76
23
4*6
26
42
42
63
66
6-6
8-4
609
901
1168
229
1321
2134
2134
2692
3353
6706
8536
12-6
12-6
14-33
14*33
31
31
46
45
45
ICO
100
667
667
6-6
6-6
14-G6
1406
20-41
20-41
2041
45*36
45-36
lb. oz.
lb. oz.
lb.
02.
lb.
lb.
lb.
lb. oz.
lb.
lb.
lb.
2 0
3 0
6-76
16
6-6
10-6
6126
9 11
14
24
31
0-907
126
2-608
0-426
2-496
4-76
2-78
4-395
6-36
108:6
14061
22L0
2600
9060
1486
2300
3000
2200
2600
3000
26:o
28:o
674
792
930
462
701
914
670
792
914
761:
853
423
686
922 218
1137
1934
1510
2109
28::8
4334
54C6
131
181-6
286-6
67-6
3621
699
467-6
653
869-6
13422
1683-4
80
10-3
13-7
—
116
17-4
121
156
193
19-4
229
203-2
261-6
3480
294-6
4420
307-3
396-2
490-2
492-8
681*6
20
20
30
15
12
12
12
12
12
9
9
10
10
10
12
12
12
12
13-5
14
16
254
264
264
306
305
305
306
343
356-6
406-4
40
46
60
40
40
45
60
46
45
40
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
31
36*26
88-26
48-5
61
69-3
66-7
76
85
105
31497
86832
38864
49278
61818
60252
67770
77216
86360
1C6:80
450
600
600
850
850
850
850
1250
14:0
2230
20412
226-8
226-8
S85-66
885-56
386-65
385-65
567
636
998
lb.
lb.
lb.
lb.
lb.
lb.
lb.
lb.
lb.
lb.
86-6
167
200
166
260
260
285
296
324
890
89-24
76*76
90-7X
70*31
117-93
117-93
129-3
134-26
147
177
2400
2800
8000
24C0
731
2650
28:o
dOOD
2700
2700
2340
731
868
914
808
863
014
823
823
713
17973
27181
81203
88D49
41386
46208
53046
63187
707TO
83530
6566
8418
9668
10513
12816
14310
16428
19568
21916
25868
299
887
44-1
67*5
486
47-8
62-6
51'3
582
604
T59*4
IOQB'4
112W-1
052*5
1107-4
l2Ctt-4
18ad'5
13DB-0
1391-8
1280
8
8
8
2
t
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i^04
ViCKERS GUNB
Naval Guns. (This
table is
37 m/m.
37 m/m.
3-pdr.
6-pdr.
3 in.
Semi-
•
Auto.
SOoal.
42-5 cal.
60 cal.
50 cal.
60 cal.
Diameter of Bore • in.
1-457
1-467
1-85
2-244
3
Length of Bore . • in.
43-5
62
92-5
112-2
160
Length of Gun • . in.
73-76
94
98-9
118-6
166-996
Weight of Projectile lb.
1
1-25
3-3
6
12-6
cwt
cwt.
cwt.
owt.
owt.
Weight of Gun ....
3-75
6-42
6-63
9-29
19
Muzzle Velocity . . F.S.
1800
2300
2800
2600
2700
Muzzle Energy . . F.T.
22-5
45-85
179-4
281
632
Penetration of Wrought
Iron Plate at Muzzle.
1-9
3-3
6-7
7-6
9^66
Gavre formula . . in.
■
Penetration of Hard Steel
Plato at 3,000 yards.
—
-.
—
—
..
Garre formula . • in.
Rounds per minute 1 .
300
300
SO
28
26
Weight of Mounting com-
c. q. lb.
0. q. lb.
c. q. lb.
0. q. lb.
t. c. q. lb.
plete with Shield . .
4 1 10
4 3 20
11 2 0
18 1 0
1110
in.
m.
in.
I'hickness of Shield . .
•1875
•le
•25
•26
•26
c. q. lb.
c. q. lb.
0. q. lb.
0. q. lb.
c. q. lb.
Weight of Shield . . ,
0 3 11
0 1 22
10 0
12 8
2 10
Angle of Elevation . .
16«'
15"
20**
20°
20°
Angle of Depression . .
25°
20°
20°
10°
10°
7-6 in.
8 in.
9-2 in.
9-2 in.
10 in.
50 cal.
50 cal.
46 cal.
50 cal.
46 cal.
Diameter of Bore . in.
7-6
8
9-2
9-2
10
Length of Bore . . in.
376
388-75
429-3
460
460
Length of Gun . . In.
386-7
400
442-35
473
464-6
Weight of Projectile lb.
200
216-7
380
880
478-4
tons
tons
tons
tons
tons
Weight of Gun ....
16-0
14-6
26-85
2781
34-86
Muzzle Velocity . . F.S.
3003
3090
2800
2850
2850
Muzzle Bnergy . . F.T.
12506
14350
20660
22930
26946
Penetration of Wrought
Iron Plate at Mnzzle.
30-75
31^5
36-3
280
38-9
Gavre formula . . in.
Penetration of Hard Steel
Flat* at 3,0(X) yards.
11-4
126
141
16-2
16-8
Gavre fozsaula . . ^in.
•
Bounds per mintitd . .
8
6
4
4
3
Weight 'of Mounttng tfola-
t. c. q. Ih.
plet!e Vidi 8bi^d . .
ThicktieSB of Bliield . .
Pi
0) .4 O
Weight of Shield . . .
qSs
Angle of Elevation . .
'
Angle of Depression , .
885
AND Mountings
•
>
supplied by the Manufacturers.)
4 In.
Semi-
Auto.
4 in.
4-7 in.
4-7 in.
4*7 Naval
Howitzer
6 in.
6 in.
7-6 in.
40 cal.
60 oal.
45 cal.
50 cal.
18 cal.
45 cal.
60 cal.
46 cal.
4
160
166-6
4
201-15
208-45
4-724
212-6
220
4-724
228-45
236-2
4-724
85
89-9
6
269-6
279-2
6
300
310-07
7-6
337-5
349-2
31
31
45
45-14
45
100
100
200
f
owt.
25
2300
1137
cwt.
41
3030
1975
tons
8-18
2800
2445
tons
3-2
3050
2910
c. q. lb.
11 1 14
1200
450
tons
7-42
2900
5830
tons
7-8
3100
6665
tons
1402
2875
11465
10-8
16
13-9
17-8
—
22-6
24-8
28-75
—
—
—
5-0
—
7-6
8-8
10-6
f 20
15
12
12
10
10
10
8
t. c.q. lb.
1 10 2 0
in.
•028
t, c. q. lb.
2 6 3 0
none
t. 0. q. lb.
3 13 3 0
in.
2 and -313
t. 0. q. lb.
6 5 3 0
in.
4-33
t. 0. q. lb.
4 8 1 11
in.
2
t. 0. q. lb.
9 110
in.
3
t. c. q. lb.
12 0 2 0
in.
3 and 16
t. 0. q.lb*
bO*M •
PI 0 ^
o. q. lb.
110
20*»
10^
none
15»
c. q. lb.
117 0 0
110
20°
70
t. c. q. lb.
2 8 2 0
16*»
70
t. c. q.lb.
1 9 1 14
70*
5°
t. c. q. lb.
3 110
15°
70
t. 0. q. lb.
5 5 0 0
16«»
70
10 in.
12 in.
12 in.
13-6 in.
14 in.
16 in.
60 cal.
45 cal.
60 cal.
45 cal.
45 cal.
45 cal.
10
12
12
13-5
14
15
486
640
600
607-5
630
675
600
, 496-4
66766
860
617-7
650
626-9
1250
648-4
6953
1400
148812
1720
1950
tens
tons
tons
tons
tons
tons
tons
tons
287
67-7
66-85
76126
80-25
80-25
96
96
2863
2860
3010
2700
2615
2525
2656
2500
28225
47875
63400
63190
66385
66790
84070
84510
40-2
48-3
62-1
52-8
52-0
51-6
67-2
67-5
16-4
21-0
22-2
22-8
22-9
22-6
25-1
25-6
3
2
2
1-3
1-35
1-35
1-2
1-2
CO
386
Coventry Obdnanoe
AfouQtain
3-3 in.
20-pdr.
Howitzer.
Diameter of Bore
>> »(
Leng^th of Gun .
»f »» •
Weight of Charge
>« *>
Weight of Projectile
»
t>
Weight of Gun .
»»
Muzzle Velocity
t>
>t
Muzzle Energy
Penetration of Wrought Iron Plate at
Muzzle. Gavre's formula . . in.
» (( ti °3m.
Penetration of Hard Steel Plate at
5,000 yards. Gavre's formula. . in.
mm.
*>
>*
i>
4 in.
4-7 in.
6 in.
•
. in.
60 oal.
50 cal.
50 oal.
Diameter of Bore
40
4-7
60
»» »» • • •
.mm.
101-6
1200
163-4
Length of Gun .
• m.
203
243'6
310
tf ».••••
. mm.
5283
61592
7873-8
Weight of Charge .
. lb.
11-25
160
31-0
•> *> • • <
. . kgs.
5 1
7-26
1403
Weight of Projectile
. kgs.
1403
2041
43-33
ft »» • «
. lb.
31
45
100
t. 0. q. lb.
2 2 0 0
6. 0. q. lb.
fc. c. q. lb.
Weight of Gun ....
• •
3 14 2 0
8 15 0 0
»» »i • • •
. kgs.
2134
3785
8890
Muzzle Velocity
. F.S.
3000
3000
2950
»» »».••«
M.o.
914
914
900
Muzzle Energy .
M.T.
599
870
1869
»» }»••••
.F.T.
1934
2810
6034
Penetration of Wrought Iron
Plate at
•
Muzzle. Gavre'a formula
• m.
16
17-4
2.3-1
" '•
„ mm.
406-4
441-9
686-7
Penetration of Hard Steel
P*ate at
6;000 yaida. Gavre's formuh
I. . in.
—
2-6
5-5
t» ft
,t mm.
"^
66
1£9 7
88
Works' Guns.
Fjbld.
12t-pdr.
2aoal.
30
76*2
76-0
1904-9
1-0
•45
6-67
12-6
o. a. lb.
15-pdr.
33-44 cal.
4-66 in.
Howitzer.
t. o. a.
0 6 0
304*8
1600
48d
68-7
222
8
30
76*2
100-34
2648-5
1-626
•74
6-8
150
« 0. q.
I 8 2
431*8
1850
564
110
356
lb
0
4-66
117-5
72
1828-8
11
*5
1701
37-5
6 in.
Howitzer.
t. c. q.
0 8 2
431*8
1000
306
80-5
260
lb.
0
6-0
152-4
101-5
2578
5-0
2-27
45-36
100
fc. c. q.
1 2 2
1143
1120
341
269
870
3 in.
40 cal.
30
76-2
123-6
3139-3
. 20
*91
5-67
12-6
3 in.
BO cal.
4 in.
40 cal.
lb
0
o.
» 12^
622
2300
701
142
258-6
7-7
195-6
lb.
0
30
76-2
154-5
3924
5-25
2-38
5-67
12-6
t. c. q. lb.
D 18 2 21
948
3C0D
914
242
780
11-25
285-7
40
101-6
166-4
42264
5-26
2-38
1406
310
t, o. q. lb
1 6 3 1(
1316
2300
701
352
1137
10-8
274-3
7*6 in.
9-2 in.
11-02 in.
12 in.
13*5 in.
14 in.
14 in.
60 cal.
60 oal.
60 cal.
60 oal.
45 cal.
45 cal.
46 cal.
7-6
9-2
1102
12-0
13-5
140
14-0
190-6
233-7
280
304-8
342-9
355-6
855*6
387-6
475
668
617*7
630
648*7
6i^'7
9842*2
12064
14427
15689
16001
16476
16476
71-0
95
270
285
290
300
305
32-2^
43-09
122-47
129-28
131-54
136*08
138-35
90-72
172*36
344*72
385-56
6670
636*0
726
200
380
760
850
1250
14IG0
1600
t. c. q. lb.
15 10 0 0
fc. c. q. lb.
t. o. q. lb.
42 0 0 0
t. 0. q. lb.
67 0 0 0
t. 0. q. lb.
76 10 0 0
t. o. q. lb
t. 0. q. lb.
28 0 0 0
81 0 0 0
81 0 0 0
16749
28449
42674
68075
77728
82300
82300
2950
2950
2950
2960
2600
2600
2150
900
900
900
900
792
792
747
^737
7101
14203
15884
18148
20322
20C19
12088
22930
46861
51290
58630
65620
66580
1
29-8
37*9
61*2
50-66
49-1
51-2
61-7
7569
%2-6
1300*3
1286*4
1247
1300*3
13132
8-6
12-3
180
18*3
18-3
19*3
20-7
216-9
312-4
467-2
464-8
464-8
490*2
626-8
388
PARTICULARS OF 15 IN. GUN.
(^Continued from p, 379.)
Elevation of
Qua.
Remaining
Penetration in
Range in yt&rds.
Velocity in feet
inches into Modevu
per second.
Armour Plate.
6,280
1** 63'
2,090
18 0
6,560
3** 69'
1,890
15-6
9,840
6' 24'
1,700
131
18,100
9** 19' .
1,630
11-3
16,400
12'' 41'
1,420
10 '
19,700
16** 36'
M30
91
Note.^^The penetrations are obtained by formula ; those
given for armour of greater thickness than 12 inches are
probably Tinder-estimated.
Abuour Bolts.
For 4!' to 7" armour use 3i" bolts ; for 8" to 12" use 3}" bolts.
Blinimum distance of centre of bolt from edge of plate, 12".
Arrange one bolt to each 7 or 8 square feet of plate.
For weight of bolts add 1} (thick plates) to li (thin plates)
per cent to the weight of the armour taken as unperforated.
N0TS8 OH MACHIHXSV.
Batzno of Hotor-boats.
Bating
-60V-| + /V/| + 2S.
L =
N
area of greatest immersed section in square feet,
length in feet at a distance 4 inches above Ij.w.l.
beajoa on L.W.L. to outside of planking at the position
of the greatest immersed section.
* motor-power.*
7 X Va X N f or 4-stroke motors.
10 X Va X N for 2-stroke motors,
area of exhaust orifice in square inches,
number of cylinders.
Ratinq of Motor-enoines.
(fioyal Automobile Club.')
H.P. sA number of cylinders X square of diameter fn
inches*^ 2*5.
Note. — This formula assumes a piston speed of 1,000 feet
per minute.
MACHINEIIY.
389
'
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890
MACHINERY,
WEiaHTS OF MABINE ENGINES* AND BOHiEBS.
{Compiled by Percy A. Hillhouse, B.Sc, M.I.N. A^y and reprinted \
by permission from Kempe's *^ Engineer^ i
i Year Book''*
.)
Type of Ship.
Type of Engine.
Type of Boiler.
Horse-
power
per ton
totftl
Weight
of
Engines'
Tons
Weight
of
Boilers.
Tons
WJtnU.
per H.P.
per H.P.
Cargo
Beciprocating
Cylindrical
4.67
.107
ai2
Steamers
Geared Tm bines
ti
4-50
-106
ai7
Mail
Beciprocating
l»
6-56
•085
•095
Steamers
Direct Turbines
«>
9-72
•037
•066
Geared ,,
»»
8-00
-050
-075
»t »f
Oil-fired
water-tube
9-35
•050
•057
Channel
Direct Turbines
Cylindrical
13 00
•022
•056
Steamers
a »»
Water-tube
16-00
•022
•045
Geared Turbines
Cylindrical
11-60
•033
-054
it »»
Water-tube
13-60
•033
•041
♦ Published by Crosby Lockwood A Bon.
Weight op Warship's Machinery.
In large turbine-driven ships take 16 to 20 H.P. per ton ;
in destroyers up to about 60. About one-half the weight is in
the eng-ine room.
Pounds of Coal per H.P. per Hour.
For all purposes at maximum power.
Average values.— Battleships 1*7 T, 2*2 R ; cruisers 1-6 T,
22 R ; destroyers 16 T ; high-speed passenger vessels,
1*5 T, 1-611 ; cargo vessels 1*75 T, 1-8 R ; steamboats 2J to
3, exceptionally as low as 1*3 is obtained. (T = turbine,
R = reciprocating.)
For oil fuel take '7 the weight of coal.
For wood fuel take 8 if damp, 6 if dry.
For internal combustion engines (petrol or paraffin) take
•8 lb. per B.H.P. per hour.
Number of tons of coal per 24 hours = -0107 X H.P. X
number of lb. per H.P. per hour.
DESIGN. 891
NOTES ON DSSION.
Determinatiox of Dimensions.
As a first; approximation the dimensions are determined
from t|hose of a fairly similar ship^ using the same block
coefficient of fineness.
The length is then examined from the point of view of
(a) the sum of the longihs of the neoessary compartmemlts
and (5) the minimum length for econonucal propulsion. In
slow ships (i) is of little importance ; the necessary lengths
of machinery spaces, holds, and (in warships) of magazines,
which form the midsliip portion of the ship, together with
the same lengths of bow and stern as have been previously
adopted give on addition the totad length of ship. In vesseU
of nigh or even moderate speed, the length foT economical
propulsion is determined by experience with the help of the
tablo on p. 171.
Subject to these requirements the length is kept aa small
as practicable, for' aoiy increase of length leads to additional
hull weight and cost ; in armoured ships the relative armour
weight also rapidly increases with the length, while
manoeuvring power mminishes. It may therefore be advisable
to accept a length smaller than that desirable from propulsive
considerations.
The product of the beam and draught is known when
displacement, length, and coefficients of fineness are det^-
mined. The draught is frequently limited by the service for
which the vessel is intended ; in that case the beam is then
at once found. Usually the ratio of beam to draught is
determined from considerations of stability.
When trial dimensions have been decided on, the design can
be roughly worked out, and the weights of the various com-
ponents approximated to. A rough approxiUiation to the power
gives the weight of machinery when the H.P. per ton (p. 390)
is known. Generally it is found that some small alterations
are then required in the dimensions. The effects of these or
of changes in the design conditions are dc?.lt with below.
Chanqb op Stability.
If the metacentric hei;^ht is found to bo too great or in-
sufficient, it is readily adjusted by the process described on
p. 129. If the beam and draught are both free, the length
need not be altered ; if the draught is fixed, either the length
must be changed to suit the change necessitated in the beam
or the shape of the midship section altered. The displacement
is assumed to remain constant. Alternatively, by changing
the form of l.w.l. aft, it is possible to modify the position
of the metacentre without changing the principal dimensions.
If "this is admissible, it can be left to a later stage. (See
" Preparation of Lines ".)
392 DESIGN.
CiiANQE OF Dimensions due to Addition of W£iauT.
This may be necessitated by the total weiglit being found
to be in excess or defect of the assumed displacemcfnt. It is
assumed that all dimensiofns increase in proportion.
The total weight w is divisible into certain items (p. 102) ;
these are grouped into two parts, the first of which includes
items whose weight varies as the displacement, and the second
items whose weight is constant. Hull would be included in the
first part, and load (passengers, cargo, or armament), equip-
ment, and usually coal would be included in the second part.
Machinery might all be included in the first part, or its
weight might be assumed to vary as w^ ; armour varies partly
(main belt) as w^, partly (deeks) as yf^, aad is purtly
constant. One -third of all weight varying as wi and two-
thirds of all weights proportional to w* should be included
in the first part ; the remainder should be put in the second.
Call the first part KW, and the second p, w = Kvr + p ;
or w = p/(l — K).
If p is increased by p, w must therefore increase by
p/(i-^K). Generally each ton added to the constant term
or load necessitates an addition to the displacement of 2 to
4 tons, taking the higher number in vessels of high speed.
Al-TEnATTON OF SPEED.
K =3: coeffioient as above (including machinery weight) ss
*6 for many ships of moderate speed.
n =a index of speed at which power varies, 8Bj 4*5 for
most ships, to 3 for very fast or very slow ships.
\ =3 number of H.P. per ton of weight of machinery and
boilers (p. 390).
I = original H.P.
w ss original disj^Iacement in tons.
V = original speed.
i; £= increase of speed (sappofled moderate).
Increase of H.P. = nvj^lj - e[i + wMl-K)]}
Increase of displacement =n"-- to if 1 - K+ ^ ^^ •)
where iv is the original machinery weight (=i/x).
Example, — A ship of 20,000 tons displacement has a speed
of 21 knots, H.P. 25,000 ; 60<>/o of the weight varies as tlie
displacement, the remaining 40 ^'/o being constant. The
machinery weight is 2,200 tons. Find the displacement and
H.P. required (a) if addilSotiai weight aggregating
1,200 tons must be added, (d) if no weight is atkled, but
the speed is increased by 1 knot.
DESIGN. 993
(a) K=« •6,i> = 1200.
1200 1-200
Increase of displacement = ^ = — r- = 3,0(50.
1 — 'D '4
New displacement; = 23,000 tons.
NewI.H.P. = 25,000 X (II)*
= 25,000 (1 + § X 1^) approximately,
= 27,600..
(6) * =± 4-5; v = 1 ; \ = 25,000/2,200 = 11-4.
By ftrsft formula, increase of H.P. = 4-5-r
/ 1 2^5 \
^n25:00b"6[25,000+.4xll.4x20,000]p ^^""^ ^'^^'
By second formula, increase of displacement =
So that the new H.P. is 31,000, and the new displt\cem«nt
21,060 tons.
Preparation of Lines.
rt is sufficient to take sections spaced l/10 apwtt (l =
length b.p.), together with two additional sections sitiuited
l/20 from either end. The principal dimensions are supposed
to be now fixed.
f^irst determine and draw the curve of seotioaal areas
Qp. 91) and the load water-line. This is best accomplished
by taking ordinates from a successful des%n of fairly similar
proportions and corresponding speed, and modifying them in
constant ratios that will ensure the displacement and beam
desired. Find the longitudinal position of the centre of
buoyancy from the area curve.
Then roughly estimate the longitudinal position of C.G.
of the ship (its vertical position and the total weight should
have been previously calculated in order to fix we dimen-
sions). If this agrees with the position of C.B. as fount?
abovci no alteration is necessary ; but in general the C.G.
land O.B. will not be in the sam>e vertical plane. This oan
be remedied in two ways : —
(1) Shift O.B. by altering the curve of areas. Tliis should
be aone J)referably in the after body (see p. 176). It is
advisable not to attempt too large a shift forward as the
propulsion may be seriously affected, particularly at moderate
or fairly high speeds^ but usually a reasoniible shift aft
can be made .without loss. At slow or very high speeds
more latitude can be given, provided that fairness is main-
tained. A convenient way of effecting this shift is to move
the midship or largest section, say, aft through a distanoci
d equal to h/(l — 2l/L) or about 25 h, where A is the shift
894 PREPARATION OP LINES.
aft of C.£. desired, and I the distance between the C.G.s of
the forward and after portions of the curve of areas. The
perpendiculars remain where they were ; but the fore body
is uniformly stretched and the after body contracted, the
shift of any section originally distant a? from Jf being 27ia;/L.
(2) Shift C.G. by altering weights. Usually the machinery
(and in warships the whole citadel) can be moved forward
or aft as may be necGi3s(a,ry to g^t the C.GK in its correct
position. This method is often preferable to (1), particularly
when C.G. is found to be before O.B., since the propulflion
is not involved.
Modify the L.w.L. aft if necessary to adjust the stability
(p. 391). This should be done cautiously, though more latitude
is allowable here than in alterations to the curvo of areas.
Sketch a body, using a planimeter to ensure that the sections
have the areas determined. Roughly fair with a bilge
diagonaL
Check the " critical sections ", viz., through propellers to
ensure adequate clearance 12 in. to 15 in., and through ends
of engine-room, and fore end of boiler-room to ensure there
being sufficient space ; also in warships through the end
barbettes and magazines. Find where the shafts leave the
ship ; the unsupported outboard lengths should not be exces-
sive. The propellers should clear, when possible, in trans-
verse view ; and care should bo taken that the lines permit
a ready flow of water to them from forward. The position and
extent of the side docking keels (if any) require consideration.
The stem and stern can then be drawn in, and the sections
carried above the l.w.l. to the upper and forecastle decks.
A midship section can be prepared showing the construction
and scantlings. Where necessary the longitudinal strength
can be investigated (taking a coefHcient for the bending
inoment from a ' previous similar ship), and the scantlings
revised if necesaary. When finally decided upon, a second
approximation to the weight and vertical C.G. can be made
(all items but hull and machinery are definitely known); if
this u)nfirms the previous estimate, the design can be pro-
ceeded with.
The final complete calculations for the design usually
occupy considerable time,, and are not finished until the design
is nearly completed. If the previous approxiniate calcu-
lations have been properly carried out, and the effect of
all important alterations introduced when working out the
design have been carefully considered, it is improbable that
any further modifications in the dimensions or lines will be
necessary.
FANS.
895
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FANS.
Fans fob Ship Ventilation.
(See table on p. 395.)
Fia. 282.
^1
(O;
B
i
Notes.—The outputs by air in the above table ar^ sligbtty
lower, and the horse-powers higher, than those obtained by
test, thus providing a margin against slight reductions of
efficiency. The B.H.P. of motor is that corresponding to
free inlet and outlet ; at increased pressures the actual B.H.P.
may be only | of this.
The discharge pressures given' are the ' side ' pressures,
i.e. those measured from a water-gauge flush with the sido of
the trunk.
The trunks for 20" and large fans are in all cases suffi-
ciently large to reduce tho difference between ' side ' and
'head on' pressure to a negligible amount. For 12|" and
17i" tans the total heads, or *liead on ' pressures are, for side
pressures of 1", 2". 3", Bh" water-gauge, 1-62", 2-52", 3-33". and
3*69" respectively. For a 7i" fan, 1" side pressure corresponds to
l»r total head.
The sizes of suitable trunking are calculated on a basis of
velocity of 1,600 feet per minute, with outputs at about IJ"
to.g, for all the larger fans and at about 2^' tif»g. for the
lt\[* and 17}" fans. When the latter fans are used against
high resistances — such as air coolers or heaters — ^the trunks,
if short, may be 10" and 14" square respectively ; trunking
of unusual length or conta'nin? many bends or abrupt changes
of section should be increased in size.
Qeneeal Notes on Ventilation and Movement of Air.
Air weighs at standard barometric pressures and
temperatures and with 70o/o humidity 0*08 lb. per cubic foot,
whieh is equivalent to 13 cubic feet per lb.
VENTILATION. 897
If p = weig-ht in lb. per cubic feet, at 70Vo humidity ;
b = heig'ht of mercury barometer in inches ;
T = temperature in degrees Fahrenheit ;
ff = acceleration due to gravity = 32'2 ;
\/^=
then 'Y ~ =42-25 -0-56+0-03T approximately for mo«5erate
changes of pressure and temperature.
For air moving in ventilating systems, where changes of
height, temperature, and pressure are small, the total head
is equal to the sum of the pressure and velocity heads.
Expressed, as usual, in inches water-gauge, it is the pressure
registered by a gauge which faces directly the current of air.
Pressure head is that due to * side ' pressure, or that
roistered by a gauge connected to a hole in the side of the
trunk.
Velocity head is that due to the kinetic energy of thci
air. The velocity corresponding to 1 inch water-gauge ia,
in feet per second, equal to ^/ 62*5/12 X y/lglp. In feet per
minute this becomes 5,790 + 4'lT — 68*5J approximately ; with
a 30" barometer and a temperature of 65° F., this velocity is
4,000 feet per minute. With any other velocity v feet .per
minute, the velocity head is equal to (jaaa/ inches w.g.
For air moving along a tube without resistance, the total
head (i.e. side pressure -{- velocity head) is constant. The
side pressure therefore increases where the velocity head'
decreases, i.e. where the tube is enlarged ; and vice versa.
On emerging into the atmosphere the side pressure is zero ;
it may thence be determined at any other point, being equetl
to the differenee in tlje velocity heads. (This is irrespective
of any change of volume due to the changing conditions.)
In actual trunking various kinds of resistance are
experienced, each of which is equal to the velocity h€od
mmtiplied by a coefficient F.
(a) Frictional resistance. — ^In average trunking, square or
circular, F = 1 for a length of trunk equal to 40 times the
diameter or eide. For shorter or longer lengths take F
proportional. For rectangular trunks of sides a and b take
an equivalent side equal to — t"? *
(6) Resistance due to bends, — ^For a right-angled bend
Fssl*5. F<» bends of inner radius equal to the depth of
trunk in the plane of bend the resistance is fairly small.
After all bends the flow of air is concentrated on the oatier
side of the bend.
898 VBNTILATION.
(«) Re»i8tane0 due to ehange» of $eeiion» — ^For a oon*
traction the sides may be sloped 1 in 2 without appreciable
loss. For an enlargement loss occurs if each side is sloped
more than about 1 in 12. In the worst case, when the
enlargement is sudden, F (referred to the smalletr velocily)
is 1 :; where n is the area ratio.
(d) Resistance due to obstructions. — ^These vary according
to nature and area of obstruction. For armour bars of
ordinary spacing f = *72. For a nest of closely spaced tubes
in a cooling tank it may be as high as 6. For a suction trunk
built closely round the eye of the fan and of depth equal to
the side of discharge orifice, leading off perpendicular to eye
axis Fs=l*5. For a diaphragm constricting area to 70<yo,
P = l-2; to 400/0, P = 8; to 20o/o, F = 60.
Adding these losses tos^^ther we can express —
Final total head = Initial total head + head lost.
This equation enables the velocity of the air in any system
of trunking to be calculated from the difference of pressure
at the ends.
POWEB IN AlE ClECUITS.
Air horse-power is equal to the supply of aar in cubic
feet per minute multiplied by the total nead in inches, w.g.,
and divided by 6,340. If measured at the fan delivery, it
represents the effective horse-power of the fan. The eleomcal
horse-power supplied to the motor is commonly 2 or 3 times
the air horse-power. Multiply this by 746 for the number of
watts.
Example. — A compartment requires 2,000 cubic feet of idr
per minute, which is supplied through trunks of aggregate
teingth 60 feet, and 10 inches square in section. Bends, etc.,
in the circuit give a total value of 2'5 'to F. Det(ermine the
pressure (total) and horse-power.
F due to friction is ^r^^ = IS. Total F = 2*5 + IS
40 X 10
«4-3.
Pressure head = o at outlet.
Velocity head throughout = (r-r — rpv^) = *36 inches w.g.
Head lost due to resistance = *36 X 4*3.
Total pressure at fan = '36 X 53 = 1*9".
.. , 1-9x2000 ^
Air horse-power = — tttt; — = '6.
oo4U
Electrical H.P. = -6 X 25 (say) = I'S ; watts = 1*5 X
746 = 1,120.
Note, — ^In the above if a bell-mouth be added to the
dischar&ro trunk of size 12^' square, F is increased bv *2
VENTILATION. ^99
(say), but the velocity head at outlet is now (JS)* = about
4 it8 former amount. Total pressure at fan is '36 (4*5 -f- -o)
•«= 1*8" or about 6^h less ; and the output of air would be
eorrespondingly increased.
Position of Thunks in Compartments.
Air entering a compartment will, unless the velocity be.
very small, travel a considerable distance across the compart-
ment in almost a straight line in the direction in when the
trunk is pointed. On the other hand, air leaving a compart-
ment travels radially from all directions to the exhaust oriifioe^
its motion bein(^ independent of the direction of the trunk.
Supply trunks should therefore be pointed away from
exhaust orifices to ensure a good circulation of air. They
•hould, when supplying cold air, be directed horizontally
just beneath the beams ; the density of the cold air is
8n£5cient to cause a good circulation on the floor. In very
hot compartments, such as engine-rooms, they may be directed
towards the spaces where men are generally working. Whepi
supplying heated air they should, on the other hand, point
downwards ; in such oases it' may be necessary to have a
shifting mouthj)iece whose direction can be changed.
Exhaust trunks should have their orifices as high as
possible.
The efficiency of the installation is improved if bell-mouthe
of sufficient size to reduce the air velocity to 1,200 feet peir
minute or less be fitted to both supply ana exhaust openings.
That on the exhaust can be short, sloping 1 in 2 on each
side. That on the supply must taper gently— not more than
1 in 12 each side ; and draughts are reduced if they can be
made of sufficient length if> reduce the velocity well below
1,200 feet per minute, since even 200 feet per minute is
perceptible.
Quantity op Aib eequieed.
BoUer-roomB.—AMoyr under forced draught 18 lb. of air
or 280 cubic feet per lb. of coal burnt ; with oil fuel allow
20 lb. of air or 260 cubic feet per lb. of oil. Under natural
draught multiply these amounts by IJ^
Sleeping tpaoe3,—'Eitty cubic feet of air per man per
minute is an ample allowance. In messing and similar spaces
occupied intermittently by men, this allowance is unnecessary;
about one-half is generally sufficient (see Board of Trade
Begnlations below).
Miscellaneous spaces, — ^These, if packed with men, may be
dealt with as above ; otherwise it is usual to allow so many
minutes for complete renewal of the air in the compartment.
400 VENTILATION.
In U.S. Nav^r, this allowance b as follows :^
Space.
Number of minutes foi
complete TeD6\ral of air
Officers' quarters and crew space
ontside armour ....
10 to 12
Do. inside armour .
4 to 6
Sick-bay
8
w.c.'a . . • •
4
Store-rooms .
10 to 15
Magazines ....
4 to 8
n (for cooled air)
6 to 12
Engine-rooms .
2
Steering gear compartments .
8
Workshops . . . ,
4
Dynamo-rooms
1
Switchboard-rooms
6
Ice-machine rooms
4
Efflux of Aib theouqh an oeipice.
Pi = initial gauge pressure In inches of water.
J) ss actual gauge pressure in inches of water after time
t (seconds).
y ss yolume of compartment in cubio feet.
A «= area of orifice in square inches.
Q s= rate of efflux in cubic feet per minute.
T = time in seconds required to reduce pressure from
Pi to atmospheric.
For moderate pressures pi and p —
Q = 20a Vi? for a plain orifice.
* = v(Vl?i- Vjp)-f70A.
ff = V Vi?i-r70A.
Note,— The time necessary to reduce pressure
amount is about one-third that for pressure
atmospheric.
This formula could be used to detect the size of any opening
causing leakage in a -W.T. compartment by putting it under
air pressure from a fan and noting with a water gauge the
rate at which the pressure drops when the fan is cut oft
and the trunk closed.
Board op Trade Eules for Ventilation op Steerage Com-
pabtments and hospitals in emigrant ships.
Natural ventilation. — ^Each compartment to have cowls,
or equivalent, having an aggregate area of 5 square inches
to half its
to become
VENTILATION. 401
per adult* accommodated. Area to be measured at narrowest
part ; half to be inlet and half outlet. In vrings adjoinii^
engine- and boiler-rooms this area to be increased by 33 o/o .
Cowls to be carried to a convenient height above deck and
to be clear of obstruction ; area of cowl to be at least 50 o/o
more than that of pipe. When the pipes are bent or kneed,
increase area as follows : —
(a) Curved bends (inner radius greater than diameter). —
Angle of bend up to 30°, no addition ; from 30^
to 60° add 6«/o for each bend ; from 60** to 90** add
10 o/o for each bend.
(6) Curved knees (inner radius less than diameter). — As
above, substituting 16 o/o for 5 o/o, and 36 o/o for 10 o/o.
Each cowl ventilator to project 3" below roof of com-r
partment, and downcasts are to be fitted with a canvas pipe
extending to about 12'^ above floor. Two cowls td be fitted
in compartments containing up to 75 adults ; three from 76
to 125 adults, and four 'for more than 125 adults.
Ventilators, or that portion leading to any one compart^
ment, should not exceed 314 square inches in area (20"
diameter). The minimum diameter (except for hospitals and
sanitary arrangements) is 10".
A velocity of air of 800 feet per minute is assumed ; the
allowance to each adult is then 830 cubic feet per hour
(18*8 cubic feet per minute). Trunkways built solely for
ventilation and carried to sufficient height, may be accepted
instead of cowls as either inlets or outlets, but not both ;
their area must be doable that required for a cowl.
Hospitals should be ventilated independently and to the
open sir ; the aren required is 5 square inches of inlet,
and the same for outlet per adult, with means for controlling
the size of the openings. Sanitary arrangements should be
ventilated to the open air.
In general no ventilator must pierce a transverse W.T.
bnlkheMl.
Ariifldal ventUuHon, -^The above rules hold for the general
arrangementa of Trunin, and for the quantity of fresh air
to be supplied (880 cable feet per hour per adult + 83 o/o in
maoUnery wings ; 1,660 oubio feet per hour per adolt in
hospitals).
6^#ii«f«^— Inlets and outlets to be placed at oppositfe ends
of compartments. Tho TeutiUAion of eaoh passenger deck is
to ho independent ; that of tho hold must not affect
that of any of the passenger compartments.
* i.e. perflon of 12 years or more.
Dd
402
HYDRAULICS.
HTDBAULIC8.
Duty of Ship's Pumps.
1
{Engineer Commander J, E. Mortimer ^ M.I.Meeh.E,, \
M.LN.A.)
•
Bilge>
Centrifagal.
Three-throw
Portable
Type of Pomp.
Reciprocating
Reciprocating
Single-acting.
Doable-acting.
Capacity in tons per
hour
50
10 5
6
Diameter (inches)
4
3J 2i
3i
Stroke (inches) .
4 4
41
[Height
3' 3"
3' 7" 3' 7"
I' 6"
Including
motor
Floor space
5' 0" X 2' 0"
2' 4" X 2' 3"
1' 8^x3' 9"
iweight(cwt.)
13
H 7f
84
B.H.P. • •
17
2 li
U
Delivery (gallons per
minute)
. 187
37 19
19
Delivery pressure (lb.
per square inch)
60
30 30
13
Suction lift in feet .
25
25 25
20
Size of delivery pipe
(inches)
4
2 2
li
Table op the Pressure of Water at Different Heads.
H » head in feet, p - pressoie in lb. per sqnaie foot, p « pressure
in lb. per square inch.
R
p
P
H
p
p
R
p
P
1
62-4
•4333
5
3120
2-1666
80
1872-0
13-0000
1-26
78-0
•5416
6
374-4
2-6000
40
2496-0
17-3333
1-5
93-6
•6600
7
436-8
3-0333
50
3120-0
21-6666
1-75
109»2
-7683
8
499-2
3-4666
60
3744-0
26-0000
2
124-8
•8666
9
561-6
3-9000
70
4368-0
30-3333
3
187-2
1-3000
10
624-0
4-3333
80
4992-0
34-6666
4
249-6
1-7333
20
1248-0
8*6666
90
5616-0
39-0000
Discharge of Water from Sluices and ORinoBS.
V = theoretical velocity due to head of water in feet pep
second.
H = head of water in feet.
A s= area of aperture or outlet in square feet.
HTDRATJUCS.
408
Q
a
9
V
k
quantity discharged in oubio feet per second,
quantity discharged in gallons per minute,
force of gravity = 32'2.
velocity of real discharge in feet per second,
coefficient for different diameters of sluices.
V= \/2gfH = 8-025 ^H
= •01553V*
V
Q = Kk\/2gK = 8*026Afc v'h
G = 375Q = 3010Afc Vn
v-hx^^gk =8-025A;V'H
Table of th£ Values of Coefficient &. j
For Short Square Tabes.
For Short Cylindrical Tubes. ]
Luth.
ft
Lgth.
k
•69
•65
•62
Lgth.
ft
Lftth.
Dia.
ft
Uth.
Dia.
1
ft
Lefh.
ft
Dia.
0
2
10
Dl«.
Dia.
Dia.
•617
•814
•75
20
30
40
60
60
100
•59
•56
•48
1
2
4
•62
•82
•77
13
25
37
•73
•68
•63
49
60
100
•60
•66
•48
Time eequired to Fill ok Empty a Compaetment.
A = volume of compartment in cubic feet.
A ss area of pipe or aperture in square feet.
K 3= coefficient (see above).
hi^ 7t2 = initial and final heads of water in feet.
Time in minutes = 15A/a( V^i+ ^/h^jK, if aperture is under
water.
Note, — ^Wlien filling a compartment, such as a magazine,
by means of a pipe whose open end is part way down the
side, calculate separately the volumes above and below thq
level of the open end. The above formula gives the time for
the upper volume ; that for the lower portion = 7*5 A /ak ^/h
where h is the head to end of pipe.
If the pipe is long, or if it contains bends, valves, eto.,
replace K 'by i/(i -\- f), where f has the value given on p. 404.
Flow op Watee through Pipes.
h s= difference between heads of water at ends of pipe in
feet.
^ SB diameter of pipe in inches.
404
BYDEAULXC8.
/ =
Fas
Q =
V =
leai^th of pipe Lb feet,
coefficient of f rioiion for pipe.
sum of all coefficients ox resistanee referred to dia-
meter d,
cnbio feet of water passing per teeond.
number of gallons per minute,
velocity of water at diameter d in feet per second.
32*2 feet per (second)^.
^(l+fl/d + F).
Sx/h/(l+fl/d + ^).
Q = vd^/lBZ
'^ 28^ i + nid+
■{■flid+v
O — 2*05 «<{3 — l%4^^h/(l + fi/d + P).
d.
/ for new iron pipes.
/ for old iron pipes. 1
r«l
vmlQ
r«l
on 10
8
6
12
16
24
36
48
60
•0024
•0023
•0021
•0019
•0018
•0014
•0012
•OOU
•0021
•0019
•0016
•0014
•0012
•0011
•0010
•0009
•0051
•0045
•0036
•0029
•0025
•0020
•0019
•0018
•0041
•0034
•0028
•002S
•0020
•0017
•0015
•0014
For steel riveted pipes take f ^ i value for new iron pipe.
For brass or lead pipes take / = '8 value for new pipe.
YAiiU£S OP r.
Sadden enlargenient (referred to small diameter), _ 1
area ratio \ .^~\2
Gradual enlargement, angle 0 f as for sudden enlargement x aiu 6
Diaphwigm, hole S pipe area . . . . . F= -76
... F = 3-9
Fs=X8
F = 49
F as witb diaphragm x 1 • 8
F= -86
F = l
F = 1.86
Bend, ,, 90^« inner radius = ^(2 . F= '14
Note. — If any portion of the pipe is of diameter D (different to
d)t multiply F for resistances in that portion by (4/d)^
1*
>>
hole I
hole 'Z
hole -3
Cock paitly open •
Elbow, angle OC*
90^
120"
»♦
I*
»♦
»♦
)«
HSAT.
1106
HEAT.
Thehmal ^opeeti£:s of MAxmuLs.
Coefficient of
t
Material.
•
Meltingr-point
in degrees
Tahr.
' conduction—
B.T.U. per
square foot
per hour for
1 in. thi«k-
ness.
Linear expan-
sion per decree
Fanr. x one
million.
Specific
heat.
Air (62«P. constant
pressure) .
—
-^
680
•24
Aluminium .
1,250
1.000
12-8
•212
Antimony
810
125
6-2
•061
Asbestos millboard
_
1-2
-^
—
Bismuth . «
600
61
7-6
•030
Brass • . » .
1»700
670
10-6
•090
Bronze ....
1.690
^
10-4
•08C
Copper ....
2,000
2,100
100
•097
Cordite (density 1*57) .
-m
.^
-m
•36
Cork ....
.mm
•48
__
—
Glass ....
.^
46
12 to ^18
Gold ....
2,200
•^i*
80
•032
Gun-metal
1.860
«.
10*4
•088
Ice ....
82
16
21
•504
Iron, cast
2.200
480
60
♦112
Iron, wrought
2.950
6S0
6-4
•113
Lead ....
620
240
16-9
•031
Mercury
39
4$
330
•033
Nickel ....
2.810
_^
71
•109
Petroleum
.^
_
185
•61
Platinum . • •
3,080
_
60
•032
Silicate cotton
1 —
•66
— .
Silver . . ...
1,860
3,200
11^
•056
Steam (212<' F., constant
pressure) .
^^
.^
780
•477
Steel ....
2,460
320
6-4
•118
Tallov ....
92
-^
.^
Tin ....
446
440
m
•056
Water ....
—
—
266
1-00
Wood (fir) .
—
104
3*0*
lO'Of
170
•16
Zino ....
750
900
•094
*A!ons: fibre.
t Across fibre.
Notes ok Heat.
1. One British Thermal Unit (B.T.U.) is the amount of
heat required to raiie the temperature of lib. of water by l^F.
2. One B.T.U. is equivalent to 778 foot^lb. of work.
406 AEKODYNAMICS.
8. The loss of heat per square foot from a heated body is
proportional to the difference between its temperature and
that of the surrounding air. This loss is divisible into two
parts— loss by radiation and loss by oouvectioa.
4. The loss by radiation in B.T.U. per hour per square
foot for 1** F. excess of temperature is *03 for highly polished
surfaoee, '06 for polished brass or tin, *75 for oil paint, wood,
01^ canvas, '65 for dull metallic surfaces, and *82 for dull black.
5. The loss by convection depends on the shape and position
of the surface. In still air the loss in B.T.U. per hour per
square foot for $^F. excess of temperature is about
K$ . (e/23)'28S, where K is—
'363 -\- 1'05/r for a sphere of radius r inches.
'421 -j- '307/r for a horizontal cylinder of radius r inches.
. 204( • 726 + -^) (2 . 43 + ^^^) for a vertical cylinder,
radius r inches, height h feet.
*861 + '2SZ/^Jh for a vertical plane surface of h feet.
6. By assuming an approximate value of $ for (5), the
total loss per square feet per degree per hour due to radiation
and convection can be estimated. Call it E.
7. Similarly the loss due to conduction for 1^ difference
and t inches thickness is found by dividing the coefficient of
conduction in the table by t. Call this c.
8. Then the total loss of heat passing through a lagging
material and including the loss at two surfaces is, for a total
temperature difference of $, equal to 6/{- +- ) per square
foot per hour. If there are air gaps and several materials
2 1
find - for each air gap and - for each material ; the sum
makes the denominator of the above expression.
9. The temperatures (P. = Fahrenheit, O. = Centigrade,
and B. = Reaumur) are connected by F. = 1*8 C. -f 32 : R. i»
•8 0. ; F. =2-25R. + 32.
10. Absolute temperature = F. -f 461 or C. + 273.
AEBODTNAUIGS.
A. W. Johns, Esq., M.I.N.A., R.C.N.C.
FoECES ON Plates.
The resistance of a&r to a plate moving in a direction
normal to its surface is given by a ss kav^ where E is in
pounds, A is in square feoife, V is in miles per hour, and K
is a coefficient.
The value of k as determined by various authorities is as
given in the table below.
AEEODTNAMICS.
407
Experimenter.
Size of
plate in
inches.
Square pMtea.
Stanton * . . .
Hagen
Borda
Dines
Finxi and Soldati .
Eiffel
Desdouits . . • .
Hagen
Cailletet and Colardeau
Langlev . . . .
Eiffel
Cailletet and Colardeau
Eiffel
t» , . . . .
Institute of Koutohino
Dines
Finzi and Soldati .
Thibault . . . .
Eiffel
Dines
W. Froude
Eiffel
Marriote . . . .
Eiffel
Reichel . . . .
Le Dantzec
Eiffel
Didion . . . .
Von Lossl
Finzi and Soldati .
lleichel . . . .
Stanton . . . .
»»■ i . . .
Paris
Rectangular plat&a.
Nipher
Rateau
Eiffel .
Canovetti
Hutton
Ecrnot
St. Loup
Eiffel .
Ceaufoy
Langley ^
Canovetti
Stanton
t*
L-77xl'77
2x2
8x8
4x4
4x4
4X4
4X4
6x6
6x6
6x6
6x6
8)x8}
10x10
10x10
12x12
12x12
12x12
13x18
14x14
16x16
18X18
aoxao
21x21
28x28
88x33
89|x
doix
X8!
Xb9i
X
60x60
120x120
notknown
86X48
12x20
6x9
54x92
4X8
4|x8
4x8
9|xl9i
6x12
6x12
6X12
89'4X78-8
60x120
1X8
o s u
2 js » o
> »
19
4
21
22
20-33
44
4
65
40
20-33
60-64
20-33
40-80
2-14
40
22
38-76
21
8-10
38-76
38-76
90-110
38-76
15
2
22
70-90
0-26
0-25
0-90
20-60
80
20-33
12
6-8
12-27
85
40-80
20-33
3-14
20
14
0-26
14
Value
of K.
00287
003
0039
0035
0031
00265
0063
0032
0029
0030
0027
0028
0027
00286
0035
0029
0033
0948
0031
0035
0039
0030
0049
0031
0042
0034
0032
0032
0042
0042
0042
0032
00322
0049
'0025
0027
•0027
'0036
•0033
•0033
•0029
0031
•0029
•0045
•0033
•0087
•0032
•0029
Method of experiment.
»>
*>
Plate in a current of air.
Whirling machine, 8 ft. radius
28
i». »» 1^ »i
Plate in a current of air.
„ carried on a train.
See above.
Plate falling under gravity.
Whirling machine, 30ft. radius
Plate in a current of air.
See above.
Plate in a current of air.
„ falling under gravity.
„ in a current of air.
See above.
»>
Whirling machine, 4ift. radius
Falling plate.
See above.
Plate carried on truck of
experimental tank.
Falling plate.
Whirling machine, 4if t. radius
Falling plate.
Whirling machine, 21ft. radiui^
Falling plate.
i»
Whirling machine,
ft
»
,^ft.
radiu*'
21
Plate m a natural wind.
*t
*>
»
Plate carried on a train.
t, in a current of air.
»» , »» »♦
,, sliding down a sloping wire
Whirling machine, 4«t. radius
it
>*
10
»» -^ »»
Falling plate.
Plate in a current of air.
WhirliAg machine, 9 ft. radius.
»» » 30
See above.
Plate in a natural wind,
in a current of air.
ft
408
AEKQDYNAMIG8.
Experimenter.
Eiflfel
Institute of Koutohino
Finzi and Soidati .
Dines
Langley . . . .
Dines . . . . .
Eiffel
»»
Stanton
Eiffel .
Dmes .
Hagen .
Dines .
Eiffel .
»» •
Stanton
Circular plates.
Stanton . . . .
Institute of Koutchiixo
f*inzi and Soidati .
Dines ....
Mannesman
Eiffel ....
Cailletet and Colardeau
Hut ton
Eonard
Dines ....
Eiffel ....
Finzi and Soidati
Dines .
Eiffel . . .
»
O'Crorman .
Finzi and Soidati
Oanoyet^i •
Von Lossl
Size of
plate in
inches.
6x18
4X12
4X13
4X16
6x24
6X24
7X28
6X84
6x36
4X86
I'9xl9
•3x4-5
1-6x24
3X48
1X16
1x16
1-85X27
*86x^
-15x9
2"dlam
3^'
4"
4i"
5-8"
6"
f
61"
8"
9^'
11-8" ;,
12" ..
18-54" „
16"
22r
82"
86"
89-8r' „
39-87" „
43-8* M
I*
t*
O d h
O ki o O
9 tt a
>9"
20-33
9-18
22
21
12-22
21
38-76
20-33
20-33
20-33
20-33
14
20-33
40
4
21
20-33
20-33
14
16
2-14
22
21
45
20-33
47
7
O-lOO
21
40-80
20-33
22
21
40-80
-10-80
40-80
22
17
2
Value
of K.
•0029
•0029
0032
•0036
•0033
•0036
•0031
0030
0030
•0031
0031
•0034
0034
•0036
0032
0039
•0036
•0040
•0043
•0029
•0033
•0032
•0036
•0049
•0027
•0028
•0034
•0042
•0034
•0029
0028
•0042
•0035
•0029
•0031
•0032
0037
•0042
•0031
•0042
Method of experiment.
Plate in a current of air.
Sec above.
t>
»» »•
»» »»
Falling plate.
Plate in a current of air.
»» »»
»»
>» »i
»»
*t »)
**
>> M
it
Whirling maehiue, 28ft. radius
Plate in a current of air.
>* M
»»
** M
• »
»
See above.
M
*»
t*
>»
Whirling machine, Ij ft. radius
Plate in a current of air«
Falling plate.
Whirling machine, 4Kt. radius
«* >* 28
Falling plate.
Plate in a current of air.
Whirling machine, 15ft. radius
♦> *> ^
Falling plate.
ti
»
n
>>
Plate towed through air.
See above.
Whirling machine, 3^ft. radius
Notes on above. — ^The value of K determined by Marriote
is that upon which Smeaton based hid^ well-known formoia
H == -OOdAV^. It gives too high a value of K, since the length
of the whirling arm is small compared with the size of plate.
The same remark applies to the results- obtained by Thioault^
Eeichel, Von Lossl, Beaufoy, Eenard, Mannesman^ Huttos,
and for the largest plates of Finzi and Soidati. The values
given for the mt two are for the medium line of the plate
AERODYNAMICS.
409
and not for the wholei, and for this reason the value of e is
greater than if taken over the whole plate. The larg^o value of
K determined by Paris ia probably due to too low a speed of
the wind being taken. This was measured by an old-fashioned
anemometer.
The value of K now usually employed for calculations of
pressure on square or circular plates of medium size is '003ty
in place of '005 (tihe value given by Smeaton) formerly used.
For large plates it is '0032 ; for rectangular plates of medium
size it is approximately given by K = '003 -I- •000025», where
n is the ratio of the longer to the short side.
VAIitJES OF THB COEFFICIENT K FOR OTHEB BODIES.
Body.
Sphere, liO^ diameter
»♦ »t •
•t l* »» •
Hollow hemisphere, base flmfc
Hemisphere, onrred part first
Cone, vertical angle 60°, height
= diameter, point first
Cone, vertical angle 80^, base
» Itf', poimt first .
Cylinder, moving base first
II »t »i
Smooth wires .
Stranded wires .
Perforated plate
•»
ft
t*
II
t .
•I
Orating .
Metal gatuee
Trellis work
Struts
Value of K.
•00015
•00045
•00100
•0035
•0008
•00013
•00006
•00255
•0021
•0023
(•0028 to)
{•0032 I
•003
•0027
•0024
•00225
•00180
•0005
•00165
•00225
f -00046 to I
t-OOB
;
Authority.
EiflCel
Benard
Beaufoy
Eiffel
»•
fN.P.'li.A)
1 Prandtl )
Dines
OanoTetti
II
N. P. li.
Bern arks.
Length=diameter.
Length =2| diameter.
/Plate 12"xi2f with
8-1" holes.
(PUte 112" X 12", with
77 holes to sq. in.,
nett area 61 p.c.
Holes 12 to sq. in..
nett area 56 p.o.
Plate 8i sa. ft., per-
• forated with 8,000
, hole«,nettarea96p.o.
(Plate 8i sq. ft., per-
forated with 106
holea.nett area70p.c.
Nett area 18 p.c.
1. M VO P.O.
.1 I. 60 p.c.
Depending on shape
of section*
In the above the resistance is obtained by multiplying the
coefficient by the (speed) ^ and maximum section perpen-
dicular to the direction of motion. For the perforated plates,
grating, gauze, ete., the coeflficient is for tlic area of tho
containing figure.
410 AERODYNAMICS.
In the case of struts the resistance varies considerably
with the shape of section in the direction of motion. The
best result obtained by the N.P.L. was with one whose
length of section was six times the breadth^ the greatest
bre^adth being one-third the length from nose. The breadth
half-way between nose and greatest breadth was *9 the latter,
and half-way between tail and greatest breadth was *72 the
latter. The nose and tail were sharp, and the area of section
was 70 per cent of the containing rectangle. This gave the
coefficient '00045. Experiments carried out by N.P.L. for
Ogilvie gave a qmaller resistance than above with a strut whoso
length of section was 2^ times the maximum breadth. It
had a flat nose and a somewhat bluff tall. UnfortunatelyS
both series of experiments were made at only one speed of
air current, and it is quite possible that the best form at one
speed may not be the best at another. Eiffel's experiments on
struts show the coefficient of resistance diminishes as the speed
increases, the diminution being different in different shapes.
Besults are therefore uncertain, and a great amount of experi-
mental work is still necessary. The same remarks apply to
the greater number of experimental results for bodies fotr
which the eddy resistance is a large part of the total resistance.
Parallel plates moving normally. — ^Eiffel made experiments
on circular discs, 12 inches diameter, one behind the other.
The total force on the two is less than on one for intervals less
than two diameters, the minimum being at about IJ diameters
interval where it is about three-fourths that on a single disc. For
intervals greater than two diameters the total force is greater
than on a single diso, bat even at three diameters interval
the total force is only 40 per cent greater than on a singly
disc. Similar results have been obtained by Stanton on 2"
and \\" damcter plates. For the former size the total force
on both plates is twice that on a single plate when the interval
between is about five diameters. For the 1^ in. plate the same
is not obtained until a much greater interval is reached. The
experiments show the extent of the shielding effect.
Frictional or skin resistance in air. — ^The most extensive
series of experiments on skin resistance of bodies in air was
made by Zahm. He used boards (maximum length 16 feet)
with surfaces of dry varnish, wet sticky varnish sprinkled
with water, calendered and uncalendered paper, glazed fabric,
and sheet zinc. AH gave the same results. Coarse buckram
cloth gave 10 to 15 per cent greater resistancey and the latter
varied as v^ for speeds up to 25 miles per hour. His result?
are expressed by
E in lb. = -0000158 x area x -757-
where l = length in feet, A = area of surface in square feet,
0rnd y U 0peed in miles per hour. Assuming the same law
ABJRODYNAMICS.
411
as regardfl variation with length to apply to greater lengthy
the following values of the coefHcient are obtained.
Short lengths .
15 feet lengths
20
100
200
800
400
600
ft
it
»
it
it
)i
it
it
»
it
K = -0000158.
K = -0000131.
K = •0000128.
Etr= -0000114.
K = -0000109.
K «= -0000108.
K = -0000104.
K = -0000103.
Expressed in terms of the area of surface and square of
the speedy i.e. b = eav^, k has the following values:—
For short lengths E = '000009.
For a length of 200 ft. E = '00000615 between 40 and 60 m.h.p.
300 ft. E := -000006
400 ft. E = -0000059
600 ft. E = -0000058
»t
it
tt
tt
it
tf
t>
-i
ti
it
it
ti
Vl-83
tt
Taking the formula E = '0000158 X area X-r.07- the fol-
lowing table gives the coefficient which multiplied by the area
gives Ihe resistance in lb.
Length of
Surface.
Valaes of Coeffloient 1
at SO m.p.h.
at 40 m.p h.
at 50 m.p.h.
at 60 m.p.h.
200 feet
300 „
400 ..
500 „
•0059
•0057
•0056
•0055
•0100
•0097
•OOJO
•0094
•0152
•0147
•0145
•0142
•0212
•0206
•0203
•0200
Thurston's experiments on skin resistance were carried out
on glass surfaces up to 4 feet long, and gave the following : —
K = -0000098 A (v2 + 32-22).
Franck*s experiments carried out on surfaces swinging
with a pendulum gave —
B = -0000124 AV».
FoBCEs OF Inclined Plates.
Before quoting results of experiments on inclined platelB
it is necessary to explain the following terms : —
Aspect ratio is the ratio of the dimension perpendicular
to the direction of motion to the dimension parallel to the
latter. Thus a plate 12 in. by 2 in., moving broadside
on, has an aspect ratio of 6. The same plate moving
endwise has an aspect ratio of j^. The dimensions of
a plate are always written with the side moving perpendicular
412
ABEOBYKAMIGS.
to the wind first, so that t^ie aspect ratio is at once seen^
i.e. a 36 in. by 6 in. plate is one mofing^ broadside on, and
lias an aspect raitio o. A 6 in., by 36 in. has an aspect
ratio }.
The lift on a surface is the force on it, due to its motion
relative to the air, in a direction perpendicular to that of
motion. The lift eoefflei&nt^ i.e. that coefficient which, multi-
plied by the product of area and square of speed to give the
lift, is usually denoted by Ky.
The drift is the force in the direction oppoeite to that
of motion, i.e. the resistance. The drift eoefficimtt is usually
denoted by k^.
The total or resultant force is the resultant of the lift
and the drift, and its coefficient is usually denoted by ff. On
flat plates the resultant force acts sensibly normal to the
surface. Skin friction and head resistance — especially the
latter— cause a slight deviation from the normal. !Blffel*8
experiments showed that/ at small angles, the deviatian was
appreciable. The plates used by him were, however, relatively
tliick — nearly one-eighth of an inch — the edges were not
chamfered, and a sensible head resistance was thus caused.
Experiments on plates with chamfered edges show the
resultant force to be practically normal to the plate.
The value , .«. expresses the aerodynamical efficiency of
the plate. The larger this ratio the greater the weight wliich
can be lifted for a given resistance. The maximum valuo
occurs at small angles varying for different shapes between
2" and 8\
Critical angle, — If the resultant pressure coefficient is
plotted on a base of angles of inclination a curve as shown
m the figure is obtained for a thin flat plate. The curve
Fio. 231.
Curve of S
Angflea of Inolinatioli.
is generally a straight line from zero inclination up to an
angle n. It then drops to a minimum at C. It then rises
from c to 90'. The angle b is the en' Heal anjle. For a
square pfate it occurs at about 38". As tlie aspect ratio
AEROPTNASflCS.
418
increases the- critical angle diminishes, and the difference
between the maximum and minimmn values also decreases.
For aspect ratios less than one the critical angle is above 38^
and moves towards 90° as the aspect ratio dmiinishes. The
portion between B and 0 also flattens out. A similar, bub
more exaggerated curve is obtained for curved plates. Since
for aeroplanes the lift would suddenly diminish if the inclina-
tion exceeded the critical angle and would lead to difficulties^
this angle is the practical limit of inclination of wings, and
its corresponding lift ooefficient defines the minimum speed
at which the aeroplane can safelv fly. In addition to the
lift decreasing as the critical angle is passed, the drift alsd
increases suddenly. The sudden change is caused by a change
in the type of flow of the air over tne surface. For angles
less than the critical the streams on tihe front all pass along
from the fore or leading edge to the after or trailing edge.
For inclinations above the critical angle the streams on Qie
front divide, some pass around the fore edge to the back
regions, others pass direct to the after edge. At l^o critical
angle the change In type of flow occurs, and at this angle the
flow is unstable, and may be either one or the other. In such
a case two different coefacients may be obtained at the critical
angle. (See Gottingen results for a square plate at 40°.)
Experimental results. — In the following ^bles the eo^
efficient of resultant pressure (k) is given. Units pounds,
square feet, and miles per hour. The lift and drift etn
efficients can be obtained by resolution in tiie directions
perpendicular to and parallel to the motion.
EiFFBL'S Experiments (La resistance de Voir et V aviation), \
Speed of air about Z7 miles per hottr. 1
"g •
1?
^IvllMW flf ^liwffi^^ffllt K«
Position of
critical angle
9
Dixnenfl
ininci
and value
OlKBt
the angle.
6«
10°
•0019
20O
•0022
30*>
•0024
40<=»
45®
60'
dO"*
36x4
•00189
•0029
•00306
lO"
•0025
6
38x6
•00111
•00175
•00212
•00237
—
—
•D028
•00302
160
•tXfil
8
18X6
•000^
•00145
•00223
•0022
— .
■— ^
•0027
•0029
20**
•00223
S
1SX6
•00075
•00134
•00963
•0020
•00224
...
•0026
•00286
21©
•00278
11
9X6
«.
•ooaii
•0033
•0023
•0024
..
•0096
•00277
26°
•0030
1
XOxiO
.<«>
•QQQ96
'9m
•0094
•0038
•00294
•0098
•00278
37°
•0039
1
6x^8
•noofla
•00061
•00147
•0096
•0035
•0035
•0031
•0029
43°
•00854
•KM
•oooa
•0041 -ooii
•ooaa
**"
•0061
•0089
•0080
64°
•0069
The plates used were '12 in. thick, with square edges.
Eiffel's experiments also showed that for similar plates
414
AEBOBTNAMICS.
the ratio of the coefficient at an angle i (K{) to ihe coeffi-
cient at 90° remained constant. He proposed the following
formula for angles of inclination up to 12° : —
Kgo \
?l'
«-2+^)i56
where i is the angle in degrees and n is the as^tect ratio.
Gottingen reatUta.Sl&tQ^ all 14 in. measured in a. direction
parallel to motion.
Aspect
ratio.
Values of CoefBcient k.
Critical angle
and valoe of
10°
15°
20°
30°
40°
50°
60°
corresponding
coefficient.
8
2
il
1
•00186
•00151
•00131
•00114
•00008
•00072
•00233
•00226
•00205
•00185
•00157
•00128
•00235
•00261
•00282
•002C0
•00234
•00188
•00244
•00231
•00224
•00891
•00377
•00329
•00275
•00260
•00282
•00292
•00471
•0029i
•0045
•00614
•00814
•0080
•0048
Mil t J
•
18°
18°
22°
33°
42°
43°
•00235
•00266
•0O295
•004J7
•00474
•00468
In these experiments the under side of the plates was
chamfered and the edg^s were very thin, and this probably
explains the differences in the values of tho coefficients as
compared with Eiffel's results.
Institute of Koutchino's results-^
Aspect
ral^o.
Dimen-
sions.
Values of Coefficient k. 1
10°
15°
20°
30°
40°
60°
60°
8
1
12" X 4"
12"xi2"
•0^15
•0U088
•0017
•0021
•0018
•0024
•0027
•0028
•0028
•0024
•0027
•0025
•0027
Mr» M, B. Froude*8 results in water divided by ratio of
densities of water and air —
Aspect
ratio.
Value of Coefficient.
Critical angle
10°
IS-
•20°
30°
40°
90°
and value of
coefficient.
1
2
•0011
•0014
•0009
1
! -0017
•C022
. -0012
1
•0024
•0023
•0018
•0034
•0022
•^038
•0029
•0024
•0088
•003
•0081
•0032
4
37°
35°
50°
•004
•0028
Centre of pressure, — The results of different experimenters
differ in the position obtained for the centre of pressure. For
motion perpendicular to the plate it must bo at the o©ntr«.
ABRODTNAMICS.
415
As tlie Inolinaiion diminislies it moves slowly towards the
leading edge of the plate until the critical angle is reached.
As the angle is further diminished it moves rapidly towards
the leading edge for aspect ratios above 1, and below 1 it
moves venr slowly until very small angles are reached.
The following shows EiffeVa results, the distance from tho
leadinjBf edge in terms of the dimension of the plato in tho
direction of motion being given :-^
"S •
o o
11
6
8
1
1
Dimensions
of plate
in inches.
Distance of C.P. from leading edge.
6«»
10»
•34
•80
•96
•SO
•82
20«
30«»
400
60«
eo»
70<»
80»
90"
S6X6
18X6
10X10
6x18
6x86
•276
•240
•330
•266
•806
•896
•408
•836
•316
•886
•406
•420
•885
•825
•335
•42
•426
•426
•846
•886
•48
•485
•486
•86
•88
-445
•445
•446
•446
•40
•455
•46
•466
•455
46
•476
•476
•47
•47
•466
•5
•6
•6
•6
•6
M, JE^. Froude^9 remits in water-^
Aspect
ratio.
Distance of C.P. from fore edge. |
io«»
16«
20"
30»
36»
3
1
i
•246
•246
•257
•27
•81
<28
•806
•880
•295
•875
•400
•820
•40
•896
•880
lH$tribution of pressure, — ^The resultant pressure on a
.plate is due to an excess pressure on the front and a defect
of pressure on the back. For small angles the latter is bv
far the greater, but as the critioal angle is passed <&e baos
pressure becomes less important.
The following results were obtained by Eiffel, and show
the pressure on the front as percentage of the total-; —
6" 10° 20° 30® 40° 450 60* 90»
Square pbxte .
26
24
23
34
29
43
65
68
Plate 34' X 6".
2J
22
22
81
44
— .
eo
67
So far as the actual distribution on the front and back
of the square plate is concerned, measurements along the
middle line show that for angles of inclination up to 20° the
pressure on the back is all negative, and on the front all
positive, the maximum values occurring in the vicinity of
the leading edge, and diminishing very rapidly to the truiling
edge. At 30° there is a slight amount of positive pressure
on the back at the after edge, and the maximum defect
pre9STir9 19 ftt about one-thira the length from foro odofo.
416 AERODYNAMICS.
At 35° tlio region of positive pressure on the back has
increased, and also the maximum defect pressure. At 40^
the same occurs. From 45° to 90° the defect pressure on
the bacic is praotioally uniform over the whole lengtiiy bat
the maximum pressure at the leading edge on the front
remains constant, and as the angle increases becomes unifbrm.
At 90° the maximum pressure on the front is to the maximiun
defect pressure on the back as 69 is to 22.
From the 34 in. by 6 in. plate the pressure on the back is
fairly uniform in distribution for angles from 20° to 90^.
For 5** the maximum defect pressure is at leading edge, and
rapidly diminisfaes to the after edge. At 10^ the maximum is
still at leading edge, but the pressure towards the after edge
is much greater than at 6°. The positive pressure on the
front is at all angles of inclination except 90° a maximum on
the leading edge and diminishes rapidly toward Uie after
edge, becoming negative at that edge for angles less than
about 30°. At 90° the maximum pressure is at the middle,
and is about 58 as compared with 22, the maximum defect
pressure on the back.
The ratios given above for the maximum defect pressure
on the back to the maximum excess pressure on the front are
for the speed employed by Eiffel, viz. about 27 miles per
hour. For other speeds it is probable that these ratios ore
altered. , , i i i -
Curved Plates.
For curved plates the inclination i^ taken as the inclination
of the chord to the direction of motion. Unlike plane plates
the curve of resultant pressure coefficient plotted on a base
of angles of inclination of the chord does not pass tiirongn
sero inclination. There is an appreciable value of the co-
efficient when the chord Is parallel to the wind, and this doeii
not become zero until a negative inclinaiion is reached. This
negative angle varies in different shapes. The resultant force
is no longer in a direction normal to the chord. Since for
aeroplane calculations it is only necessary to know the lifi^
and drift coefficients and the position of the centre of pressure,
these are the quantities measured in experiments. The
resultant pressure can be determined in amount, direction,
and position from these three quantities. The lift and drift
coefficients having been deterxoined for various angles it is
also easy to construct a curve of ^ ^. or efficiency and thus
to determine the most efficient angle for flyiag. Sinea fo^
Sraotical purposes the upper limit of the angle of inclination is
le oritioal an^le, it is usual in experimental iiivosfcigatton?
on the aerodynamical qualities of aeroplane wings to limit
observations to angles just exceeding the oritioal angle.
ABHOPYNAMICS.
417
Compared with plane platee curved plates possess far better
aerodynamical qualities. This can be seen by comparing the
lift and drift coefficients and also the values of , .... for
dritt
a plane plate 36 in. by 6 in., and one of the same size bent
into a circular arc, the camber being ^--^ of the chord. The
results are taken from Eiffel's book (X« resistance de Voir
4t I'aviation),
Values of Coeffioieata.
8»
e**
9»
IQo
160
20»
'
^38
■00013
000176
•00029
•00034
•00031
■00074
\
Plane plate,
86 in. by 6 in. '
^9
■000^5
4*2
00103
6-2
•00166
6-7
•0017
60
0020
8-7
■0019
2*6
Critical
- angle,
16«».
Onrved plate, /
36 in. by 6 in.
•00016
03020
•00032
00036
•0007
•001
%
Cambered .q. ■
•00186
11-6
0023
11-6
•0028
8-7
0029
8-1
•0031
4-4
■0028
2-8
. Oritical
angle,
16»
J
The angle at which the uiJ^yiinnTn value of
lift
drift
■occurs
is for the plane plate 5^, and for the curved plate about 3°.
At these angles the drift coefficients for the two plates are
practically the same ; the lift coefficient of the curved plate
is 87 per cent more. Hence, if for a certain speed and drift
the plane plate will lift 100 lb. the curved plate at the same
speed and drift will lift 1871b.
^ For a plane plate at angles below the critical angle the
' centre of pressure moves rapidly towards the leading edge
as the angle of inolinatioii dimimshes. This is an advantage,
since if from any cause the angle of inclination diminishes
the movement of the centre of pressure forward tends to
increase the angle. A plane plate possesses inherent stability
for this reason. On the other hand, with a cajnbered plate
the centre of pressure at inclinations below the critical angle
moves rapidly towards the rear edge as the angle is
diminishea.
For the curved plate given above the centre of pressure
moves as shown below :—
se
418
AERODYNAMIOS.
Angles of Inclination of Chord : (f fSP IdP IS* IT 20=
CP from fore edge
Length (6")
53 -44 -37 -36 -40 -43
It will be noted that at the crltioal angle the oentre of
pressure is nearest the leading edge. For angles greater
and less it is farther aft.
If such a curved plate were used as an aeroplane wing,
and for some cause the angle of inclination diminished, the
movement of the centre of pressure is such as to still further
diminish it. The curved plate is thus unstable, and stability
must be obtained by independent elevating planes. This
is common to all unicurvml plates, and is the only dis-
advantage as compared with the plane plate.
The advantac^e in the aerodynamical qualities of the curved
as compared with the plane plate is due principally to the
greater negative pressures on the back. This is best seen
from the figures below showing the curves of pressure over
the medium section obtained by Eiffel for the two plates
mentioned above and at 6° inclination.
Fio. 234.
A
Fio. 236.
I
/I
ft
f •
/ •
WIND.
-«
I
WIND.
.< .
DISTRrBUTION OF DISTRIBUTION CF PRESSURE
PRESSURE ON PLANE PLATE. ON CURVED PLATE.
The dotted lines show the curve of pressure over the back,
the base lines being the planes themselves. The lower full
lines show the pressure on the front of plate. In the plane
plate the pressure on the front becomes negative just abaft
the middle, whilst for the curved plate it is positive except
for a small portion at the after end. In both cases the
negative pressure extends over the whole back. The resultant'
pressure at any point is the intercept between the dotted
and full curves, and the shapes of the curves in the two cases
explain the difference in the distances of the centre of
pressures from the fore edge.
ExpeirimenU on Cur ued Plates,
Since the curved plate has such marked aerodynamical
advantages over plane plates, experimental investigation lias
been devoted to obtaining that section from which the best
results are to be obtained. It is evident that for a curved
AERODYNAMICS.
419
plate very manT' variations can be obtained, i.e. the aspect
ratio, the camber, the thickness of the wing, etc., can all
be varied. The following tables give particulars of results
on variation of aspect ratio :—
Authority.
Aspect
ratio.
Angle of
maximam
lift i^
drift
degrees.
Value of
maximum
lift
dx-iit
Value of
lift coeffi-
cient at
maximam
lif-;
drift
•00071 1
•00063 }
•00068 )
•0016 {
•00106
•00046
- {
Remarks.
N.P.L. .
»• •
Gdttingren
Inst. .
Eiffel
Eoutchino
Inst. .
6
4
8
4
6
1
8
4
4
4
4
?
6
7^6
7-6
7-6
7-6
6*4
6*0
9*2
Plane plates.
Plane plate, bevelled
edges.
Plane plate, square edges
Plane plate/'beyelled
edges.
N.P.L. .
»f •
ft •
t» •
Gdttingen
M
f»
•t
.» (by
Interpo
lation)
8
4
6
6
7
1
11
4
Infinite
6|
4-7
4'7
4-6
4*7
6'0
6-0
4*76
6-0
476
4-76
10*1
11*6
12-9
14-0
161
4-8
6*0
8-0
9^8
121
19-0
f
•00165
•00169
•00160 .
•0C160
•00160
V,
•00106 f
•00122
•00171
•00228
•0024 X
•0028
Oamber of upper surface
^; of lower^; maximum
camber at \ length
from fore edge ; critical
angle about 14 in all
oases. Maximum lift
ooefficiont about '0031.
Plates bent into circular
arc camber A chord.
Oritioal angles varied
from 40® for aspect
ratio 1 to 12i<> fox
aspect ratio oa lift
ooofficient at critical
angle varied from *002£
to '0036 as aspect ratio
increased from 1 to oc.
The results show that the e£Eect of increasing the aspect
ratio of carved platea is to incf^ase the effioiencv, and also
the value of the lift coefficient at the angle of maximum
efficiency. The critical angle decreases, but the value of
the lift coefficient at this angle increases. For flat plates the
efPect of increasing the aspect ratio is Small compared with
that for cnrved platos.
In practice a very large aspect ratio cannot be employed^
since it involves more supporting wires, etc., to meet the
greater forces on ;the .wings. Tne dirift is thus increased
withont alteration of the lift, and the V-n^ u reduced below
drift
that of the experimental results. It is therefore rarely the
case that in an actual aeroplane the aspect ratio is above
6 or 7.
42Q
ABBOPTN^mCS.
Effect of Alteration of Lowbb Surface.
o
0
N.P.L.
•*3
o
o
■IH
li
6
6
6
6
H 0 0
© H
4«
4°
4"
12-8
120
12-3
12-8
© w "
> 6
•00176
•00186
•00191
•00206
8^
111°
lljo
lip
lip
The results show that there is no great alteration in the
aerodynamical qualities by alteration of the lower surface.
This is almost self-evident, since the greater part of the lift
and drift is caused by the negative pressure on the back.
The form of the leading edge modifies this to some extent^
but this form is not greatly altered by changes in th^
shape of the lower surface.
Effect of Altering Thickness. 1
Authority.
Aspect
ratio.
Dimen-
sions.
Max.
thick-
ness.
Angle
of max.
lift
drift
Value
of max.
^ft
drift
Valne of
lift at
max. lift
drift
Eiffel
ft
6
6
6
inches.
86x6
86x6
86x6
inch.
•4
•56
•72
2i
8
a
11*2
10-0
100
•0016
•0014
•001^8
Effect of Varying the Position of Maximum Ordinate
(N.P.L.).
Position of
maximum
ordinate.
fiOO
^^"»*»d?Ift
Angle of
!'FWfT«»«|i^'rTr
4
292
262
220
168
4
4
4i
Value.
Lift
ooefficie^^t.
« IWJU).1.W.< -JL '
•00170
•00155
•00153
•00148
Critical anglf .
Angle.
Lift
oo#/||oi«]|t
II" V*'"'
II
IT
8i
»m m**
OOM
00286
0027
0023
0021
ABR0DYKAMIC8.
421
Plaie employed 15 in. bj 2f in. with flafc lower stiffaoe and
an npper surface cambered *25 in. The poeition of maKlmum
camber was varied, and in column 1 is given its difltance*
from fore edge in terms of' the chord.
Besults show that as the maximum ordinate is moved
lift - .
forward from the middle the maximum value of ~^Q| ^^^
lift
increases and then dimiiuahes, the angle of maximum x^ni
first decreases and then remains constant ; the critical anglo
diminishes, and with it the corresponding value of the lift
coefficient. In the case of the first and second the critical
angle was not so marked as for the others, the cnrvo of lift
coefficients to base of angles of inclination being a fair curve
up to 24**. In the others there is a marked fall at the critical
angle, especially in the case of the third and fourth, for which
the coefficients decrease 22 and 80 per cent for 1" or 2^ beyond
the critical angle, and in the next few degrees attain nearly the
same value as at the critical angle.
For the particular type of section tried the best aero-
dynamical qualities are obtained with the maximum ordinate
at about one-third the chord from the leading edge.
Sftbov of VABYiKa Shape of Leadiko and Tbaiuitq Edobb
(N.P.L.).
For effect of variation of leading edge four wings were
tried each 15 in. by 2^ in. in plan. The Upp^r and loiwer
surfaces were cambered '25 in. and •IS in. respectively at
one-quarter the chord fom leading e^^. The shape abaft this
was the same in all, the forward j^ortion alone being altered.
The fore ec^ of the first was poiated, the three others were
rounded, the lower front surface of the wings being brought
In fair with the nose. The diameters of the three rounded
lore edges wore |, 1, and |, the maximiun thickness of the
wing (*10in.).
fiedlioa »i n«8e.
"^-^
Oritical angle.
Angle.
Value.
Lift
coefficieDt.
Value.
Lift
coefficient.
Sharp . . . .
Diamet«r f thickn^aa
r
ti S 1*
4
4
4
2
128
115
HI
10-4
'00204
'00214
•00204
"0016
12
u
9
0094
•0037
•0032
•0027
422 AlEBODYNAMtCS.
The sharp-edged Eection thus gave the best remlif .
Experiments on a wing in which the section at trailing
edge was gradually fined away on the upper surface showed
that as the thickness was reduced the aerodynamical qualities
improved.
Distribution of Pressure on an Aeroplane Wing, '
Experiments at N.P.L. made on a wing 18 in. by Sin.
whose lower surface was cambered y^ in., and upper surface
*26 in., maximum camber at <^ chord from front edge.
Sections taken at (a) midd'e, (6) J, (c) t^, (d)^, and W tIt.
the span from the tip of the wing. Pressure measured at
seven positions both on upper and lower surfaces at each
section.
The results show that the maximum pressures both on
front and back occur at the leading edge at all sections
except {e) and at all inclinations, and diminish to zero at
the trailing edge. The maximum defect pressure on the
upper and excess pressure on the lower surface are on section
(a) and decrease as the tip of the wing is approached. They
increase in amount as the ang^le of inclination is increased.
At («) the section near the tip, the pressure distribution at
angles — 2**' to 2^ is nearly uniform, but for angles above 2? the
maximum pressure is near the after edge. Generally, for
all positive angles and for all sections except (tf) the pressure
on the back is negative and on the front positive. For
section («) the pressure is negative both- on front and back
for positive inclinations except for a small distance near the
leading edge.
By resolution and integration of the pressures the lift
and drift at each section were obtained. The lift at a is
} greater than at B, \ greater than at C, ^ greater than at e,
and i greater than at d. The maximum value of -j-rr. is
24 at A, 18 at B, 13 at c, 8| at d, and 5 at e, and for the
wing considered as a whole 17*8. These figures neglect skin
friction, which, when allowed for, gives a maximum value of
T-Tfi o^ ^^ A^ ^ ^^^ ^^ ^or the whole wing.
The position of the centre of pressure is different at each
section. For a it is '27 of the chord from fore edge at 12^
and ^49 at 0". For B and 0 sections it is slightly further
from leading edge at all angles, and for D still further except
at 2° and below when it is nearer than for A. For section B
it is -52 the chord from leading edge at 12"" : "66 at 0".
Eiffel's experiments were made on the model of a Nieuport
monoplane wing cambersd both front and back and whose
ABEODYKAMIGS. 428
thiokneofl diminbhed at the tip. Observations made at 2^
and 6°. Sections taken —
(a) n«ar connexion of wing to fuselage.
(6) about half-way between fuselage and tip of wing.
\^/ »> F « » » »
C^/ w 5^ » >» w »
Besults for 6^ show that pressure on the back was a
maximum for section (a) and gradually diminished to the
tip, bein^y however, a defect pressure at all points on the
back in all sections. The position of maximum defeot pressure
was at '28 chord from leading edge on section (a), the pressure
diminishing towards front and rear. For other sections the
position of maximum defect pressure was slightly further
aft. The pressure on the front or lower surface was an
excess pressure on all sections except just near the edges.
It was a maximum on section B and diminished g^dually
to the tip. The position of maximum pressure was about *3
of the chord from fore edge. At 2** the distribution was very
similar to that at 6% except that near the leading edge the
? pressure on the front or lower surface was a defect pressure
or a greater distance than at 6^. The maximum defeot
and excess pressures were less at all sections than at 6°.
Sections with Reverse Curvature at the Trailing Edge*
The foregoing experimental results deal with plane plates
or unicurved sections, i.e. the section wholly concave to the*
chord. Eiffel and the N.P.L. have carried out experiments
on sections, in which the after portion of the section was
curved in the opposite direction, i.e. convex on the lower side.
Eiffel first used a section of uniform thickness, the after
half being curved equally, but in the opposite direction to
the front half, which was concave to the chord. His results
showed that (1) the lift and drift coefficients at any particular
angle decreased as the speed increased, but the drift decreased
more quickly than the lift, and hence the value increased
with the speed, (2) the aerodynamical qualities were less
favourable than in unicurved sections, and (3) the centre -
of pressure moved towards the leading edge as the inclination
decreased. The latter effect is simiLBir to that for a plane
plate, but the reverse of that in a unicurved wing.
Further experiments by M. Eiffel on a section similar to
an ordinary aeroplane section except that it had an upturned
trailing edge gave similar results. The speed of experiment
was varied from 6 to 14 miles per second, the drift and lift
coefficients at %"* being '00017 and '00136 at the lower and
'00014 and '00122 at the higher speed. The centre of pressure
waa '34 of the chord from front edge at 20" and 07 at 0"".
424
ASHODTNAMICS.
The N.P.L. experiments were carried oat using as basis
a wing whose section was slightly cambered on the lower
sarface, and well cambeted on the upper surface. The other
seotioni were obtained by raising the trailing edge starting
from a distance '4 of the chord from the latter. In the table
the rise of tall is given in terms of the chord.
Section.
Kise of
tail.
Maximum ^^-^^
Critical angle.
Angle.
Value.
Lift
coefficient.
Angle.
Lift
coefficient.
1
2
3
4
0
Oil
•027
•057
3
3f
H
10
15*8
160
143
12-8
'00138
•00107
•00122
ooua
16
18
16
16
•0032
•002»
•0027
'0024
In 1, 2f and 3 the centre of pressure moved tdt as the
angle of Inclination decreased, bnt this movement was slower
as the tail was raised higher, and in 4 the centre of pressure
moved forward as the angle decreased. The great advantage
of reverse curvature of the tail is that it gives etability, but
Hiis is obtained by the sacrifice of other qualities.
BiPLAinB Effects.
The foregoing results have been obtained with single plates.
In the case of two similar parallel plates close to one another
as in a biplane, each has an effect on the other and the aero-
dynamical qualities are altered.
Langley was the first to experiment with pairs of similar
¥lates, 15 in. by 4 in. spaced 2 in., 4 in., and Gin. apart,
he weights for the three spacings were the same, and the
speeds at which they were self -supported were compared
with that of a single plate of half the weight at the same
angle of inclination. The results showed that for a 2 in.
spacing an appreciably greater speed was necessary for self-
support than for a single plate, and for 4 in. and Bin.
spacing the speed was the same. The apparatus employed wan
not, however, capable of very accurate measurements.
Eiffel's experiments were made on biplanes formed of
(1) two similar parallel plane plates each 36 in. by 6 ia.*,
(2) two curved plates each 36 in. by 6 in. cambered f in.
The spacings tried were 4 in., 6 in., and Sin. in both sets.
Presmres taken at various points at the mediuiQ
longitudinal sections of both plates when inclined 6° show that
for the plane plates the pressures on the front and back of
AERODYNAMICS.
iU
both pifttes ai*e for all spacings less than on a Bing^le plate
except in the ease of the lower surface of the bottom plate
for 4 in. and 6 in. spacing where in the first ease it is g^reater
and in the second the same as for a single plate. In the case
of the curved plates the defect pressure on the back surface
of the upper plate is increased for the 6 in. and 8 in. spacing.
The pressure on the front of the upper plate is also increased
for 8 in. spacing, and is the same as for a single plate for 6 in.
spacing. For the lower plate at the three spaeings the
pressures on back and front are less than for a single plate.
BB8T7LT8.
Lift ooeflScients.
Plane plates.
Lift coefficients.
Carved plates.
B'
6»
go
1<^
3°
6*'
Singrle plate
Spaced 4 in.
t» 8 »»
•00066
•00028
•0003
•00037
•00109
•0007
•00077
•00082
•00166
•00106
•001X7
•00124
•0012
•0011
•0012
•0012
•0017
•00134
•00143
•00163
•00237
•00176
•00184
•00196
The N.P.L. experiments on the same point were made on
a Bleriot section of wing 20 in. by 5 in. The table gives the
ratio of the lift coefficient to that for a single similar wing.
Spacing.
Ratio of lift coefficient to that of a single wing.
6°
go
lOo
2 in.
4 in.
Sin.
6 in.
Sin.
•61
•76
•81
'86
•89
•62
•77
•82
•86
•89
•68
•78
•82
•87
•90
The lift coefficients are thus smaller than for a single
similar wing even at the largest spacing. Since, however,
increase of spacing between the wings involves increase in
the lengths of struts and bracing wires thus increasing the
drift) when this is taken into account, the best spacing appears
to be equal to the chord of the wings.
Hi
AERODYNAMICS.
Method of representing Experimental results.
The method employed b^ Eiffel offers many advantages.
It was first employed by Lilienthal, and consists in taking
two rectangular axes ox and OT, and setting off a vector from
the origin o in magnitude and direction equal to the coefficient
and the resultant pressure on the wing. The lift coefficient
Ey and the drift coefficient k^^ are then the projections of
the coefficient o¥ the resultant on the axes of 07 and ox
respectively. If this is done for various angles of incUnation
a carve as shown is obtained.
Fio. 386
Polar diagram of resultant
pressure.
The angles marked on the curve correspond to the angles
of inclination of the plate, and are generally not on the
lines drawn throug^h the origin at an angle from oy equal to
the inclination, since the resultant pressure does not generally
act in a direction normal to the chord.
In addition to the above curve a second set of curves
showing the resultant pressure coefficient (k), the lift co^
efficient (k^) and the drift coefficient (Kj.) i^ drawn to a base
of angles of inclination. A third set of curves shows the
position of the centre of pressure at various angles of
inclination.
If in the curve shown above a tangent is drawn from the
origin touching the curve this line gives the direction of
the resultant force for -t-ttt to be a maximum, and the
drift
tangent of the angle this line makes with ox gives the
maximum value of - .^ '. The actual angle of inclination
drift ^
Vi found by measuring the corresponding value of the resultant
AERODYNAMICS. ii^
coefficient setting this up on the curve of K to a base of
inclinations and measuring the corresponding angle.
Method of Caloulation,
In an actual aeroplane there are, besides the wings, other
parts which offer resistance to motion without contributing
any support. The result is that the drift coefficient for the
whole machine is much greater than that of the wings, whilst
the lift coefficient is that of the wings alone. The resistance
of the wings alone is termed the active resistance, since it is
a necessary complement of the lift. The resistance of - the
remainder of the machine and from which no lift is obtained is
termed tho passive resistance. It is generally expressed in
terms of the area of a square plate which^ moving normally
at the sam-e speed, will give an equal resistance. This
equivalent plate is termed the detrimental surface. If, there-
fore, the wing surface is s and the detrimental surface s^,
Ky the lift coefficient, K^ the drift coefficient of the wings
alone, the total resistance is given by
B = Ka.SV2 4- -0032 S^V^,
where v is the speed in miles per hour, s and s^ are in square
feet, and '0032 is the resistance coefficient for a square plate.
The lift is given by l==k^sv2.
The resistance can be expressed in the form
o arr2/.r I '003281 \
and the lift by LsSV^Ey.
-. is a maximum when 2L- — . is a maximum. The
angle at which this expression is a maximum can be deter-
mined from the resultant pressure vectorial curve of the
wing by setting off a distance 00^ ==-— - — from the origin
s
to the right along ox. By drawing a tangent to the curve
the angle at which — *' i:i a maximum can be
* s
obtained.
Mathematically it can be proved that the expression Is
a maximum when the active resistance is equal to the passive,
. ^ ^ -003281
I.e. Kj-e*
s
If the weight (w) to be lifted is known the speed for
maximum value of — can be found aince w ■« E«sv2, and thus
4dd AlOtODTKAMtCS.
vea A/ of which w, B, and K,, are known. The speed
^ KyS
being known the resistance and effective horse-power required
can he calculated.
The minimmn speed at which the aeroplane can be flown
is determined by the value of the lift coefficient corresponding
to the critical angle. If this be denoted by E^y tl|p minimum
speed is given by —
^-V^u-^rr"^"^
l^m if ^e lift ooefllci«nt at the critical angle is twioo that
lift
at the angle of maximum , , the minimum speed will be
drift
1 Mff:
-^ or 71 per cent of that at speed for maximwn -tztml
The above are simple calculations in connexion with aero*
plane work. There are others which are far more complicated
and for which reference must be made to works on the subject.
li may be mentioned that the detrimgnttU turfttee as deter-
mined by Eiffel for a fnll-^sized R.E.P. monoplane is one
square metre, and for a Nieuport monoplane two-thirds square
metre. For a BJB. 2 aeroplane it is 7*6 square feet.
Comparison between Besults of Model and Full-sized
Machines.
Hathomatioians have ^own that the two resistances
operating in airships and aeroplanes— skin friction and eddy
resistance — are in similar bodies moving in air strictly com-
parable when the product L7 is constant. L is a dimlBasion and
V the speedr If the dimensions of a model are on^^twentieth
those of the airship or aeroplane and the speed of the lattor
is 50 miles per hour^ the model results are only strictly
applicable to the fall-sLzed machiae when the model velocity
is 1,000 miles per hour. This speed is impracticable. If,
however^ another medium is employed similar motions obtain
if — Is constant where v is the coefficient of kinematic
V
viscosity. Since the value of this for water is thirteen times
the value for air, it is possible to compare results obtained
from similar models in air and water by re^pulating the speeds
in accordance with the law. So far the law has been experi-
mentally verified for skin friction, but not iot eddy
resistanee.
^ The variation of the various coefficients (lift, drift, eto.)
with the speed has not received great attention, expttrimenta
AERODYNAMICS. ^29
in each laboratory having been carried out generally at one
speed, and although the speeds may be diSereat ia differeat
laboratories other features have prevented a strict comparison
between the results. Generally speaking, the speed of experi-
ment is limited by the size of the wind tunnel. Larger wind
tunnels have, however^ been recently constrncted by Eiff^
and at the N.P.L., thus permitting of higher speeds and
possibly in* the near future much more information will be
available on this important point.
Eiffel's early experiments with plates falling under the
action of gravity showed that the coefficient of resistance for
a plate of one square metre area was 10 per cent greater
than for a plate of ^^^ square metre area. In the first edition
of his book La resistance de Voir et Vaviation, he recommended
that results obtained from model experiments should be
increased 10 per cent for application to full-sized aeroplanes.
In a later edition he recommends that the model results should
be applied without correction. This suggestion is based on
a comparison of the results obtained from an aeroplane and
those obtained on a complete model in his laboratory. The
published comparison of these results show that the maximum
difference is only 5 per cent and the mean difference about
2 per cent. The comparison is not, however, conelusive,
since it might be expected that with the propeller working
in the aeroplane and not in the model much greater differencefi
than ^ per cent would result.
Experiments have recently been carried out on a model
of an aeroplane wing at the N.P.L., the speed being varied
over a fairly large range. The wing employed was 15 in. by
2\ in. ^d the speeds 7, 10}, 14, 21, 28^/ and 85 miles par
houir. The lift and drift coefficients were measured at every
2** up to 20** inclinati(m at each speed.
The results show that as the speed increased the lift oo^
efficient increased for angles up to 6^ inclination, and that
for speeds between 14 and 35 miles per hour it remained
constant for angles of inclination from 6° to 14^. FrOm
14° to 20^ the lift coeffieient increased as the speed inoreased*
but this was apparently due to the fact that the oritioal angh»
also increased with the spaed and at 14** the oritioal angle
for the lower speeds was passed ; the lift ooeffieienta decreased
rapidly in v^lue for these lower speecbl, but uob so quioUy
for the higher speeds. The run of the curves would appear
to indieaiie that at angles above '20^ the Uft eoeffieieiit deareaiM
ap the speed iaereases, a point W]iieh will be tf lerjred to later.
Th^ drift coeiipieot decreased as the speed increased for
all angles up to about 20*. At this angle the values for 14,
21, 28, and 35 miles per hour are the same, but for 7 and 10}
m.p.h. they are less than for the others. The net effect of
the alteration in the Uft and drift coefficients is that as the
480 ABRODTNABnOS.
lift
speed increases the maximum value of ^-ttt increases fros
10*5 at 7 m.p.Ii. to 17*6 at 35 m.p.h. The angle at which
this occurs is 7"* at 7 m.p.h. and 3 at 35 m.p.h. From the
cnrves it looks quite probable that at higher speeds than
lift
35 m.p.h. the an^le of maximum , .,, will still farther
■^ ^' drift
diminish and the value of the latter increase. The critical
angle increases from 11^ at 7 m.p.h. to 15° at 35 m.p.h.
These results are similar to those obtained by Eiffel on
a model wing of a Nieuport monoplane inclined at 3% ihe
speed being varied from about 12 to 36 miles per hour. The
lift coefficient increased and the drift coefficient decreased
about 6 per cent for this range of speed. This compares
with differences between results at 7 and 35 m.p.h. of about
40 per cent in the case of the N.PJL. experiment at the same
angle.
In the case of a win? of reverse curvature Eiffel found that
both the lift and drift coefficients decreased as the speed
increased, but since the drift coefficient decreased at a greater
rate than the lift the efficiency increased with the speed.
It seems evident, therefore, that for different shapes of
sections the effect of increase of speed on the aerodynamical
Qualities may be greatly different, and no general law can be
aeduced from the results of a particular wing. A greai
amount of experimental work is therefore necessarr before
a general law can be enunciated. It seems clear, however,
that generally if model results are used for design work the
resulting aeroplane will possess better aerodynamical qualities
than the model. It would also appear that the greater the
model speed the more closely will the results agree with those
of the aeroplane.
A second series of experiments was carried out at the
N.P.L. to compare model results with those obtained on
full-sized wings in the Laboratory at St. C^ in France..
The models were placed in a current of air with a speed of
about 34 miles per hour, and the lift coefficients so found at
2*^ to 6° inclination compared very favourably with those on
the full-sised wings. The drilt coefficients were, however,
much smaller.
The N.P.L. therefore concludes from this and other
exiperimeati that the lift coefficients from model experiments
at fairly high spMds will ajpiply to the full-si^ed wings, btit
that 15 to 20 per cent must be added to the ma^dmtua j^jj-
dnft
of the models to obtain the value in the full^sized wing.
As before mentioned, the N.P.L. experiments on varia-
tion of coefficients, etc., with speed were only carried to 20°
AER0NAUTIC8. 4S1
inclination. It would appear from the run of the oarves
that for angles of inclination greater than the critical augle^
at which inclination the type of flow changes,^ the law^ of
variation with speed is different, and that the lift coefficient
decreases with increase in speed. On this point the results
of experiments made at the Institute at Koutchino on a square
plate at different speeds are of interest. At angles less than
the critical the normal pressure coefficient increa<;ed with the
speed, but for inclinations greater than the critical angle it
decreased as the speed increased.
Final Note, — The experimental results given in the fore-
going form the basis of aeroplane design. For details of
actual aeroplanes and for detailed calculations as to the
strength and other qualities, reference must be made to books
and periodicals dealing with such matters. Much fuller
experimental information ip also given in Eiffel's classical
work, which has heen quoted, and in the reports of the
Advisory Committee on Aeronautics.
The results of experiments in air are of some practical
importance to Naval architects, since the problems connected
with submarines, propellers, rudders, etc., are similar to those
in connexion witn airships and aeroplanes. The results, so
far as coefficients are concerned, are applicable to water by
multiplying their values by 832 for sea water or 810 for
fresh, these numbers representing the ratios of the densities of
water to air.
AEROKAirTIGS.
A. W. Johns, Esq., M.I.N.A., R.C.N.O.
AlBSHIPS.
Airships are divided into three classes : —
1. Non-rigid, in which the oar is hung directly by ropeai
or wires from the envelope, the ropes being so arranged
that the load is spread as evenly as possible over the greater
portion of the length of the envelope. The upper ends of
these ropes are secured to bands of fabric encircling the
envelope, or to strips of fabric fastened to the envelope
along the intersection of a horizontal diametral plane. The
strength of such an airship depends entirely upon the strengtii
of the fabric of the envelope. The spherical balloon is
the most elementary form of this type, the - car being
suspended from a network of rope endircling the upper portion
of balloon.
2. Semi-rigid, in which a framework of steel or aluminium-
alloy rods is attached to the lower portion of the envelope.
To this framework is attached the cars carrying the engines,
fuel, crew, i^nd other loads. It extends in some oases the
182 AERONAUTICS.
whole length of the envelope, and the straining actions on
the airship are to a great extent taken by it, Instead of on
the fabric of the envelope as in the non-rigid type.
In both types, i.e. non-rigid and semi-rigid, the shape of
the envelope is maintained by an internal excess pressure.
This excess pressure causes all parts of the envelope to be in
tension, and so long as the straining forces on the airship
produce compressive forces less than the tensile, the envelope
retains its shape. If the compressive stresses are greater
than the tensile stresses caused by the excess pressure, the
portion of the envelope concerned will collapse. Since, there-
lore, the whole strength of the non-rigid and the shape of the
aerostat of the semi-rigid depends on this excess pressure,
And the hydrogen with which the envelope is inflated is
always permeating through the fabric it is necessary to
provide means for maintaining the excess pressure. This
IS done by fitting expansible internal compartments termed
' ballonets ', which are connected by trunks to a fan in the
car. At the start o(f a voyage the biiUouets hstve little air
in them, but as the hydrogen is lost the volume of the
ballonets increase so that the envelope remains distended,
and the same excess pressure is maintained inside. This
excess pressure is generally from 10 to 25 millimetres of
water (i.e. 2'1 to 5*2 lb. per square foot). In some designs
there is a ballonet at each end of the airship, the totaj
volume being from one-fourth to one-seventh the volume
of the envelope. In the Siemens-Schukert design (non-rigid),
in which the envelope is divided by two transverse bulkheads
into three parts, therei is a ballonet in eioh part. In
the Forlanini (semi-rigid), two concentric envelopes are fitted,
and the annular space between the two forms the ballonet.
Safety-valves, which blow off automatically when the internal
pre.ssure exceeds the designed pressure, are fitted to the
ballonets and to the envelopes, and these valves are also
arranged to be worked from the car by ropes or wires.
They are also used when the airship rises to higher altitudes,
and the gas expands due to the diminution of the outside
pressure.
The ballonets when fitted one at each end can also be
used for steering in a vertical plaue by pumpiiig air from one
into the other. In other designs, movable weights are fitted
for the same purpose, and i^n anothe? tl^ ca? itself cap
be shifted along loqgitudiuaUy by suitable arrAngemer^t^ {n
the ff^spennion wires. A shift of the car is also ne(3ei994ry
to counterbalance the tilting moment on the airship caused'
by the line of thrust of the propeller being below the line
of action of the resistance.
3. Rigid J in which the form of the airship is maintained
by a rigid framework of transverse and longitudinal members
AERONAUTICS. 488
well connected together at the joints. The Zeppelin design
is the most suceessfal airship of this type, and the following
remark's apply more particularly to that design. A keel of
triangular section is fitted at tiie bottom^ and to this are
suspended the cars with the engines and propellers, fuel,
ballast; etc. The transverse section is not circalar aa in non-
rigids and semi-iigids, but a regular polygon of fifteen,
sixteen, or seyeiiteen sides, the lowest side being horizontajl
and forming the upper side or base of the triangular keel.
The transverse frames are from 25 to 30 feet apart, and are
built in short lengths corresponding to tiie sides oi the
polygon.
Wiire bracing is fitted between each joint of a transverse
fraine and the two lowest joints, and serves to transmit the
lift of the gas-bags. Badial bracing is also sometimes fitted.
The longitudinal frames extend the whole length of the ship,
their positions corresponding to the comers of the polygon,
at which they are strongly connected to the transverse frames.
At the bow and stern they are connected to conical platen
or castings. Wire bracing is fitted in the rectangular spaoeb
formed by adjacent longitudinal and transverse frames.
Additional partial transverse frames and wire bracing are
fitted in wake of the propeller brackets.
A light fabric cover is fitted on the outside of the framing
to form a fair surface. Cylindrical gas-bags are fitted
between the bracing of the transverse frames. The lift of
these gas-bags is transferred to the framing by a rope neft-
work fitted on the inside of the framing. Since adjacent
fas-bags are separated by the wire bracing of the tratisverse
raiftes this has to take the strain when one gas-bag becomes
deflated and the adjacent one is inflated. Valves are fitted
in each of the gas-bags, and these work automatically and
by wires from the cars. Pressure gauges showing the pressure
of gad in each bag are fitted in the oars.
In the rigid type of airship the strains coming on the
structure due to differences in distribution of the buoyancy
and weight are taken by the keel by transmission through
the frames. The transverse forces due to the wind or othet
causes, are taken on the framework or its wire bracingSr
No stresses other than those due to the excess pressure of
fas in the bas-bags comes on the material of the latter^
0 far as the shape of the airship is concerned the gas-bags
may be fully or partially inflated. In the latter case the
distribution of buoyancy is altered.
Sitice the total buoyancy and weight of the airship are small
it necessarily follows that the framing must be light and)
efficient. The longitudinal frames, which are only supported by
the transverse at every 25 to 30 feet, tend to deflect outwardi*
by the excess pressure of the gas-bags. Tensile forces on
Wt
484 AERONAUTICS.
these lonj^itadinals are not difficult to meet since they tend
to diminish the deflection of the frames due to the gas
pressure.
Compressiye forees are those which are most difficult to
provide against, since the thin material of the girders tends
to buckle under eompression, and the tendency is inoreasei]
by the deflection due to the pressure of the gas. To with-
stiand these compressive stresses it is necessary that the inertia
of the girder section and also the modulus of elasticity of the
material should be as large as possible for the weight.
In the Zeppelin airship a light aluminixim alloy, said
to be wolframium, is used for the framing. The girders, both
longitudinal and transverse, are either of triangular or
trapezoidal section about 7 to 8 inches deep. In the triangular
section, an angle section is fitted at each corner, and these are
connected together by bracings of stamped section inclined
at 450, and connected by two rivets at each end. In the
trapezoidal section the shorter of the parallel sides is a channel
section, angle sections being fitted at the extremities of the
longer parallel side. The channel and the angles are con-
nected together by stamped braces as already described. The
apex member of the keel \a either a tube or a rectangular
f-irder, which is connected to the two lower main longitudinal
rames of the airship by circular struts about 8 or 10 feet
apart. Wire bracing is fitted in the panels formed by the
apex girder, longitudinal frames, and struts.
In the Sohutte-Lanz airship the material of the framing
is said to be white Russian fir, which is moulded and pressed
into channel and angle sections. The frames in the first air-
ship of this type were worked diagonally instead of transversely
and fore and aft as in the Zeppelin. In a later design just
completed it is understood they are worked transversely and
fore and aft.
In the Speiss (French) airship the main framing is of
wood and the transverse and longifcudinal frames are worked
very similar to the Zeppelin design, i.e. the transverse frames
in short lengths forming the sides of the polygon, and the
longitudinals running fore and aft between transverse frames,
and connected to them at the corners of the polygon. The
portions of the framing are tapered, the sectional area being
largest in the middle and smallest at the ends or joints.
The wire bracing in the Zeppelin is of high strength
steel, and it is understood this material is also used in the
Schutte-Lanz and Speiss rigid airships.
In the British Naval airship, built by Messrs. Vickers some
years ago, duralumin, a light aluminium alloy, was employed.
This has a tensile strength of about 23-30 tons, yield-point i
22 tons, extension 10-18 per cent, a modulus of elasticity of |
about 10*5 X 10^ lb. units, and a specific gravity of about 2*8.
AERONAUTICS. 485
Steel has been proposed as the material for the framing
of rigid airships. The modnlus is 30 X 10^ and, although
for equal weight to aluminium alloys, the products of
modulus and inertia for the same shape of girder are the
same, thus ensuring equal resistance against buckling, yet
the extreme thinness required for the steel framing must
result in great difficulties in construction. An aluminium alloy
of moderate tensile strength, easily worked into the necessary
construotion sectix>ns, capable of withstanding atmospheric
conditions without deterioration, appears to be the most
suitable material, since the modulus of most of these alloys
is in the neighbourhood of 10*5 x 10^ lb. inch units, and their
specific gravity between 2*7 and 2*8.
Stabilizing Planes, Rudders, etc.
Experiments carried out at the National Physical and also
at the Gottlngen Laboratories show that for airship forms the
resultant force when the airship is. inclined slightly to thei
direction of the resultant wind will act at some considerable
distance forward of the centre of gravity, and thus tend to
further increase the inclination, i.e. the model is unstable.
In the cases tried the line of action of the resultant force at
small angles was far forward of the nose of the ship. The
addition of plane fins placed at the tail of the model had
a very marked effect on the line of action of the resultant
force, the effect being greater as the area of the fins was
increased, and after a certain area was exceeded the line
of action of the resultant force passed aft of the centre of
gravity, and thus tended to diminish the inclination, i.e.
it rendered the ship stable. Stabilizing planes are therefore
fitted in nearly all airships, and since motion takes place in
any direction, both vertical and horizontal fins are necessary.
In the latest Zeppelin airships the shape of the fins in profile
or plan is practically that of a rectangle of the same
height or breadth as the maximum diameter of the ship,
ana finishing at the after extremity. In some of the French
and British airships four or more external ballonets of
spindle shape have been fitted at the after end to act as
stabilizers. In the Paraeval airships vertical and horizontal
fins are fitted at about a quarter the length of the ship from
the after end.
The steering and elevating rudders in the latest airships
are placed at the stern on each side, and are balanced and of
the box type actuated by a curved sector and wires from
the car or oars. In some cases elevating planes are fitted
forward as well as aft, and this was the case in the e(]irlieQr
Zeppelins. In the Forlanini elevating rudders are fitted
just forward of amidslups and also aft at the tail. In
4B6
AEROKAV'TtCS.
some cases also the radders and elevating |>laA)fts are of
tlie flexing type, i.e. the fore edres are fixed and the trailing
edges are pulled one way or other
Fig. 297.
In the Forlanlni air^ip (fig. 237), 236 feet long and 69 feet
diameter, the area of the raddier id 409 square feet, the
stern elevating planes are 323 square feet, and the forward
ones 215 square feet. In a Zeppelin airship, 490 feet long
and 40 feet diameter, the rudders have an area of 330 square
feet and the elevating planes 400 square feet. In a Farseval,
190 feet by 30 feet, the rudder is 80 square feet, earried at
the after end of a stabilizing fin of 200 squ|are feet. The
horizontal stabilizing planes have aii area of 340 square feet.
No elevating planes are fitted', the vex'tical steeriilg being
obtained by pumping aur f^om oiie ballonet to the othei'
or by longitudinal shift of weight. In a later Farseval
a central oallonet only is fitted and elevating planes ate
fitted towards the f()re end.
There is very little information as to the steering oapa-
bilities of airships. That published for the Zeppelin SafMdben
is as follows :—
Condition.
Time of*
turning.
Speed of
ship.
Diameter of circle in
lengrihs of ship.
3 motors, rudder hard-a-port
2
Starboarid motor, staiisoard
turn
0ec8.
133
134
265
ra.p.s.
12-4
9'6
4|
3*8
5-8
Lifting Power.
At O^G. and 760 mm. pressure, 1,000 cubic feet of air
weigh 80*7 lb. At the same temperature and pressure 1,000
cubic feet of chemically pure hydrogen weigh 6' 7 lb., giving
a lifting power of 75 lb. per 1,000 cubic feet. Oomilkeroial
hydrogen is not pure, ana under the same conditions 1,000
eubie feet will lift about 741b. According td Pie1»ker
(German I.N.A.), in practice 1,000 oubio feet of hydrogen
Will lift about 72*7 lb., whilst if the fabric is not of the best
material this is reduced to 69 lb. For continuous running the
same authority gives 70'6 lb. per 1,000 cubic feet, whidi at
15** C. and 760 mm. reduces to 69 lb. The usual figure taken
is 1,000 cubic feet of gas-bag capacity liffcs 68 to 70 lb. A
pise of t«niper|ktuB.e pf l^'C. reduces the lifting power by
37 per centr Increase in altitude diminishes the density of
the air and the theoretical lifting power diminishes as shown
below : —
Lifting: power per
1,000 onbio -feefe.
lb.
At sea-Ievel at 0* and 760 mm. . , 70
671
1,000 feet high
2j000
9f l»
ff ft
ft ft
65-^
62*8
60-6
58-5
56-3
48-7
3,000
4,000
5,000
6,000
10,000
Actually the lifting power is also affected by .pthor circum-
stanoas, sijace it genexAlly happens that due to radiation the
hydrogen is at a mgher temperature than the air. In Zejppelin
airships this difference has been measured and sometimes
amounts to 11^ C. Taking the above figures it will be seen
that the lifting ppwer of an airship of capacity 640,000 cubic
feet which is 20 tons at ;iea-lev)el is at 5,000 feet altitude
only 16*7 tons, ai;id thus $*3 tons of disposable weiglit luiust
be got rid of to allow the airship to attain and remain at that
height. Since at this height the volume of the hydrogen is
increased by about 16 per cent the valves must be of such a size
that the gas can escape whilst the ship is attaining the height.
According to Pietsker, rain or snow will increase the
weight of a Zeppelin by 1 to 1} tons, whilst a very da^ip
atmosphere may increase it by about 1,S00 lb.
The net lifting power, i.e. the lift exclusive of the fixed
weight of ship (hull struoture, envelopes, engines, rudders,
fittings, etc.) which can be utilized for the carriage of fuel,
ballast, crew, stores, armament, wireless gear, etc., varies
in different designs and must necessarily depend on the speedy
since this determines the weight of the engines. In thel
Forlanini semi-rigid Citff of Milan it is 32 per cent of the
gross lift. In a passenger-carrying Zeppelin it is 25 per cent.
In a Zeppelin for war service it is stated to be 28 per cent,
whilst in the Parseval design it is from 29 to 88 per cent.
The loss of hydrogen by permeation through the fabric of
the envelope will be always diminishing the total lifting
power, and therefore also the net lift. Thus, taking an airship
with a surface of 24,000 square feet and with a fabric whose
permeability is 10 litres per square metre in twenty-four
hours, the loss of hydrogen by permeation is 800 cubic feet in
twenty-four hoars. This represents about *6 per cent of the
total gas capacity of the envelope whose surface has been taken.
488
ASRONAVnCS.
BESTSTArrcc op Aibships.
There is little information available as to the reslstaooe of
airships. The results of trials on two—the La Francs of
Renard and the Zeppelin Schwahen — have been published.
The former, one of the earliest dirigibles, was 50*4 m. long,
8*4 m. diameter, and 1,864 cable metres capacity. A Gramme
motor of 9 nominal H.P. gave a speed of 6*5 m.p.s. A two*
bladed tractor screw was used.
If B is the resistance in lb., A the area of the largest cross-
section, v the velocity in miles per hoar, and K the coefficient
of resistance, and BssKAV^, in the case of La France,
K = propulsive coefficient X '0014.
Taking the propulsive coefficient as 40 por cent«
K= 00056 or r=00056av2.
Schwaben. — ^In the case of the Sohwaben the trials were
more extensive, and the following results were obtained : —
Date.
Total Power
Speed.
Propnlsive
Cosfficionfc.
xn.p.s.
29-6-11
454
19-6
•65
n
290
16-8
•65
8-7-11
288
16-4
•55
9»
136
11-4
•38
15-7-11
454
19-34
•55
n
290
16-63
•65
ft
136
11-30
•38
The propulsive coefficients were deduced from trials on
the motors and propellers. The published dimensions of the
Schwaben are 459 feet long, 46 feet diameter, and 629,000
cubic feet capacity. From the above results the resistance
(b) in the same units as before is given by B =» '00070 av^.
This result was confirmed by stoppinff tests, in which the
motors were stopped and the decrease of speed over measured
times observed.
Model Expebimekts.
Experiments on models have been carried out by O'Gormao
at the Royal Aircraft Faetory and by Prandtl at Gottingen.
The former used model balloons of goldbeater's skin, the
models being towed at speeds of from 10 to 25 f.p.s. The
table on p. 439 shows the results.
Little information is given as to the method of the ex-
periments, but the value of the coefficient for a circular
plane 3 feet diameter, obtained by the same method, is 14
per cent greater than that given by Eiffel, and 16 per oent
greater than that given by Stanton.
AESONAOTTCt.
439
e
u
1 G^
v«>'
1 ctS
Model.
Length i
feeti.
Diamete
in foet.
Volume i
cubic fee
'43 S
'Surface i
square fef
N.P.L. (12) ..
19i
3A
90-5
•66
141^4
•380
Olement-Bayard
18
3A
88-5
•69
1874
•388
Beta
13J
H
62-3
•60
102-6
•340
Gamma .
16A
H
78-4
•64
i22^9
•380
B.P. (36) .
17i
m
75-4
•62
121-8
•342
Lebaudy .
17A
^t^
45-2
•71
960
•229
Mayfly
B.JP. (32) .
17
18
28-2
•80
76-6
•173
18}
3A
94-0
•67
143 0
•377
The total resistance was found to vary as the 1*98 power
of the speed. Expressing the resistance as before in terms
of the area of greatest cross section the coefficient of tota](
resistance (k^) is given below. If the frictional resistance
is calcnlated by Zahm's formula and subtracted from th9
total, the head or form resistance is obtained. The value of
Ka the coefficient for this resistance is also given.
Name of model.
N.P.L. 12 .
Clement-Bayard
Beta .
Gramma .
B.r.36 . .
tebaady
Mayfly • . •
B.P. 32 . .
Value of K<r, the total
resistance coefficient.
•000285 at 20
•000275 at 20
•000245 at 25
•000261 at 10
•000267 at 25
•000275 at 10
•00026 7 at 25
-000274 at 10
•000376 at 20
•000448 at 25
000466 at 10
'00027 at 20
f.p.s.
*t
>>
i>
Value of Ky or
skin resistance
coefficient. .
•000125
•000160
•000109
•000126
•000148
•000246
•000316
•000126
Value of Eh, the
bead or form
resistance
coefficient.
•000160
•000125
•000136
•000141
•000119
•000130
•000133
•000144
These results show that the head or form resistance varies
in the different models from 30 to 55 per cent of the total,
the smaller percentage being for the ship having the largest
-T-, — ^-r— ratio, the Mavfli/. For the next smaller ratio —
diameter * .^/ »
Lebaudy — ^the percentage is about 35. For the full-sized
naked models the head or form resistance will be a larger
percentage of the total resistance, since the skin friction
becomes relatively less. Fdr a naked full-sized Mayfly the
percentage is increased to 35 per cent.
440
ABRQNAUTIGS.
GOTTIHOEN KeSULTB.
llxe models employed by Prandil were of thin metal with
copper deposited surfaces. They were surfaces of revolution
and stream line form placed in an air ourrent whose speed
varied from 2 to 9'8 metres per second. The total resistanoe
was first measured. Afterwards small holes were pierced
along a meridian line oin the model, and these holes were in
turn unplugged, the pressure in the interior measured and
taken as beinff the pressure on the surface at the point
considered. These pressures were resolved fore and aft and
gave the fwrm resistance. The difference between the total
and form resistance gave the frictional resistance.
Model.
Length
mm.
Diameter,
mm.
Volume,
cubic m.
Surface,
square m.
Prismatic
coefficient.
«H
K»
I
II
III
IV
V
VI
VII
1300
U25
1032
1062
1146
1056
1160
200
194
200
188
188
188
200
•038J
•0182
•0182
•0182
•0182
•0182
•0182
•746
•479
•479
•479
'479
'479
•479
:$3
•66
'56
•61
'57
•62
•50
•00030
(•00043)
•000195
(00020)
•00022
(•00026)
•00024
(•00027)
•00016
(00022)
•00018
(•00023)
•00015
C00020)
•000147
•000113
•000116
•000134
•00010
•000106
•0001
000164
•000082
•000104
•0OO106
•00006
•000074
•00005
The coefficients Et^ t^w* ^^^ ^r ^^^ ^^ coefficients giving
the resistances in pounds in terms of the maximum area in square
feet and miles per hour, Kt being the total, Kg the head
or form, and Kf the skin friction resistance. The values of Kt
given in brackets are for the lowest speeds. The iorm or
head resistance at the maximum speed varies in the different
Imodels from 47 to 66 per cent of the total, which is greater
than the O'Gorman results. It has been stated that the
skin resistance for polished copper surfaces is about one-
half that for balloon fabrics. This is hardly in agreement
with Zahm's results, which gave i^e same results for all
smooth surfaces. If such a dSerence exists this may account
for the relatively larger percentages of the form resistance
as compared with the O'U-orman results.
Compared with the results for skin resistance as calculated
by Zahm's formula, the experimental results are very mucl^
ABJUWAOTICS.
HI
smaller and vary with different powers oi the spaMl. Tlie
power of the speed with which the frictional xeaistancA vftruM
is 1*81, 1-74, 1*78, 1*81, 1'49> 1'54, and 1*48 in the savAxi
models. Fuhrmann, however, points out that small arrom
in the pressures measared involve large diifereiioflB in the
frictional resistance dedaoed.
Fig. 238.
.nifvuwt Cw**
c
The models are shown In fig. 288 and in three of ^hese
the curves of pressure are shown, the base-line of those
curves being the axes of the models. OrdlDates beW the
442 AXBxmAVTim.
axis repreaent defect pressares and tiiOM above exoese
preesares. The pressure at the extreme forward end was
m all cases foand to be that equivalent to the head due to
the speed.
Other experimental results which have a ■light bearing
on airship resistance are g^ven below :—
Authority. Shape of model. Talne of Xr-
Eiffel {''y]^^%J?* hemispherioalj .^^^
fCone with vertical angle 20°^ «* io — -v i. AAAa>7\ u
i with hemisDhore U" rad I** ^^ m.p.h. -OOOSTXhaae
" 1 rnbafie) ~-J at 86 m.p.li. -00026/ first.
•00022}S^-
Be-^f'S^bo'Sc"^^^^^^^^
{Sur&ce formed by revolving\ ^^^. .
parabohc curve (L = 3B) / '^^^^
Fusiform bodj (l = 2b) * 00013
OoMPAfiisoN OF Model akd Fmx- sized Besttlts.
The exact law connecting the resistance of the full-
sixed airship with its model is not yet fully understood.
Theoretically the law connecting speeds and dimensions gives
LVs=a constant for airship and model, and this law would
necessitate the corresponding model speed being very much
greater than the airship speed, an impracticable condition.
Moreover, it would appear that as the speed of model
increases, the coefficient of resistance (assuming the v' law
to apply) diminishes. If the Lv law of comparison is the true
one, the coefficients of resistance for the model taken in the
following comparison are therefore too large when applied to
the full-sized ship. In any case the comparisons given below
can only be taken as very rough approximations.
Taking the O'Gorman Mayfly model as representative of
the Zeppelin airships, and the head resistance coefficient
'000133 to be the same in the model and full-sized ship, but
the frictional coefficient to vary with length in accordance
with Zahm's formula, the total resistance of a full-sized
Zeppelin » '000383av> compared with *000448av2 for the
model.
If this is compared with the experimental result on the
Sohwaben b==*00070av' it must follow that the difference
*00082av' must be due to the resistance of the appendages, i.e.
to the car, keel, struts to propeller, etc., all of which were
absent in the model. That is the model results modified for
▲ESONAUTtCS. 448
the difference In skin friotion must be increased by 80 per
cent to give the total resistance of the actuai airship. Taking
the propulsive coefficient as 50 per cent one arrives at the
result that the indicated resistance of airship =s 2 X 1*8 X
•000383av2 = •00136av2, which agrees fairly well with the
results obtained by comparison of the H.P., estimated speed
and dimensions as published for Zeppelin airships.
Model IV of the Qottingen experiments is practically the
same form as Farseval airskips.
The total resistance for Model IV = '00024av>. If this is
multiplied by 3*6 as for the Z^pelin, one arrives at the
indicated resistanoe s±s '00086 av^^ which compares with
'OOOOav' deduced from the H.P., estimated speedy and
dimensions of the latest Farseval designs. No deduction has
been made for the difference in frictional or skin resistance.
This should amount to about 50 per cent of the ooefficient of
the latter in the model, but as already ]pointed oat the skin
resistance of the polished copper models is stated to be mvxh
less than that of balloon fabrics, and the increase due to
this would probably counterbalsmoe the decrease due to
difference in lengths.
The Forlanini airship is very similar to O'Gorman's model
Beta, The total resistance of the latter when modified for
decreased skin coefficient due to length becomes = '00022av2
as compared with an actual indicated resistance =s: *00036av'.
The propulsive coefficient must be very high, and the
appendage resistance very small, therefore, to obtain agrcjs^
ment since the model resistance is only 60 per cent of the
indicated resistance. Possibly the published H.P. is under-
stated.
Published dimensions and speeds of the Beta airship give
in indicated resistance = '00082 A v^, which is just 3'5 times
the model resistance modified for the difference in skin friction
coefficient due to difference in length. In the case of ixamma
a similar comparison gives 3*3 as the ratio.
Prom the above figures it would therefore appear that
the indicated resistance, i.e. the resistance as deduced from
I.H.P., is roughly given by taking the nearest model results,
modifying the portion due to skin friction for the difference
in lengths by Zahm's formula and multiplying the result by
3*6, or if model results are not available the indicated
resistance for a Zeppelin ship is given by
B = 'OOISOavZ.
This value would also appear to apply fairly well to
lenirtu • , . .«
airships whose fineness, ratio, i.e. -tt ~ , is great, i.e. tn*
Speiss, Siemens-Schuhert, and Sohutte-Lanz airships.
444 ABaoNAuncs.
'For th* non-rigidfl ftnd lemiorigids, wIuMe ratio of length
to diameter varies from 4 to 6, the indicated resiBtance is
approximately given by B » *0008 to "OOIav^.
The figares which have been given showing the difference
between the naked model resulta and those of the actual
airships show the large effect of the resistance due to wires',
struts, oars, eto. It is, therefore, important for best speed
results to make tiiis as snu^ as practicable. In the Astra-
Torres airship the section is tralobnlar, and the wiring is
placed inside the envelope. The jnnctions of the lobes are
fitted with longitudinal wires connected aeross by vertical
wires. This to a great extent aocoants for the good apeed
results obtained in this design.
Subsequent to the writing of these notes Ei-geiL has pnl^-
lished in Nouvelies recherohes 'de la risi^ance de Voir et
I'aviatien xesults of experiments on models of three dirigibles^
vis.: (1) the Clement-Bayard, (2) the Fleurut, and (3) the
AMttra-Torres. The models were of poUshed wood and were
tried bare and also fully equipped. The lengths varidd from
1 to 1^ metres and the speed of experiment from 9 to S25>
metres per second. For this range of speed the resistance
varied as the jquare of the speed.
Expressing the resistance as before in terms of the
inazimum oross-eeotional area the values of the coeffioieBt
of resiatanoe for bare hulls in British units are : —
Clement-Bayard .... K = '000296
. FUurus b: = -000198
AUra-Torres K = '00024
The first compares with K = 000275 determined by O'Qorman
for a much longer model of the same design.
For the fully equipped models the values became '00067,
•00073, and '00072 respectively, but it is quite probable these
values are excessive since the reproduction to accurate scale
of the ropes and other small fittings in the models is very
difficult. In the case of' the Astra-forres it was found that
the rudder, elevating planes, and fliers were alone responsible
for a coefficient lof resistance equal to •()0018.
FOBM GIVENT TO AlfiSHlPS.
The form of the envelope in various designs of airships
are shown in the figure. In the rigid type a fairly long
parallel middle body amounting to from 50 to 60 per cent
of the overall length is allowed. Before and abaft tiiis
the form is tapered, the radius of the head varying from
1} to 2 diameters of the ship, and the radius of the tail from
7 to 9 diameter^.
AERONACnCd.
445
For non- and semi-rigidfl thci*e is no or very little piarftlfel
body except iii the Lebandy design, the head is strtM^ ta
a radius of 1 to 2 diameters of the ship, ilild the tail tb*
ft radius of 5 to 7 diameters.
viQr. aso.
-O «^ 54
^
'^JO^^iSUU^JXu.
\ "^-^^-y/^
xJbi
-^^ 32
Materials of ibe Envelopes.
The material to be used 'for the envelopes of airshipi#
must be lights strong, impervious to hydrogen and moisture,
and fairly durable. The material is usually rubbered cotton,
i.e. fine cotton material in one, two, or three layers witih
a thin coating of rubber on the inside and between the layers.
In some small airships gold-beater's skin, an animal tissue
obtained from the large intestines of the ox, has been
employed* CRiis material is coatly, but from the point of
view of permeability is the best.
So far as the rubbered cotton fabrics are concerned tiio-
double thickness is usuallv arranged with the threads parallel,-
but in some cases the wreada are crossed diagonally. The:!
material is manufactured in rolls about 100 yards long and
3 to 4 feet wide. The edges of adjacent lengths are lapped.
U6
AKRONAUTICS.
itnck together with rubber insertion, the lap doable-etitched
iurooffhy and a rubbered oottoa tape atuok on over the lap
)oth inside and out.
The followinfi^ is taken from the report of teats on balloon
fabrios by the National Physical lAboratory : —
Fabric.
Weight in lb.
per BQ. yd.
Tensile
Btrength,
lb. per ft. width.
Permeability,
cubio ft.
per sq*. ft.
per 34 hours.
Warp.
Weft.
Single rubbered fabric .
Double fabric, one layer rubber
Parallel double, two „ „
iJ ft 1) 11 >t
»» »» »» »» »i
,, ,, (coloured yellow)
,, „ three layers rubber
Diagonally,, two „ „
,, ,, tnree ,, ,,
(Bed rubber outside.)
Treble fabric ....
Goldbeater's skin, four layers .
,, ,, five ,, •
,, „ eight ,, .
•417
•499
•417
•444
•433
•612
•599
•407
•604
•582
•168
•204
•554
488
1008
637
708
1030
913
697
456
1548
515
732
620
731
768
752
384
1476
• 0415
•0442
•0436
•0282
•0302
•0367
•0380
•0240
•0138
•0424
•0009
•0013
•0003
Exposure to the weather for fifty days reduced the strength
\}j about 20 to 30 per cent, and increased the permeability
considerably.
The following table shows the material of the envelopes
stated to be used in various designs of airships : —
Makers of
Airships.
Lebaudy
Astra
Italian
Astra and
Clement
Parseval
Zeppelin
}
ja • "
608
700
728
-478
Type of Fabric.
Double parallel
ti
»»
I*
If
diagonal
parallel
Tensile
Strength,
lb. per ft.
width.
1008
1340
806
605
$K>*
2 layers
each.
•159
205
205
094
^ *^.
(4
h^
290
290
318
-290
AEBONAUTICS. 447
The French Government Specification for fabrio requires
two thicknesses of cofcton each 86 grammes per sauare metre,
to be worked warp on warp and weft on weft. Outside
thickness coloared yellow wiUi lead chromate. Between the
two is worked a layer of rubber weighing 83 grammee per
square metre. This layer of rubber consists of two separate
layers, one unvulcanlzed (50 grammes per square metre) next
outer thickness of cotton and the other vulcanized and
weighing 33 grammes per square metre. On the inside of inner
thickness of cotton a layer of vnloanized rubber 75 grammes
per square metre is worked. Total weight 330 grammes
per square metre ('608 Ib^ per square yard). Limits allowed
820 to 340. Rubber to be pure Para vulcanized under heat ;
5 to 7 per cent sulphur^ no other mineral. Tensile strength
1,500 kilos, per metre width (1,0081b. per foot width) in
warp or weft ; limit 1,350 kilos. Permeability to be less than
10 litres per square metre ('033 cubic feet per square foot) in
twenty-four hours. Another account gives the constraction
cotton layer coloured with lead chromate 85 grammes,
onvulcanized Para rubber 45 grammes, yulcanized rubber
40 grammes, cotton layer 85, Para rubber vulcanized 75.
The German fabrio is stated to be cotton layer coloured witb
aniline 110 grammes. Para vulcanized 130 grammes, cotton
layer 110 grammes. Para vulcanized 30 grammes.
In some airships the upper portion of the envelope has
been covered with aluminium powder to reflect the rays of
the sun. In the latest Zeppelins it is stated that the gas-bags
are of single cotton rubbered on the inside, and lined with
goldbeater's skin to decrease the permeability.
Tensions in the Envelope of a Non-rigid Airship.
The longitudinal and transverse tensions in the fabric are
connected with the pressure p by the relation : —
where B^ and B^ are the principal radii of curvature at the point
considered.
T^ is obtained from consideration of the equilibrium of the
portion of the airship cut off by a transverse plane through the
Fia. 240.
448 AERONAUTICS, ETC.
point. Besolving along the axis the resultant tensile force around
the circumference must equal the whole presnure on the section
in the direction of the axis, or 2inr . Tj cos 9 = irr^ , p. Now
r
cos ^ ~ n vhere n » leogkk of nonnal intercepted between corire
and axis ; hence T| ■= ~ and H « Bj ; hence Tj = N^? I 1 -- — J •
T a*0 when if = o, which is the case at the extremities, and
= Np when Bj » 00 , which is the case when the curve is a straight
line, i.e. where there is a puallel portion. At &e mn.-riTTi^nn
diameter N = -^, where D is the diameter, and the value of T^
beflomos = "^ I 1 ""TZ" J *nd T| a= *^ , showing that the tensions
increase with the diameter if the pressure p remains constant.
Fot the airshipi to maintain its shape under a bending moment
H (taken for comparison at the greatest diameter d) there must be
no compressive stress on the fabric. This condition is satisfied
vpt>^
wn<m M == -f^ ) 01 since in similar ships m should vary as d/p, the
excess ^essure p should vary as D and the tensions in the material
will thus v&rj as D^. In the above the bending moment has been
considered. If the shearing force on a section due to vertical
loading is considered, this shearing force is equivalent to com-
pressive and tensile forces acting in a direction at 45° to the
veriical. So long as the equivalent compressive force is less than
the tension on the fabric the airship will retain its form, lliis
condition is less difficult to satisfy than that due to the bending
moment.
BOABB OF TRADE BEOTTLATIOirS FOB MABIHS BOILERS,
Etc., IN PA88EK&BB STEAMSHIPS.
Hydbaulio Test.
All new boilers to be tested hydraulically, previously to
their being placed in the vessel^ to double the working
pressure. Old boilers to be tested to IJ times the pressure
after important repairs.
Test Pieces fob Matebials.
For plates and sectional bars to be about 18" long, of which
at least d" to be planed down parallel to take gauge marks
8" apart ; the width of this portion to be 1^" if over i" thicki
2" from §" to J", 2i" under i". (Designation A.)
liength*
Gauge lenglih,
in.
in.
H
2
3g
3
4
3i
BOABD OF TRADE REGULATIONS FOR MARINE BOILERS. 449
For round bars, rods, and stays to have enlarged ends separated
by a parallel portion of length not less than 9 diameters, to take
8-diameter gauge marks. (Designation B.) Alternately, bars
over 1" diameter may have 4^ diameters parallel length to take
4- diameter gauge marks. (Designation F.)
For forgings and castings to be circular (the ends being
enlarged), with dimensions as follows : —
Parallel
Dasignation* Diam. of parallel Area in
portion in in. sq. in.
c .664 J
I> -798 i
K -977 i
For bending tests, pieces from plates or sections should be at
least li" wide ; from round bars should be of full section (those
more than 2" diameter may be turned down to 2") ; ttom. forgings
and castings to be rectangular T'xJ", machined, with corners
rounded to ^' radius, to be bent oyer the smaller section.
Tests fob Materials.
Those regarding boiler material refer to ordinary mild
steel. In all cases the test pieces and gauge lengths described
above are assumed.
Plates (ordinary). — ^Tensile, 27 to 32 tons per square inch ;
elongation, 20o/o above §" thick, with reduction of 3o/o for
each J" less than §". Bending through 180^ with inner
diameter three times the thickness.
Plates (to be worked in fire or exposed to flame). — ^Tensile,
26 to 30 tons per square inch ; elongation, 23o/o, with reduc-
tion as above. Bending, as above, but with tempered strips.
Stays, Angle, and Tee Bars. — Tensile, 27 to 32 tons pet
square inch ; elongation, 20o/o, except for combustion i'hamber
stays, where test is 26 to 32 tons per square inch, with
elongation 23o/o. On test piece F the elongations should be
240/0 and 280/0 respectively. For sectional bars under f
thickness deduct 3 0/0 from elongation allowed for plates.
Bending tests as for plates.
Bivet Bars. — ^Tensile, 26 to 30 tons per square inch, with
elongation 250/0 on B or 30 0/0 on F.
Mivets. — Shanks bent cold, and hammered right over to
180^ without fracture. Heads to be flattened, when hot, until
diameter is 2} times that of shank, without cracking at edges.
Tensile (on length. 2} times diameter), 27 to 32 tons per
square inch ; contraction of area about 60 0/0. '
Solid-drawn Steel Steam Pipes, Boilfir Tubes, etc., subject
to internal pressure.— Tensile, 23 to 30 tons per square inch,
with elongation 20 0/0 in S'', or 18 0/0 if thickness is leaa
than i". All tubes to be tested to a suitable hydraulic pressure.
450 BOARD OF TRADE BEGUIATIONS FOE MARZNB
Solid-drawn Steel Tubes, subject to external pressure.—
Tensile, 23 to 30 tons per square inch, with elongation as
above. Hydranlio test.
Steel Lap-welded Tubes, subject to external pressure. —
Tensile (on strips from which tubes are made), 23 to 80 tons
per square inch ; with elongation 20o/o in 8". Hydraulic teat.
If no allowance over iron is required the test for the two
last may bg omitted.
All Tubes, -r-The hydranlio test should not exceed 8 times
the working pressure, or 4 times that given by
6,000 X thickness in inches
— i — rj — J-. 7 — » — ! — r — = pressure
inside diameter in inches ^
for lap-welded tubes, or 5 times that pressure for solid-drawn
steel tubes. Steel to be open-hearth acid steel. Solid-drawn
tubes whose thickness is more than i" should be finished by the
hot-drawn process.
Forcings, — ^Tensile, not exceeding 40 tons per square inch ;
with elongation 17o/o ; for lower tensile strengths the sum of
this and of the elongation percentage should not exceed 57.
Bending, through 180^ with internal radius i" up to 32 tons
per square inch, f" from 32 to 36, and §" from 36 to 40.
Castings. — ^Tensile, 26 to 40 tons per square inch ; with
elongation 15o/o. For pistons and important parts elongation
shomd be 20o/o if tensile strength is less than 35 tons per
square inch. Bending, if tensile 35 to 40, through 60° ;
otherwise through 90^ ; for important parts through 120** ;
internal radius of bend 1 inch. All steel eastings to be
annealed.
The above tests for steel castings may be omitted if the
scantlings be those appropriate to cast iron,
STfiEL Boilers.
Plates used to be not less than ^" thick. Xlivet holes to
be drilled. Plates drilled in place to be taken apart and the
burr taken off, and the holes slightly countersunk from outside.
After local heating plates should be annealed. When the
eylindrical shells of boilers are of material, tested and
approved, with all the rivet holes drilled in place, and aU the
seams fitted with double butt-straps, each of f the thickness of
the plates they cover, and all the seams at least double-riveted
with rivets having an allowance of not more than 87| per
cent ever the single shear, provided that the boilers have been
open to inspection during tiie whole period of eonstruction,
then 4*5 may be used as the factor of safety, the minimum
actual tensile strength of the plates being used in calculating
the working pressure.
WWHi^TJl' lllJLUJL AILUULrATlUJNS JfUK MAliIx\£ BUILERS. '101
Table givino the Constants to be added to the I
FACToa OF Safety foe Cylindrical Boilers.
Mark
B
B
P
H^
M
M
Uo'ii-
stants
15
15
1 a
15
2
2
3
Circumstaaoes in which the constants have to be added
When the holes are fair and good in the longi-
tudinal seamsj hut drilled out of place after
bending.
When the holes are fair and good in the longi-
tudinal seams^ but drilled out of place before
bending.
When double butt-straps are not £tted to the
longitudinal seams, and said seams are lap
and double-riveted.
When double butt-straps are not fitted to the
longitudinal seams, and the said seams
are lap and treble-riveted.
When only single butt-straps are fitted to the
longitudinal seams, and the said seams are
'double-riveted.
When only single butt-straps are fitted to the
longitudinal seams, and the said seams are
treble-riveted.
When any description of joint in the longi-
tudisiai seams is single-riveted.
When there are two or more belts of plates^,
and the seams are not properly crossed.
When tibe holes are fair and good in the
circmnferential seams, but dnlled out of
plaoe «fter bending.
When the holes are fair and good in Hie
circumferential aeams, • but drUled before
bending.
When the circumferential seams are fitted with
single butt-straps and are double-riveted.
When the circumferential seams are fitted with
single butt-straps and are single-riveted.
When the circumferential seams are fitted with
double butt-straps and are single-riveted.
When the circumferential seams are lap joints
and are double-riveted.
Wh&n. the circumferential seams ai-e lap joints
and are single-riveted. *
When the boiler is of such a length as to fire
from both ends, or is of unusual length, such
as fine boilers, and the seams are fitted as
described opx)oeite E, M, and n ; but when
the ciroumferential seams are as described
opposite L or 0, P '3 will become p 'i.
• The allowance may be increased still further if the workmanship
or material is very doubtful or very unsatisfactory.
d
452 BOARD OF TRADE REGULATIONS FOR MARINE BOILERS.
When the above conditions have not been complied with,
the additions in the scale (p. 451) should be made io the
factor of safety, acoording to the circomstanoes of each case.
Strength of Joints and Pressure on Safety Valves in
CyHndrical Boilers.
Formula (Inch-ton units).
P Bs percentage of strength of plate at joint as compared
with the solid p&te.
P' sr percentage of strength of rivets as compared with the
solid plate.
p = pitch of rivets.
a = area of one rivet.
d = diameter of rivets.
n = number of rows of rivets.
8^ = minimum tensile strength of plates.
Sj = shearing strength of rivets to be taken as 23.
c s= 1 for rivets in single shear ss 1*875 for double she&r.
t = thickness of plate.
(p-d)xlOO , (axn)x 100 SsO
p pxt . Si
If percentage strength of rivets in longitudinal seams is
less than calculated strength of plate, then pressure on shell
should be calculated from each percentage
Si X 2y240 X o/o strength of joint X 2t
~" D X factor of safety X 100
where D =* inside diameter of boiler. The smaller of the two
pressures is that to be allowed ; take factor of safety ss 4 5
for rivets or as found from the table for the plates.
For steel plates and iron rivets, take Sj =« 17*5. For iron
plates and rivets take s^ »s s^.
Where alternate rivets are omitted in the outer rows
P— — iEz — I + ~ where p is the greatest pitch, i.e. at the
outer row.
The diameters of the rivets should not be less than the
thickness of the plates.
Mivei Spacing, etc., in Jointi,
From the above formulae the value of any type of joint
can be calculated.
In addition let
y SBB perpendicular distances between rows of nvets.
E =s distance from edge of plate or butt-strap of centres
of outer row.
Ix = thickness of butt-strap (supposed of same material
as plate).
BOABI> OF TRADE REGULATIONS FOR MARINE BOILERS. 453
(a) Joints where no rivets are omitted.
ti = it for double straps = It for single straps.
B = -^ , V = 2(2 * for chain riveting.
V = ^V(llp + 4d) (p + 4d) for zigzag riveting.
(6) Joints where alternate rivets are omitted from the outer
row, •(??== pitch at outer rows : Vi = distance between adja-
cant xows both containing the full number of rivets ; V =s
distance when one jow has alternate rivets omitted.)
** = Q^^£) *^' **'''^^^ '*'*P' = l\p^2) *°' ^«^^ "^^^P"-
If, in treble-riveted straps, the alternative rivets are
omitted also fnpm. the inner rows,
ti = 1^ for double straps = it for smgle straps.
dd
« = T •
V (chain riveting = 2dt or ^ V (ll2? + 4<i)(|)+4d), whichoTer
is greater. Vi = 2d.t
T (zigzag riveting) =
Vi (zigzag riveting) = J^ v^(llp+8d)(i>+8<l).
Maximum Pitohss for Riveted Joints.
In inch units, let i = thickness of plate ; p »» maximum
pitch of rivets; and c a constant from the following table:— >
Number of rowi
of rivets.
Constanta for lap
joints.
Constants for double
butt strap Joints.
1
2
3
4
5
1-31
2-62
347
4*14
1-75
350
4-63
662
6-00
Then p = c< + lg.
p should never exceed 10}, and should preferably be
rainer less than that given above.
* In treble riveted bntts either the outer edires of both stnp and plates
or those of tiie strap only may have alternate rivets omitted*
f fid + i is preferable.
454 BOARD OF TRADE REGULATIONS FOR MARINE BOILERS.
Openings in Shells, Ifoort, 0to.
In cylindrical boilers openings in the shell shonld have
the shorter axis placed lonffitndinally. Compensating rings of
the same effective sectional area as the plates oat out, and ot
the same thickness, should be fitted aronnd all manholes and
openings, and efficient stiffening should be otherwise provided.
It is desirably that these rings should be of L or T-bar when
round openings on flat surfaces. When alternatively tbe plate
is flanged, D, the depth of flange, should be at least equal to
\/ width of opening x thickness of plate. Cast-iron doors are
not allowed.
Ends.
Hemispherical ends subjected to internal pressare may be
allowed twice the pressure suitable for a ^liader of the same
diameter and thickness.
Ends of steam receivers which are dished and flanged
hydraulically under one heat need not be stayed if radius of
end is no more than diameter of shell and does not exceed
4 feet ; the outer radius of flange at root should be at least
3 inches, and steel should be of usual quality, and annealed
after flanging. The working pressure allowed without stays
90 000 T^
should not exceed — ^-ra — where Tsend thiokness, D— diameter
D'
, /i = R - \/r3 - ^ i where X a inner radius of end, all
in inches.
If the dished ends require stayS, but are sufficient for the
pressure when considered as portions of spheres, the stays, if
of solid steel, may have a nominal stress of 18,000 lb. per
square inch. Otherwise the ends should be stayed as flat
sajfaces. For iron take 14,000, or 10,000 if welded.
Stays for Flat Surfaces.
Solid steel stays may have a working stress of 9,0001b.
per square inch. Welded stays, except strong tubes welded
longitudinally, are not allowed. When the threads of longi-
tsdinal stinrs are finer than sir per inch, the depths oif
the external nuts should be at least 1^ the diameter of sta^.
For iron stays, stress allowed is 7,000 lb. per square inch,
or 5^000 if welded. In combustion chambers 9,000 is allowed,
if iron has a tensile strength of 21^ tons per square inch,
with elongation 27 per cent in 8 inches.
To find the area of any diagonal stay, find the area of
a direct stay needed to support the surface ; mul^ply this
area by the length of the diagonal stay, and divide the product
by the length of a line drawn at right angles to the surface
supported at the end of the diagonal stay.
of shell
BOARD OF TRADE REGULATIONS FOR MARINE BOILERS. 455
TfiOie.-^Whetk gusset stays are nsed their area should be
in excess of that found by the above rule«
Steel stay tubes may be allowed a stress of 7,500 lb. peir
square inch (6,000 for iron), but their net thioimess should be
at least one-quarter inch.
If no allowance over iron is required, stays and smoke
tubes should be made of the sizes required for iron.
Girders far Flat Surfaces.
When the tops of combustion boxes, or other parts of a
boiler, are sapported by Eolid rectangular girders, the following
formula may be used for finding the working pressure to be
»Uewed on the girders, assuming that they are not subjected
to a greater temperature than the ordinary heat of steam, and
In the case of combustion chamibers that the ends are fitted to
the edges of the tube plate and the back plabo of the
corobustion box : —
FOEMULA.
F«s working preesore.
L :a lengtili of girder in feet.
T =: thickness of girder in inches.
D SB depth of girder in inches.
w sa width of oombustion box in inches.
^ss pitch of supporting stays in inches.
d SB distance between the girders from centre to centre in
. inches.
K = number of supporting stays.
^_ NX 1,^20 , „. ,,
when N IS odd.
N+1
(n + 1)1,820
* 'I > .
N + 2
C X d" X T
when N is even.
(W-j3)dxL
Fqr iron substitute 1,200 for 1,320.
Plates for Flat Surfaces,
The pressure on plates forming flat surfaces may be found
by the following formula : —
FORUULA.
w ssm working pressure.
T s= thickness of plate in sixteenths of an inch.
8 i= surface supported in square inches.
0 == constant for steel \ according to the following circum-
e SB constant for iron / itanees : —
466BOAKD OF TRADB BEGULATIONS FOR MilBINE BOILBBS.
0
240
192
0
tf !
0
0
e-
0
0<
0:
0
e
G
c
01
0
e
210
168
165
132
150
120
112i
90
77
70
75
60
67-6
'■ 54
100
80
66
60 1
I
c
c
39-6
36
wh^ the plates are not exposed to the impact
9f heat or flame and the stays are fitted with nuts
on both aides of the plates, and doabling' strips
4 not less in width than two-thirds the pitch of the
I stays, and of the thickness of the plates, are
I seeorely riyeted to the outside of the plates they
Vcover.
{as above, but with washers of diameter equal to
two-thirds pitch of stays in lieu of doublings
plates.
as preceding^, but with loose washers outside,
three times &e stay diameter, and two-thirds the
plate thickness,
r when the plates are not exposed to the impact
-I of heat or flame and the stays are fitted with nuts
(on both sides of the plate.
[when the plates are not exposed to the impact
1 of heat or flame and the stays are fitted with cuts
I only.
when the plates are not exposed to the impact
• of heat or flame and the stays are screwed into
^ the plates and riveted over or expanded.
{when the plates are exposed to the impact of
heat or flame and steam in oontact with the
plates, and the stays fitted with nuts and
washers, the latter being at least three times the
diameter of the stay and two- thirds the thickness
of the plates they cover.
'when the plates are exposed to the impact of
heat or flame and steam in contact with the
plate, and the stays fitted with nuts only.
(when the plates are exposed to the impact of
heat or flame with water in contact with the
plates, and the stays screwed into the plate and
fitted with nuts.
when the plates are exposed to the impact of
heat or fiame with water in contact with the
plate, and the stays screwed into the plate
having the ends riveted over to form a sub-
stantial head.
when the plates are exposed to the impact of
heat or flame and steam in contact with the
plates, with the stays screwed into the plate
and having the ends riveted over to form a sub-
stantial head.
W=:
C or c X (t + 1)'
8-6
BO^BD OF TBADE REGULATIONS FOB MABINB BOILEBS. 457
If thickness of doubling! plate be Xi in sixteenths of an
inchy then with unexposed plates
c(orc) (T+l)'-fc(or c) (Ti+1)'
^~ S-6
In calcalatingi the working pressure of the portion of
tube-plates between the boxes of tubes, take 2s = D^ + ^>
where d and d are respectively the horizontal and vertical
pitches of the stay tubes in inches.
Compressive Stress on Tube Plates.
Let D = least horizontal distance between centres of tubes.
d = inside diameter of ordinary tubes.
T = thickness of tube plates.
yr = width of combustion-box between tube plate and
back of fire-box, all in inches.
^ ,. (d-^)Tx 28,000 (22,000 for iron)
Workmg pressure « ^ — • '
' WxD
Furnaces.
For circular furnaces with longitudinal joints welded or
made with single straps double-riveted, or double straps
single-riveted,
-„ I. 99,000 X (plate thickness)'
Wcakmg preasure = (lengfe in f^t+ljx diameter
--- , . 1. ij i , 9,900 X thickness
Working pressure should not exceed jz :
*^ diameter
The thickness and diameter are in inches. For iron take
9C,000 and 9,000. With ordinary lap joint take 92,500 ; for
bevelled joint 88,000 ; for other joints or inferior workman-
ship the number in the first formula is varied. For the
upriffht fire-boxes of a donkey boiler deduct 10<Vo from both
numbers.
For certain types of corrugated furnaces with plates at least
A" thick
Working pressure =
14,000 X thickness at the bottom of corrugation ,. , .
outside diameter at bottom of corrugations ^ ^'
The pitch of corrugations should not exceed 6" (Fox) or 8"
(Morrison and Deighton) ; . the depth extreme should be at
least 2^'.
For ribbed and grooved furnaces D = diameter over plain
part ; the ribs should be at least l^V above the plain parts,
the depths of grooves not more than {", and thei spacing not
over 9',
458 board of trade regulations for marine b0iusb8.
Iron Boilers.
Cylindrical Shells.— Take tensfle streno^th as 47,000 lb. per
square inch, with the grain, and 40,000 lb. across ; shearing
strength of rivets as the same as the tensile strength of
platea ; take 5 as the factor of safety subject to the other
additions specified for steel. If, however, elongation in 10*^
is lio/o with and 8<V6 across the grain^ take 4*5 as factor
instead of 5, and use actual minimnm tensile strength of
plates for calculating the working pressure.
Ordinary Smoke Tubes. — ^Thickness =
««-. t working pressure X outside diameter . . ,
•085 -I zi5-£. K-QQQ in inch units.
The* other differences from steel have been indicated above.
Superheaters.
Iron cylindrical superheaters are to be designed as boilers,
but using 30,000 (or 22,400 where the flame impinges nearly
perpendieular to the plate) instead of 47,000 as the tensile
strength.
For a superheater with a tube subject to external pressure
treat by the rules for circular furnaces^ but reducing the
oonslailtB in the ratio 30 to 47.
If. it eoiaiats of a nest or ooil of tubes subject to internal
pressure, these should be made of solid-drawn steel ; other-
wise steel is inadvisable in superheaters.
Superheaters must be flttea with drain-pipes and a safety-
valve of statutory size (at least 3 inch diameter).
Evaporators, Feed Heaters.
If of iron or steel oonstmot as for boilers.
If of CMt iron least thiekness should be g", if of gun-metal
j", tiie tensile siopengi^ being 10 tons per square inch ;
t =3f i^icknesfl, D a diameter, 8 = side, all in inches.
0 t=a 4,000 for oast iron ; 6,000 for gun-metal.
c^ a 24,000 for cast iron ; 30,000 for gun-metal.
0^ c=« 16,000 for oaat iron ; 20,000 for gun-metal.
B = working pressure.
c Ct— 4)
For cylindrical shells B = — ^^ 5i
For eircular flat surfaces B = -^
CoT*
For square flat surface b = -^-g-
If, however, oast steel of J" minimum thickness be used, c,
Ci, and o2 become 10,400, 62,000, and 34,700 respectively.
The pressure should not exceed 15 if the main body is
a single casting.
BOARD OF TRADB RSOUtATIOKS V0% ttARTKE BOILERS. 469
Steam Pipes.
Board of Trade Rute» for the Diameter and ThicJknesi of
Steam Pipee.
(i) For copper pipes when the joints ai^e brazed,
6000x(t-A)
p= 5
where P = wor^ing^ pressure in pounds per square inch.
T = thickness in inches.
D =B inside diameter in inches.
When the pipes are solid drawn and not over 10 inehes
diameter substitute in the foregoing formula ^ for ^,
(ii) For wrought-iron pipes made of good material and lap-
TTcIded,
6000 X T
This formula does not apply when the thickness is less
than I inch.
All now coppcj: steam pipes should be hydraulically tested
to 2i^ to 3 times the working pressure ; iron and steel pipes to
3 to 5 times the pressure.
Safety-valves.
PravUiona of the Act aji regards Safetg^-valvee,
Evezy steamsliip of which a survey is required by the
Act must be provided with a safety-*valve upon eacli boiler,
80 consfaructed as to he out of the eonf^ifol of the engineer
when the steam is up* ; and if such valve is in addition to
the ordinary valve, if shall be so constrcioted as to have an
area not less, and a pressure not greater, than the area of
and pressure on that valve.
Area of Safety-valves*
When natural draught ia used, the afea per square foot
of fire-grate surface of the looked-np safefy valve should
not be less than that given in the table on p. 460, opponte
the boiler pressure intended, but in no case should the valves
be less than two inches in aiameter.
When the valves are of common description, and are
made in. accordance with the tables, it will be necessary to
fit them with springs having great elasticltiy, or to provide
other means to keep the accumulation within moderate limits.
To find the fire-grate area, the length of the grate to be
measured from the inner edge of the dead jplate to the front
of the bridge, and the width from side to side of the furnace
on the top of the bars at the middle of their length.
When forced drai^ht is used, the area required is that
found from the tables multiplied by one- twentieth of the
60 BOAKD OP TBADS RE0ULATI0N3 FOB MABINS BOILEBS.
Safety YAiiVE Abeas.
(Natural Draught,)
BoUer
Preasure
Area of Yalve
per 8q. Foot
of Fire-crate
U
10
17
18
U
SO
81
39
38
34
35
98
37
9B
99
10
81
88
88
84
86
86
87
88
80
40
41
49
48
44
46
46
47
48
49
60
61
63
68
64
68
66
67
68
SO
00
61
02
68
64
66
66
07
08
60
70
71
T3
78
74
76
76
77
IB
1-3B0
1-90O
1171
riae
IVK,
1-071
1D41
1-018
*088
-061
•987
•914
-809
'679
*8U
TW
•781
•766
•780
•786
•781
^
-681
•660
•667
-646
•686
•614
•604
•606
•086
•676
•668
•660
•661
•648
•686
•630
•618
•606
•600
•498
•487
•460
•474
•468
•463
467
461
446
-441
-486
•481
•496
•431
•410
•413
•407
•406
Boiler
Preasure
79
80
81
83
68
64
86
66
87
90
91
93
98
04
■ 96
96
97
98
99
100
101
103
106
104
106
106
107
108
109
110
lU
113
118
114
116
116
117
118
119
130
181
133
138
194
136
136
127
138
189
180
181
1S3
188
184
186
186
187
188
189
140
141
148
Area of VaXve
per 8q. Foot
of Fire-crate
•dM
-880
•866
•878
•876
•871
•867
-864
-860
-887
•868
-8S0
•847
•844
•840
•8W
•884
•881
•838
•836
-890
•817
•816
•813
•809
•807
■804
•809
•800
•907
•396
•999
•390
•3S8
•386
•384
•381
•379
•277
•976
•278
•271
-967
•365
•264
•263
•960
•858
•356
-266
-258
•251
•350
•348
•346
•246
•248
•241
-240
•988
Boiler
Preesnre
148
144
146
140
147
148
149
160
161
168
168
164
166
156
167
168
160
160
161
108
168
164
166
iS
166
169
170
171
179
178
174
176
176
177
178
179
180
181
188
188
184
186
166
187
288
180
190
101
193
196
194
196
196
197
198
160
800
906
910
890
880
840
860
Area of Yalye
per Sq. Foot
of FIre-gtmto
•887
•986
•284
-981
•880
•987
•986
•894
•881
V9
•918
•910
•316
-814
•810
•iu
•810
•909
•80?
•806
•9D4
•901
•an
198
•197
-196
106
•194
198
199
191
190
•288
186
•187
*186
166
•184
•188
189
•181
181
•180
•179
•178
177
•178
•176
•176
174
•170
'166
169
•158
•147
'141
BOARD OF TEADE REGULATIONS FOR MARINE BOILERS. 461
estmiated consumption of ooal par sqnare foot of grate in
pounds per hour.
Th« safety-valvo to be fitted with lifting gear, so that
two or more valves on any one boiler can be eased together
without interfering with the valves on any other boiler.
The lifting gear to be arranged so that it can be worked by
hand either from the engine-room or stokehole ; safety-valves
to have a lift equal to one-fourth their diameter.
Spring Safety-valves,
Spring safety-valves may be fitted in passenger steamers
instead of dead-weighted valves, provided that the following
conditions are Complied with : —
1. That at least two separate valves are fitted to each
boiler.
2. That the valves are of the proper size. .
3. That the spring and valve be so cased in that they
cannot be tampered with.
4. That provision be made to prevent the valve flying oS
in case of the spring breaking.
5. That screw lifting gear be provided to ease all the;
valves, if necessary, when steam is up.
6. That the springs be protected from the steam and
impurities issuing from the valves.
I. That when the valves are loaded by direct springs, the
compressing screw abuts against a metal stop or washer :when
the load sanctioned by the surveyor is on the valve.
8. That the size of the steel of which the spring is made
is found by the following formula : —
FORMULA.
d s=z diameter or side of square of the wire in inches,
p Bs diameter of the spring, from centre to centre of wire^
in inches.
8 =ss load on the spring in pounds.
k «= constant s= 8,000 for round and 11,000 for square steel.
9. That the springs have a sufficient number of coils to
allow a compression under the working load of at least |
diameter of the valve.
Note. — ^The accumulation of pressure should not exceed
10 per cent of the loaded pressure. .
The number of coils required for a given compression, or ,
the compression due to the load, is given by the followiiig
formula : —
K X c X d^ 8 X D^ X Ig
^"= 8Xd3 ®' ^"^ OX<i*
462 BOAKD OF TRADE REGULATIONS FOR BCASmS BOILBR8.
where n^b number of free ooila in spring.
X ss compression in inches.
d »= diameter of steel or side of aqnare in $ixteenih*
of an inch,
0«=22 for round and 30 for square steel.
8, D are as above.
The steel of these springpi should not be generally less
than ^ inch diameter or side.
All safety-valves to be tested under full sfceam and full
firing at least fifteen minutes with the feed-water shut off^ and
the stop-valye oloaed.
BOILEB MoXTN^DINaS, SeA CoNNEX)0K6, ETC.
No arrangement is permissible where the esoapa of steam
from the sa^ty-valve is wholly or partially intercepted by
another valve.
A stop-valve shou,ld be placed between each boiler and
steam-pipe, superheater, or steam receiver.
Bach boiler should have a glass water-g^uge, at least three
test-cocks, and. a steam-gauf e. If fired from both ends, or of
iMinsnal width, an additional water-gauge and set of test-oocks
should be provided.
Each boiler shoqj[<l have a suitable check-valve between
it and the feed-pipes ; all new 'boilers to have additional
separate feed arrangeniants.
Outlets of water*olo8et, soil, senpper, lavatory, and urinal
pipes below the weather deck should have an elbow of Bub-
staatial metal other than oast iron or lead. The pipe
connected with it should have sufficient bend to allow fw
working and expansion, as should all pipes connecting the
ship's side with the deck, closet, or other fitting^. The pipes
and Valves should be protected from the cargo by a substuitial
wood or iron casing. Flans of pumping closets below the
water-line should be submitted for approval.
All inlets or outlets in the bottom or side of a vessel, near
to, at, or below the load water-line, except those above
referred to, must have cocks or valves fitted between the pipes
and the ship's side or bottom. Such cocks or valves must
be attached to the skin of the ship, and be so arranged that
they can be easily and expeditiously opened or cfosed at
any time.
All blow-off cocks and sea connexions are to be .fitted with
a guard over the plug, with a feather- way in the same, and
a key on the spanner, so that the spanner cannot be taken out
unless the pluff or cock is closed. One cook is to be fitted to
the boiler, and another cock on the skin of the ship or on the
side of the Kingston valve.
In all cases where pipes are so led or placed that water
can run from the boiler or the sea into the bilge, either by
60AKD OF TAADE REGULATIONS FOB MARINE BOILERS. ^68
accidentally or inientionally leaving a cock ob valve open,
they should be fitted with a non-return valve and a Bcrew,
not attached, but which will set the valve down in its seat
when necessary. The only exception to this is the firenmn'f
a.sh-cock, which must have a cock or valve on the ship's side
ana be above the stoke-hole plates.
The exhaust pipe for the donkey engine must not be led
through the ship's side, but must be led on deck or into the
main waste-steam pipe, and in all cases it should hav« a drain-
cock on it.
Spare Gear and Stores to be Carried.
Steamers eoming in for survey under the Fass^iger Acts,
and other steamers performing oceui voyages, must carry at
least the following spare gear, or its equivalent| which mi^sti
have been fitted and tried in its place :—
1 pair of connecting-rod brasses.
1 air-pump backet* and rod with gnlde.
1 cironlafcing-pamp bucket and rod.f
1 air-pumjj^ head valye seat, and ffuard.f
1 set of india-rubber yalves (or ocue-taird sefc metsU) for air
pumps.
1 circulating -pump head valve seat, and ^nnxd
1 set jof india-rubber yadyetf (or oo.«-third set metal) for circu-
lating pumps.f
2 main bearing bolts and nuts.
2 connecting-rod bolts and nuts.
2 piston-rod bolts and nuts.
8 screw-shaft coupling bolts and nuts.
1 set of piston springs suitable for the pistons.
1 set of metal feed-pump yalves and- teats.
3 sets, if of india-rubber, or I seb if of m'etal, of bilge-pump
valves and seats.
Boiler tubes, 3 for each boiler.
IQO iroQ asserted bolts, nuts, and wasjiers screwed, but need not
be tamed.
12 brass bolts and nuts', assorted, turned, and fitted.
60 iron „ „ „
50 condenser tubes and 1 hydrometer.
100 sets of poking for condenser-tube emdtf, or an •qaivalent.
At least one spare spring of each size for escaiKt vaives.
1 set of water-gauge glasi^es.
^ the total number of fire bars necessary.
3 plates of iroA» and 6. bars of iron assorted.
1 complete set of stocks, dies, and taps, Buitat>le for the eogiaef.
Bachet braces and suitable drills.
1 copper or metal hammer and 1 smith's anvil.
1 screw jack and 1 fitter's vice.
Suitable blocks and tackling for lifting weights.
1 dozen files, assorted, and handles for the same.
1 set of drifts or expanders for boiler tubes.
1 set of safety-valve springs, if so fitted, for every four valves ;
if there are not four valv^iT, then at least one sot of springs must
be carried. More than 6 spare springs of the same si4e need not
be provided.
* If valvelesB, a spare rod and guide only.
i If pump is centrifugal, a spare spindle and disc are required in lien.
464 BOAKD OF TRADE REGULATIONS FOR SIZE OF SHAFTS.
And a set of engineer's tools suitable for the service, in-
cludiAfi^ hammers and chisel for vice and forge, solder and
soldering-iron, sheets of tin and copper, spelter, muriatic acid
or other equivalent, etc., etc.
Size of Shafts.
Main and tunnel and propeller shafts should be of at least
the diameter given by the following formula : —
F^krmula for Compound Condennng Engine with two or tnore
. Cylinders, when the crttnks are not overhung,
s = diameter of shaft in inches.
d^ ss square of diameter of hi^h-pressure cylinder in inches,
or sum of squares of diameters when there are two or
more high-pressure cylinders.
d' =s square of diameter of low-pressure cylinder in Inches,
or sum of squares of diameters when there are two or
more low-pressure cylinders.
p Bs absolute pressure in pounds per square inch, that is,
boiler pressure plus 16 lb.
0 = length ot crank in inches.
k = constant from following table (p. 465).
S
Formula for Ordinary Condensing Engines with one, two, or
more Cylinders, when the cranks are not overhung,
8 s= diameter of shaft in inches.
d' «b square of diameter of cylinder in inches, or sum of
squares of diameters when there are two or more
cylinders.
Tis* absolute pressure in pounds per square inch.
0 Bi length of crank in inches.
^ = constant froni following table.
, J/o X p X D^
3 X fc X s'
With one crank, use the constants for 180°.
The portion of the propeller shaft forward of the stern
gland, and all the thrust shaft except that in the thrusfc
bearing, may have the same diameter as the intermediate
tunnel shafting.
REGULATIONS FOfl KEPRIGBRATOHS, ETC. 465-
FortwoCrankft
Angle between'
. For Gisalt and Tfanut
For Ttmnel
For Propeller
Cranks.
Shafts.
Shafts.
Shafts.
k
k
h
90°
1-047-
1-221
890
100°
For paddle 966
1-128
821
110°
engines of 904
1-055
768
120°
ordinary type, 855
997
727
130°
multiply con- 817
953
694
140°
stant of this 788
919
670
160°
column suit- 766
894
651
160°
able for angle 751
877
638
170°
of crank by 1*4 74d
867
631
180«
740
864
629
For thne Cranks.
*v
•
120°
1,110
1,295
943 1
Formula for Turbine Engineai
s = diameter of shaft in inches.
I.H.P. = estimated maximum indicated horse-power trans-
mitted through shaft.
R =3 number of revolutions per minute.
K = 60*3 for tunnel shafts ; 82'8 for propeller shafts.
8
■</
I.H.P. XK
Refeigeeatohs and Distillers.
Machines of ammonia>compression type should be plliced
in a well- ventilated, isolated oompartment^ preferably on deck,
bat< an ammonia-absorption' maohine may be placed iii an
engine-room if 8atifl£aetorily. v^itilated. A G O2 maehine may
similarly be placed in the engine-room if the charge that
might be released by a breakdown does not exceed 300 lb.
In emigrant 8hip% the boiler for supplying steam to the
distillers shotild be built in- aocordanoe with the regula-
tions governing the main boilers. The steam for this purpose
flhould not bo taken from the main boilers, and no exhaust
steam should enter the condenser. The boiler should not
be -filled or fed with water from the main surface condensers;
the introduotioB of lubrieants, tallow; or oil must be avoided.
The presenee of zine in such boilers is objectionable. There
must be a suitable filter charged with animal charooali
Bh
466 REGULATIONS FOR MOTOR PASSENGER VESSELS.
8tore9 to he earried with IHtUllin(^ Apparatus,
The following list of tools and material must be provided
for distilling apparatus :—
1 set of stoking tools.
1 scaling tool.
1 spanner for boiler doors.
1 set of fire bars, suitable for boiler.
1 14 in. flat bastard file.
1 14 in. half-round file.
1 10 in. round file.
8 file handles.
2 hand cold ohisels.
1 chipping hammer.
1 pair of efficient gas tongs.
1 soldering iron.
10 lb. of solder.
2 lb. of. resin.
6 gauge glasses.
24 india-rubber gauge-glass wasbers.
80 bolts and nuts, assorted.
1 slide rod for donkey pump.
5 lb. of spun yarn.
10 lb. of cotton waste.
1 deal box with lock complete.
2 gallons of machinery oil.
1 can for machinery oil.
1 oil-feeder.
1 small bench vice.
1 ratchet brace.
4 drills, assorted.
1 set of dies and taps suitable for the bolts.
2 elass salinometers.
1 hydrometer and pot.
1 shifting spanner.
1 lamp for engineer.
Animzu charcoal sufficient to charge the filter at least twice.
And other articles that the particular distiller and boiler
supplied may, in the surveyor's judgment, require.
BOARD OF TRADE REQULATIONS FOB HOTOB PASSEKOER
VESSELS.
The regulations governing passenger steamships apply, as
ixr as they are applicable, to motor- and electrio-boats which
carry more than twelve passengers. The following special
requirements apply to boats using petrol or other grade of
petroleum : —
Oil Tank. — To be well and substantially constructed, and
of reasonable size. If of iron or steel, to be galvanized
externally. Tank and connexions to be quite oil-tight, and
to be tested hydraulically to a head of 15 feet of water.
HEOUtATlONS FOR MOTOR PASSENGER VESSELS. 467
The tank should be securely fixed on a lead-lined or metal
tray, above the deep load-line, with drain-pipes leading
overboard.
The arrangements for filling should prevent oil readily
spilling iirto or lodging in any part of the vessel ; the petrol
vapour displaced when filling should be led overboard. The
wood deoky if any, surrounding the inlet pipe should be
covered with sheet metal. Each inlet or outlet to the tank
should be covered with a removable wire-gauge diaphragm ;
the filling pipe should have a screwed cap. The tank to bo
filled when no passengers are on board. No loose cans of
petrol to be carried in the boat.
An open pipe with gauze, a light spring safety-valve, or
a fusible plug to be provided for relieving the pressure in
case of m:e in the tank.
Pipe Arrangements,— The pipe conveying the petrol to the
carburetter to be of solid-drawn copper, with a flexible bend,
and with a cock or valve at each end, one on the tank and the
other on the carburetter ; the joints to be accessible so that
they can be kept quite oil-tight. Soft solder joints are un-
satisfactory.
The air inlet to the carburetter should have a wire-gauge
diaphragm, and be carried to the ship's sid^e or to a reason-
able height above the carburetter, so that there wilt be no
danger of ignition of any petrol vapour that may escape
when the engine is stopped.
The carburetter should desirably be of such a type that,
when the motor is stopped, « the supply of petrol to the
carburetter will be shut off automatically. A suitable
receptacle may be necesaary to the carburetter to prevent
an overflow of petfrol from the la;tter into the launch whei^
the engine is stopped ; this should havet a narrow neck with a
wire-gauze covering at the mouth with means of draining it.
The exhaust pipe should be efficiently cooled to prevent
danger.
Ignition, — ^An exposed irpark gap is not permitted in the
engine-room, and the leads from the accumulators or generators
to the sparking plugs should be efficiently insulated, well
secured, and protected from moisture, particularly when the
hiffh tension system of electrical ignition is adopted. Ignition
tubes -should not be passed unless oil having a higher flash
point than 78** Fahrenheit is used. If blow lamps are used
for this class of oil, they must be flxed and the flame enclosed.
Motor Compartmentf Veniilationf etc, — ^If the motor, or
petrol tank, is situated below deck, it should be confined
within a separate water-tight and well-ventilated compartment,
in which no stove or other apparatus for containing fire
should be placed. The compartment should have at least
two cowl ventilators, arranged to prevent the accumulation
468 BOABD 07 TRADE KEGUIATIONS FOR MOTOR UiUNCSES.
of oil vapour in tho loweor paaik of the sp^oe^ to wiiiiBh ]^art
one of the Tentilators should extend. Any eaoIoBed space
within which the motor, or tank, is placed should be simi&rly
ventilated exoept in small open launches where louvres^ or
other suitable openings, can do prorlded, in which case end
oowl ventilator may be sufficient. In such a vessel) the- apace
occupied by the motor, petrol tank, eto., should, pTeferiQ>Ij^
be at the after end of the boat, and separated from tiie apace
allotted for the accommodation of pasaengera and orew by
a substantial bulkhead as high as the seats, and water-tight for
at least the lower half ; but, if it is specially desired t«^ place
the motor amidships, or forward, either arrangement may
be allowed, provided a bulkhead, formed in the manner stoied,
is placed between the motor spaoe and the passenger or
prew space.
Tray for Motor.— ^f the vesael is o& wood^ a metal tpay
which can readily be cleaned should be fitted, under the
motor ; if there are flooring boardn, they ^ould be oloeely
fitted, but removable to facilitate eleaning, etc.
MiseeUaneoua, — ^The machinery to be fixed where necessary
to protect persona in the boat. The cylindera to be
hydraulically tested to twice tiieir maximum working pressure,
and the silencer and exhaust pipe to one-fourth of that
applied to the cylinders, t
Boats less than 30 feet long should carry at least one
efficient chemical fluid fire-extinguisher, and a box of sand al
one cubic foot capacity with a suitable scoop. In lai^er
boats or in special circums^^uiGes additional appliaaees naay
be required. Eull direotione should be attached, to Uie extin-
guishers ; and these should be protected, but placed readyv
lor immediate* use. The extinguishing medium should be
harmless to the person.
Motor-launches .
Special certificates are issued for open motor-launches tQ
proceed on short excursions at sea, not more than 3 miles, from
the starting-point ; the boats may then only ply in summer
during daylight and in fine weather. The general require*
menta for passenger ships apply as faJ^ as they are applicable.
Number of Paaaengera.—Tki^ must not exceed the clear area
of the space available in square feet, divided by four. In
measuring the length of the space, that neoessary forward
for anchor and cable and aft for steering arrangements is
to be deducted as well as the overall distance apart of the
bulkheads enclosing the motor spaoe. The brea4th9 are to be
taken between the backs of the side benchet^, or the inside of
the half deck, whichever is least. In any ca^e the number of
passengers sh'^.uld cot exceed the seating accommodatioxi, which
is equid to the total length of fixed seats in feet divided by
BOARD OF TRADE REGULATIONS FOR SHIPS. 469
1*5. The breadth of th« boat should be sufficient to satisfy the
snrveyor that all the passengers can be safely carried. .
Freeboard,-~Whefik the boat is loaded with weights equiva-
lent to 140 lb, for each passenger or member of crew, together
with the complete outfit and necessary fuel, the clear height
of side above water at the lowest point should be not less than
15 inches for boats -20 feet long, or less, 22 inches for vesdels
40 feet long, and proportionately for lengths between 20 and
40 feet. The length is that from side to stem to after side
of sternpost. The clear side is measured from top of covering
board, wash strake, or half deck coaming, whichever is the
highest.
JSeiffht of Sides and Rails, — ^The top of the covering board,
wash strake, or upper e^e of covering should not be less than
30 inches above tne flooring boards in boats 20 feet in lengl^h,
or less, d6 inches in boats 40 feet long or more, and propor-
tionately for lengths between 20 and 40 feet. If necessary
a rajl is to be fitted above the covering boards sufficiejatly
high to comply with the above regulation.
Lif-e-^aving Appliances. -^T^ms^, together with sound signals,
are to be provided. Also two chemical fire-extinguishers^
sand, a compass, anchor and cable, at least three oars and
rowlooiks, boat-hook, painter, heaving line, bailer, and (for
large boats) bilge-pumps.
At least two competent mcn-a seaman and an engine-
driver— should be employed in each boat.
BOARD OF TRADE REOUIATIONS FOU SHIF8.
These certificates are granted as follows : —
1. Foreign-going steamers.
2. Home-trade passenger steamers (i.e. between Great
Britain, Ireland, and within the limits of Hivor
Elbe and Brest).
3. Excursion steamers plying along the coast during
daylight ^nd in fine weather between April 1
and October 31, within the limits stated below
(see C after each port).
4. Steamers plying in partially smooth water (see
B after each port).
5. Steamers plying in smooth water (see A after each
port).
Note. — A "sea-going** vessel includes surveys 1, 2) or 3.
Plytoo Limits assigned to Pokts in the United Kingdom.
Note, — ^Af ter each port (in italics) follow the smooth-water
limdts (denoted by A), the partially smooth limits (denoted
by B), and the excursion limits (denoted by O).
470 BOARD 07 TRADS REGULATIONS FOR SHIPS.
Eitslern Coaat of Scotland.
Cromartj/: (A) In Cromarty Firth bat not below Cromarty,
Inverness: (A) Fort George to Ghanonry Point to Fort William ;
(0) Lossiemouth or Dnnrobin. Banff: (0) Peterhead or Lossie-
mouth. Peterhead: <0) Aberdeen or Banff. Aberdeen: (A)
Inside the Harbour ; (C) Peterhead or Montrose. Uontroae :
(C) Dundee or Aberdeen. Dundee: (A) Dundee to Newpottt
Ferries ; (B) Broughty Castle to Tayport ; (C) Montrose or
Leith. Queensfcrry: (A) Above the Forth Bridflre ; (B) Kirk-
caldy to Portobello ; (C) Berwick -on-Tweed or Dundee. Leith:
(B) Kirkcaldy to Portobello ; (0) Berwick-on-Tweed or Dundeo.
North-Eaatarn Coast of England.
Bertviok-on-Tweed: (A) Spittal Point ; (C) North Berwick
or Newcastle. Amble: (A) Amble Bar; (O) St. Abb's Head or
Middlesbrough. Blyth: (A) Inside the Pier Heads. (C) Berwick-
on-Tweed or Whitby. Newoaatle, North and South Shields: CA)
Inside the Tyne Pier Heads; (G) Berwick-on-Tweed or Scarboroasfh.
Sunderland: (A) Inside the Sunderland Pier H^tds ; (G) Berwick-
on-Tweed or Scarborough. Seahatn: (G) Berwiok-Oii-Tweed or
Scarborough. Hartlepool, East: (A) Hartlepool Bar; (G) Amble
or Bridlington. Hartlepool, West: (C) Amble or Bridlington.
Stockton: (A) Fourth Buoy; (G) Amble or Bridlington. Whitby:
(A) Inside the Whitby Pier Heads ; (C) Bridlington or Newcastle.
Eastern Coast of Enjland.
Scarborough: (C) Newcastle or Hull. Hull: (A) In Winter,
Whitten Ness to Brough ; (B) in winter^ New Holland to Faoll ;
(G) Lynn or Scarborough. (A) In Summer, above Hull and New
Holland ; (B) in Summer, Cleethorpes Pier to Patrlngton Church.
Ooole: (A-C) Same as Hull. Gainsborough, Lincoln, Nottingham,
York: (A-B) Same as Hull ; (G) Spurn Point or Donna Hook.
Orimsby: (B) In Summer, Cleethorpes Pier to Patrington Church ;
(C) Same as Hull. Boston: (A) Inside the New Out ; (O)
Cromer or Hull.
London District.
Wiabeeh: (A) Inside Wisbech Cut; (G) Oromer or HuU.
King*B Lynn: (A) Inside Lynn Cut; (G) Cromer or HuU.
Norwich or Yarmouth: (A) On all the inland navigation from
Norwich to inside the piers at Yarmouth or Lowestoft ; (B) S.W,
Barnard Buoy to the North Cockle Buoy inside the Banks ;
(C) Oromer or Walton-on-the-Naze, Lowestoft: (A-G) Sam*
as Norwich or Yarmouth. Aldeborough and Or ford: (A)
Inside the Rivers Aide and Ore. Harwich or Ipswich: (A)
Inside LandgQard Fort: <B) Walton-on-the-Naze to Landgoaid
Fort ; (C) London or Yarmouth. Maldon: West Mersea Point
to Bradwell Point at the mouth of the River Blackwater.
Bumham-on-the'Crouch: (A) Hollywell Point to Foulness Point;
(B) Glacton Pier to Heme Bay Pier ; (0) Dover or Harwich.
London: (A) Gravesend ; (B) NorUi side — for vessels of
approved construction and of not less than 15 knots speed,
from April 1 to September 30, Girdler Lightship to the North-ea^t
Gunfleet Buoy, and thence to Walton-on-the-Naze ; for other
vessels, Glacton Pier to Heme Bay Pier ; south side — Southend
Pier to the Girdler Lightship and from tbe Girdler Lightship
to Foreness ; (C) Dover or Harwich. Rochester: (A) Shecrness
and Whitstable inside Sheppey ; (B) Glacton Pier to Heme Bay
Pier; (G) Dover or Harwich. Dover: (B) For tenders — within
a radius of two miles from the outer eni of Prince of Wales
Pier, during line weather only ; (C) Newhaven or Sheemess.
BOABB OF TBADB REGULATIONS FOR SHIPS. 471
Folkestone: (0) Newhaven or Sheemess^ Neivhaven: (0) Ports-
mouth or Dover. Littlehampton : (A) Above Littlehampton Pier ;
(O) Poole or •Bye. Langaton and Chichester : (A) From a line
dtawn from the north point of Cumberland Fort to Gunner Point
across the entrance of Langston Harbour to a line drawn from
tbe East Saltern to the Watch House : across the mouth of
Chichester Harbour. Portsmouth: (JO Inside Portsmouth Harbour;
<B) St. Helens and the Needles within the Isle of Wig^t and to
Langston Harbour ; for small launchos not carrying boats — in
«nmmer, a line from Brading Harbour to Langston Harbour
inside the Isle of Wight to Hurst Oastle ; in winter, Spithead ;
(G) Newhaven or Weymouth. Southampton: (A) Galshot' Castle ;
(B-C) Same as Portsmouth. Cowes: CA) Between East and West
Oowes within the Biver Medina. Chriatchuroh: (A) Within
the Bar. Poole: (A) Inside the Harbour ; (C) Weymouth or the
Najb. Weymouth: (B) Portland Harbour ; (C) Portsmouth or
the Start.
South and South-West of England.
Exeter: (A) Inside the Bar ; (G) Weymouth or Plymouth.
Teignmouth? (A) Within the Harbour: (G) Weymouth of
Plymouth. Torquay: (G) Weymouth or Plymouth. Dartmouth:
(A) Biver Dart; (G) Weymouth or Plymouth. Plymouth: (A)
From the inside of Drakes Island to Mount Batten Pier ; the
Biver Yealm within a line from Warren Point to Misery Point; ;
CB) Gawsand to Breakwater and Breakwater to Staddon Pier ;
for tenders to ocean-going steamers — ^Rame Head to Stoke Point,
during fine weather only ; (G) Bzeter or the Lizard. Fowey:
(A.) Inside the Harbour ; (C) Falmouth or Plymouth. Par:
(G) Falmouth or Plymouth. Falmouth: (A) Zoze Point to
Pondennis Point ; (B) in summer, during daylight and in fine
fine weather only — ^Nare Point to St. Anthony's Point ; (G) Start
Point or Penzanoe. Pemanoe: (0) Falmouth or St. Ives.
St. Ives: (C) Padstow or Penzance. Padstow: (A) Padstow
Harbour, above Gun Point and Brae Hill ; (B) Stepper Point to
Trebethorick Point ; (G) St. Ives or Barnstaple, including Lundy
Island. Barnstaple: <A) Inside the Bar ; (G) Padstow or
Bridgwater, including Lundy Island.
South Wales.
Bridgwater: (A) Inside Stert Point; (B) Within the Bar;
CO) Ilfracombe or Swansea. Bristol: (A) Avonmouth Pier to
Wharf Point ; (B) in summer, Barry Dock Pier to Steepholm,
thence to Bream Down ; in winter, for tend<er3 to ocean -going
steamers— ~to King's Roads and not below Walton Bay, during
fine weather only ; (C) Ilfracombe or Swansea. Olouoester: (A)
River Severn or Avon to Sharpness Point, via Gloucester Canal ;
(B) in summer, Barry Dock Pier to Steepholm, thence to Bream
Down ; (0) Watchet or Barry Dock. Chep$tow: (A) River Wye
above Chepstow ; (B) same as Gloucester. Cardiff: (A) Low-
water Pier Head to the Lifeboat House near Penarth Dock
entrance ; (B) same as Gloucester ; (0) Tenby or Ilfracombe.
Barry Dock: (A) Inside Dock ; (B) same as Gloucester ; (G)
Milford or Ilfracombe. Neath : (A) Inside the Bar ; (O) Barn-
staple or Milford. Swansea: (C) Barnstaple or Millwall, including
Lundy Island. Milford: (A) Hubberston Beach to Angle Point ;
(B) South Hook Point to Thorn Island ; (G) Swansea or Cardigan.
f^ishguard: (B) For tenders — within a radius of 3 miles from
the outer end of the breakwater in Fishguard Bay, during fine
weather only ; (G) Barmouth or Tenby, Cardigan : (A) Inside
the Bar; (0) Portmadoo or Milford. Barmouth: (A) Inside
Bftrqxoath Herry ; (G) Cardigan or Bardsey Island.
472 »"tD OF TUSB EHGUIATIONS TOR SHUW.
Idverpool District,
Portmadoc: C^) Inside the Bar Buoy ; (0) Oardigaa or
Carnarvon. Holyhead: (A) Inside the Breakwater | (G) Liverpool
or Portmadoc or round the Island of Anglesea. Carnarvon: <A)
Henai Straits to Aber Menai or Beaumaris ; (B) Meaai Straito,
from Carnarvon Bar to Puffiin Island ; (O) same ac Holyhead.
Conway: (A) Mussel Hill to Tremlyd Point ; (C) same as Holy-
head. Cheater: River Dee, not below Connah's Quay ; (B) inaide
the West Iloyle Bank ; (C) Barrow, Holyhead, or Camanron.
Liverpool: (A) The Rock Iiigrht House ; (B) in «ummer, Fonnfoy
Point to Hilbro' Podnt ; for tenders to ooean-going steamem,
within a radius of 3| miles of Formby Lightrihip, during fine
weather only ; the Bell Buoy and Bar Lightship for tugs ;
(C) Barrow, Holyhead, or Carnarvon. Pretton: (A) Lytham ;
(B) Southport or Blackpool, inside the Banks ; (0) Llcuidudno
or Barrow. Fleetwood: (A) Low Light to Enotend Pier ; within
Fleetwood Harbour, for tugs plying as tenders ; (C) Whitehaven
or Liverpool. Lancatter: (A) Sunderland Point to Chapel Point ;
(C) Whitehaven or Liverpool. Morecamhe Bay: (A) For tenders,
within a radius of 3 miles of Heysham Piers ; (B) in summer,
from Heysham to Sunderland Point and to Itorecambe and
Orange ; (0) Whitehaven or Liverpool. BownsBs: (A) Anywhere
on the Lakes. Barrow: (A) Inside Walney Island ; (O) White-
haven or Liverpool. Douglas: CA) From Battery Pier to Victoria
Pier; (C) round the Island. Whitehaven: (C) Barrow or Carlisle.
Carlisle: (A) Above Port Carlisle ; (B) Southemess to Silloth ;
(C) Whitehaven or Port Whithorn.
Western Coast of Scotland,
Dumfries: (A) Inside Aird Point and Glenhaven Point; (B)
Southemess to Silloth ; (G) Wigtown or Whitehaven. Wigtoxon:
(0) Stranraer or Dumfries. Stranraer: (A) Inside Cairn Ryan ;
(B) Loch Ryan, from Bkinnaird Point to Millcur Point ; (O)
Wigtown or Greenock. Ayr: (A) Inside the Bar; CC) Stranraer
or Glasgow. Ardrosaan: (C) Stranraer or Glasgow. Glasgow:
(A) In winter, Cloch Lighthouse to Dunoon Pier ; in sumnmr,
Bogany Point, Isle of Bate, to Skelmorlie Castle and Ardlamonc
Point, inside the Kyles of Bute ; (B) Skipness to Fairlie Head
round the Island of Bute ; (C) Stranraer to Campbeltown.
Campbeltown: (A) Inside the Harbour, but not outside Davaar
Island ; (C) Glasgow only. Oban: (B) Inside the Island of
Kerrera to Dunstaffnage Point ; (C) Crinan, Tobermory, or Fort
William. BallachuUsh: (A) Within Looh Leven and not eatside
Peter Straits. Vort William: (A) On the Canal to Invemeai ;
(C) Crinan or Tobermory. Kyle: (B) Through Loch Aleh io the
Head of Looh Duich.
Ireland,
Larne: (A) Lame Pier to the Ferry Pier on Inland Hagee.
Belfast: (A) Holy wood to Hacedon Point ; (B) in summer,
Oarrickfergus to Bangor ; for tenders to ocean steamers only,
within a radius of 3 miles from Oarrickfergus, durinff fine
weather; (G) Rathlin Island or Eillough. CarUngford Lough:
(A) Greencastle Point to Greenore ; (0) Drogheda qt Strangford
Lough. Drogheda: (A) Crook Point to Burrow Point ; (C)
Dublin or Warren Point, Oarlingford Lough. Dublin: (A) Inside
the Pier Heads ; (B) in summer, Dalkey Island to Bailey Point ;
(C) Drogheda or Arklow. Wexford: (A) Inside Ely House ;
(B) Raven Point to Rosalare Point ; (0) Arklow or Waterford.
W-aterford: (A) Passage ; (B) in summer, Dunmore to Hook
Point ; in winter, Geneva Barrack to JDuncannon Light : CO)
Wexford or Youghal. Youghal: (A) Ferry Point to Green Park ;
<0) Waterford or Kinsale. Cork: Camden to Carlisle ForUi ;
(B) for tenders to ocean stTean^ers onl^, within a radii^s of
BOABD OF TRADB REGULATIONS FOR 9B£BB. %^Q
8 xai\i6a from Roches Point, duriog fine weather ; (G) Dongarvan
or .Galley Head. Bantru Bag: (A) Inside Bear Island, inBi^e
Whiddy Island, Glengrariff Harbour, Inside Oorrid Point ; (0)
Galley Head or Valencia Harbour. JUmeriak: (A^ J*oynes ; (B)
Soattcry Lighthouse to Carrig Island ; (C) Loop Bead or
Kilmore Head. Galway: (A) Lough Corrib ; (B) Black Rock
Beacon to Eiloolgan Point ; C^) Kilkieran or Liscannor Bays
iqside the Artan Isles. Killary Bay: .(A) Inside Inidkbaraa
Islands. Sligo: (A) The Wes^rn jextreme of Oyster Is]»nd ;
(B; Raghly Point to Black Rock Point ; (C) Donegal or Ballina.
Enniekillen: (A) Lough Brne. Donegal: (A) Inside the Bar ;
CO) Sligo or Rathlin O'Birnie Island. Lough SwUly* (A)
Buncrana to Muckarnish Point ; (B) Dunree Head to Port Salon ;
(C) Portrush or Tory Island. Londonderry: (A) Magilligan Point
to Greenoastle ; (B) For tenders to ocean steamers only, within
a tradiofS of 8 miles from Innishowen Lighthouse during Hat
weather; CO Banooana in Lough Swilly or Rathlin Island.
SXAMINAXIOX OF HULLS.
Plassenger vessels carrying more than twelve passei^fers
are to be surveyed once a year in dry doc^k. Tfee flnrrey
t5on<5orns .the condition of hull and machinery ; th« equip-
ment (ot boats. Hfebnoys, Hghts, signals, compasses, «nd
shelters for deck passengers ; tho Unnts of time and place ;
the nnmber of passengers at varions seasons in each port
available ; tho certificates of the master^ etc. ; the safoty-
valves and fire-hose.
New /steamships are to be surveyed before fhe hifll is
complete, and before the paint and cement are piit o% «s
well as when complete.
An efficient and water-tight engine-room and stoke-Me
bulkhead, as well as a collision water-tight bulkhead, and an
after water-tight compartment to enclose the gtern-tube of
each screw shaft, should be fitted in all sea-going steamers.
The collision bulkhead * should be at least ^ length sbetf t
the stern. It should not be pierced for openings or pipes.
In new ships the foremost bulkhead cfhould extend to the upj)er
(deck; and tiie aftormast to a wateartight fiat, if an^, other-
wise to the upper deck. In awnlne-decked veBsds the
remaining bulkheads may be terminated at the deck b^otw
the upper deck ; otherwise they should extend to the upper
deck. (In sheiter-deck vessels this may be Uie deck hAm
the shelter deck if fairly high afbove water.)
fn eertain smooth or partially «mo64ih water-vesstds, <tfae
above arrangement may be raedtfied in special oases ; in nU new
vessels, however, eKoept steam launefaes plyng ^ ^^ nvrnm
waters^ an efficient collision boflkhead nrast be provided.
In eea-going screw vessels there should be, commmoiiig
fkmn ihio stuffing-box bulkhead, eitber a W.T. tiumd to the
* All bulkhead regulations are now subject to revision. In the report
of the Committee on Subdivision of Bhips (Foreign tmd Homefltesmers) Iftie
«pMlng of bulkheads ia deteztnined t>7 flooding ovnrei that axe constmcled
fQrftoodard ships aad can be es^tcaided to all 4^r4inary vessel^,
474 BOARD OF TRADE REGtJLAT;ONS TOR SHIPS.
after engine-room balkbiead, or a W.T. oompartment of length
12 times the shaft diameter. The fore bulkhead should have
a staffing box round the shaft ; this balkhead to be either
pierced, with a W.T. door capable of being quickly opened and
closed from the upper dectk^ or a W.T. trunk up to the upper
deck should be fitted. AH such work to be of steel or iron.
W.T. doors, which can be opened from the upper deck,
to be fitted to all openings in W.T. bulkheads. It is desirable
that their closing edges be bevelled, and that vertical doors be
used with coal bunkers.
Midship sections of all new vessels are to be submitted
unless the Surveyor considers that the scantUngs are equiralent
to the standard laid down in the Freeboard Tables.
All openings in the weather deck of sea-goin^ ships
dhould have W.rr. covers which oau be expeditiously
shipped. Those over stokeholds, around funnels^ and
engiue-room skylights should have gratings as well as
iron or steel covers. Openings in the main and lower decks
flhould also be fitted with gratings or hatch covers and
tarpaulins, which can render them W.T. The coamings of
all such openings should be of sufficient height and strength.
Side Scuttles, — ^These aud the dead lights should be of
appropriate strength. Cast Iron is unsuitable for scuttles,
except in '' smooth- water " vessels of less than 50 tons net
register ; It may be used for dead lights. Cast steel and
malleable cast iron may be used ; the former material must
be tested by bending If the centre of the freeboard disc is
less than 10 feet below the sill ; the scuttle frames should
stand bending through 20 ** without fracture. Similarly
malleable cast-iron frames and plugs below this height should
stand bending through 15^ and 30^ respectively without
fracture.
Scuttles whose sills ar^ less than 6 inches above the centre
of jdise, or Indian summer line (if any), are subjeot to the
fjollowlng special requirements : Their diameter must not
exceed 10 inches in the oleax. They most be hinged to
.a strong frame of naval brassy gunmetal, or oast steel, the
flange against ship's side being ^" thick ; the securing bolts
to side being ten in number, |" minimum diameter with
one serew in way of hlnge^ or equivalent W.T. arrangemeoit^
The glass should be 1" thick, secured in a strong holder of
gunmetal or naval brass. The deadUghts should be strongly
ribbed ; to be made of a material allowable for the frame ;
tii|T>iT^tnTifi thickness to be |" ; they must be W.T. The glass
holder and deadlight should be secured by five |" naval
brass sequring bolts — ^preferably three for the former-
hinged on similar pins, with plainy square, or hexagonal
nuts. An outer cover or plug cut from f" steel plate to
be made and machined to protect the glass ; this must be
BOARD OF TRADS REGULATIONS FOR SHIPS. 475
■hipped from Inboard, and recessed at least y below ihe>
surmee of the outside plating. The naval brass to be ^
Admiralty oomposition, with a breaking strength of 25 tons
per square inch ; that of the gnmnetol being 14 tons per
square inch. . .
Ifeadlights and Outer Plugs.-^ln addition to those above
specified they must be fitted as follows : In sea-^ing ships
(a) all scuttles below upper deck neater the forward end
than I lenfl^h ; also in forecastles unless open at after end
or eitoated below an awning or shelter deck; (b) all
aouttles below upper deck in spaces for accommodation of
crew ; (o) all scuttles in spaces adopted for stowage of
cargo, fuel, or stores are to have efficient hinged W.T. dead-
lights.
Vessels on foreign-going or winter home-trade service are
to have similar deadlights to all scuttles whose silU are less
than A- tho registered breadth above the deepest load Una
in salt water. In vesseils less than 40 feet or mmje than
00 feet broad this distanoe to be 4 and 9 feet respectively^
All higher scuttles in spaces fitted for acconunodatlon of
passengers, pfficers, etc.; shall, if without deadlight, have
a substantial outer plug stowed in close vicinity to the scuttle.
No pther deadlights or plugs need be fitted with scuttles of
usual ^izes and thickaess. The upper deck is throughout
defined as with reference to bulkheads (p. 473).
New vessels for home summer, or excursion sei^vice, without
cargo, should have deadlights according to (a) and (b) above
All in enffine-room, boiler-room, and coal-bunkers, should also
have deadlights ; tiiose in spaces for passengers, officers, etc.,
are to have one outer plug stowed betweem each two
scuttles.
Vessels not plying outside the partially smooth limits need
have no plugs or deadlights if the glass of the scuttles is
sufficiently tmok.
Misc^ellaneous, — ^The windows of saloons should have
shutters, at least one to every two windows^ or every four
windows in home trade and excursion vessels respectively.
Cast steel for important parts, such as sterns^ rudclers,
steering quadrants, or tillers^, must be tested according to
p. 450. For side scuttles see above ; other castings are
tested for ductility.
Pumps, Sluice Valves, Stbeuinq Geab, etc.
There must be in each compartment, Including the engine*
room, a hand-pump of sufficient size which can be worked
from the upper deck. Their suctions should be at the after
end and on the middle line ; if this latter be impossible
in midship compartments, there should be one pump on
eaoh side. If the pomps are not on the upper look, they
476 80410 OIF TKKDB SBOULAT10MB fOR BBW9.
•faovM be of doted top type, with disdhargie pipos ^wtlH
jribofie ihe deep load line.
In lira of ihe hand-pnaipe, two rotary pomps, msiy he
fitted, oitfiBr ol whieh mmt be capable of tlrawin^ from any
bold or machinery compartment. Alternatively in Tonris
imtiing two leparate W.T. boiler-rooiiu and one W.T. ei^ine-
^voom, two« steam pumps in separate compartments may be
«Bed ; they mast be capable of pomping from any bilgo
imetion, aad tinee latter most be capable of being shixt off
:i]L any compartment flooded. There mast be a aoiDtdrng
tube fitied f lom the upper deck to eaoh oonpartmont. ^Pqies
4H>nnt0ted with pan^ worioed by the engines, are to be
sBvaaged so that eaoh oompKrtment ean be pumped oot
separately by the engines as well as by the deck pumps.
In new ships ike snotioaB must hate mm-Mtam valves
where necessary to prevent water lowing through ihem ibr<BM
« bilged compartment to another ; and the controi^ing calves
must be wowable from the Qpper deck. In maiehinery failgies
^lere should be mud4)0Kes idways aooesaible ; hold and tannd
well enofcions should have a suital^ rose box or strom.
A spare tiller, relieving tackle, etc., should be oarried in
ftH -sea-going steamers. The helmsman should have a dear
¥lew al^^ad. In high speed boato the heel on patting ^eLm
over at full speed should be measored.
A deep-sea Iead4ine of at least 120 faldioms, a lead of at
least 98 A. weight and a saitable reel, together with at least
two hand lead-fines of 25 fathoms eaoh, and leads of at least
'7%. eaoh, should be supplied to all f ordgn-going 'steamfors.
In home-trade steamers two hand lead-lines of 25 ftvthoms
eaoh, and leads of 7 lbs. each, must be supplied.
For a first-class certificate of registry (i.e. twelve -months)
dool>le the number of leads and lines must be supplied.
Bqnfvalent sounding machines are acceptable in lieu.
Xjead lines are usually marked as follows : —
At 2 fathoms a piece of leather split into two strips.
^ 3
„ ^ „ thr^e strips,
white "bunling.
„ 7 ,
,,10
1,
red bunting,
leather wi& a hole.
„13 ,
vl5 >
„17 ,
« 20 .
IJ
9>
»)
a strand
blue bunting.
white bunting. •
red bunting.
with two knots tied in it.
PiEE Hose.
A fire hose adapted for extinguishing fire in any part of
the ship, and capable of being oonaeeted with Uie cfigiues of
the flhip^ er with the donkey engine if it can be worked from
BQAltD C^ TRADB R1ZGULATI0KS ITOlt SHIPS; 4l77
tiie main boiler, should be supplied in all saakgoing ship*.
If metol pipes be fitted, they should hav» valves oontvoUing'
0a dock the water-supply when charging hoses; it shettl«:
be poesibie to reach any part) of the vesseFs holds^ bankers^
or Hving quarters simultaneously with two lengths of hose.
Distress Signals.
. All sea-going paflsengor and emigrant shipai must oasr7^<
(1) Orb gun 3) inches bore or. moroj ot one mortar 6ti inoheau
bore, with twenty-^ur charges (16 oz. of powder each) fooK
4X>reigii-goIng ehips and twelve for others. All aeoessories
neeoasary must be carried^ Alternatirely socket or sound sigjsa&i
loobeta. Jpi equal number may be carried. (2) TWO' deokr
flarea, Imrmng forty mmutes except for daylight eKouratom
veeaeLs. (Z) Twelve rockets or shells, each having 19 ok..
<^ eomposition. Alternatively as with 1. (4) A oontinnaas
s^oAding foj^-aignal apparatus.
Ilk- additioa nix liiebuoy lights burning forty minutesu
(o) ffuiipowder, (6) raebet^ (0) sooket: signals,, (d) flttee*'
and buoy lights, (s) other pyrotechnie ngnals must eaoh' be-
stowed in separate magaaines. The powder should be kepi
in flannel bags contained in a strong copper magaziiKa.
C0MPA89BS.
Each foreign-^oing steamer is to have three compasses and*
binnacles, of which one is to be a standard compass. Vessels in
partially smooth water are to have one compass.
Ma&tea's anp Cb£w Spaces.
The measurements for orew must not: include uselesstspaoes^
e.g. under ladderways or galleya ; the^ tumble hone^ eKovpt:
that more than 5 ft. 6 in./ above the floor must notrbe inoltidea^
The quarters must be strongply buiU, free from odour fh>m
lamp-rooma or paint' stores, or from other effluviumy aoii^.
properly, lighted (wben olear it sbosdd be possible to read a
newspaper with onertiiird of the light cut off). TJhere should,
be oomtplete protection from- weather and sea. Cables led
throii^h <b3 spaees should be cased.
VentilsMtidn should be complete and thorough^ . ivith twa^
ventilators (one inlet, one outlet) >to eaoh spaoo, one of wfaiohk*
extends to the lower edge of beams-. The^ topst of these should
be fitted preferably with, revolving oowls (wfaioh may. be>
portable), as high as the bulwarks ; mushroom ventilators^
minimum height 80 iaehes or height of bulwarks, may be-
fitted, but are not. desirable except for deck faonasB. When
practieable all o&bina should have a.' cowl or swaa-neiok
vttotilatotr. SkTiighti, souttles, companions, and doors, althenghi
frequently uaeiui as aozUiaries, cannot be accepted < as effioieiiilh
ventilators in aU weathers^ Privies should be suitably
ventilated, la vessels liable to be scurt to the Tropio$,|iroviBioi»'
478 BOAKD OF TRADE BEGULATIONS FOR SHZP8.
•honld be made for Introduoing a windsail 18 inches or more
diameier over each apaoe; thia may be lit a akylk^ht or a hatch.
Store funnels most have outlets mstinct from tne ventilators.
All iron decks in crew spaces must be sheathed with wood
at least 2) inches thick, properly laid and caulked ; no portion
ot a bunk may be placed directly over an iron fitting pre-
venting the complete sheathing of the deck. Lining under
decks or at sides is undesirable ; but bunk boards 18 inches
high should be placed to protect occupants from condensation
at the sides.
Spaces should be 5 ft. 6 in. hiffh in the clear to underside
of beams ; the lowest bunk mtist be 12 inches above the floor,
and the bottonus of the bunks must be 2ffc. Gin. from oni^
another and from the deck. Their length must be at least
6 feet.
Space must be drained by pipes, provided with plugs and
lanyards. Wood bulkheads to be tongued and grooved and
made of weU-seasoned material ; against tiie galley and privies
it should be doubled with felt between ; agaiiut a aonkey
boiler space or the engine and boiler casings a wood lining
with 3 in. space filled with non-conducting material to be
fitted.
There should be one privy for every ten men, exclusive
of officers ; if over 100 men add 4 per cent for each addi-
tional 100 or part of 100. With less than twenty men^
including officers, two privies are sufficient ; with less thap
ten, one only. With trough closets for Lascars, a linear
18 inches clear opening is equivalent to one privy. Privies
should be efficient, and well separated from crew spaces ; if
they open directly into a crew or officer's space, no tonnage
deduction can be claimed.
To obtain the number of seamen and apprentices, measure
the clear area available, excluding useless spaces (see para-
gri^h 1) and encumbrances such as hatchways, trunks, etc.
The cubic capacity is equal to the clear area multiplied by
the height from deck to deck at the middle line. There must
always be bunks and hammock fittings equal in number to the
accomodation certified, but they are not deducted as encum-
brances in the space except in cabins.
In all ships the number of men in each space must nqt
exceed one: per 12 square feet clear area, and one per 72 cubic
feet capacity, including only such spaces as are used for
sleeping. In new ships (except fishing boats and ships of
not more than 300 tons net), there must also be sufficient mess
room, bathroom, and wash-place accommodation to bring up
the total space (inclusive of these) to 15 square feet aaid 120
cubic feet per man. The latter regulation does not apply
to Lascars, out in cabins there should be IS square feet per
man exclusive of the bunk.
BOARD OF TRADE REGULATIONS FOR SHIPS.
Pas8eno£r Accommodation.
479
Gfifierdl,
Foreign and home-trado steamers to be properly lighted-
and ventilated by day and night, with proper meanB of aiocesaf
wherever pasdengers are accommodated. Spaces not naturally
lighted mast be lighted dectrically, not by oil lampa. There
should be a good air supply in bad weather under closed,
hatches. jBlectric Lighting should be arranged to minimize
risk pt fire, with the source sufficiently high to prevent the
probability pf the light being extinguished after a slight
accident. The lamp-room, if near the passengers' quartev^
^ould be separated by a fireproof bulkhead. Decks under
and above the passengers' quarters must be sheathed, if ipf
metal (except from May 1 to August 31) ; all floors b^ng
properly laid and caulked. In new foreign vessels the over-*
flow pipes to drinking tanks must not discharge into the
bUgeSy ^nd the air pipes must be led to the upper deck.
Bails ^nd stanchions must be 3 ft. 6 in. high, and no^
more ,than 9 in. apart unless provided with strong nettinflf.
The freeing ports of close bulwarks (which must also be
8 ft. 6 in. high) should be protected by grids. In vessela
plying in smooth or partially smooth water the height of
rails or bulwarks (top of rail above top of deck, not including
waterway) laihould be as follows : —
Registered Ijength of
y^H^l in Feet.
Under
60.
60 to
70.
70 to
90.
90 to
180.
180 to
170.
170
and
•
Height of
raU
Partially
smooth limits
ft. in.
} 2 9
ft. in.
2 10
ft. in.
8 0
ft. in.
8 2
ft. in.
8 8
ft. in.
8 4
Smooth water
Umits
} 2 e
2 8
2 10
8 0
8 2
8 8
Pasaengera in Foreign-going Steamen,
\ The weather deck, and the surface of the poop, forecastle,
and bridge deck, are never to be included in the measurements
for passengers ; nor are the poop, round house, or deck house,
unless they form part of the permanent structure of the
vessel.
Foreign-going steamships carrying more than twelve
passengers are to be measured as follows : —
Saloon or lat Class, and Second Cla$s, — ^The number of fixed
berths or sofas that are fitted determine the number of
passengers to be allowed.
480 BOARD OF TRADE RtGULAf tOWS "FORI SMft.
Suffioient light and Tenitlaitiim and a reasonable amonnt
of floor space most be provided.
Zrd Cla$s. — ^The number may be determined in like manner
if berths are fitted ; if n«t, the net area of the deek, multi-
l^ed b; ihe height between decks aad the product divided
D^ 72| gives the number to be allowed. The breadth of the
deek is taken inside the water-waj, or at the greatest tumble-
htooM of the side, if there is any. The height betwacoi deeka
nuiBi not be less than 6 feet.
When cargo, stores-, etc., are carried in the mioe measujred
tot pasaengersy one passenger is to be deducted for every 12
ipperfioial' feet of deok space so ooonpied.
Passengsrs in MomS'Trad^ 8ea-going Steamers.
Fore-cabin paasengers InchidQ all paasenprers except those
entered as after-K»bin or saloon passengem in the waybill*
In jMw- veitiils closets must be provided on tiie following
scale.: —
133
2
200
325
450
5
1
575
700
825
^Cleaete . • • •
1
3
4
1
*
6
7
3
8
'TTHnals or extra
closets
2
4
If, however, two or more classes of passengers are taken,
each class need have only six closets and two urinals. Two
clos^, at least; must always, be provided. A fair pro^rtion
of the closets must be allotted solely to women and children,
and reasonable privacy afforded. Closets must be dean,
well lighted, ventilated, drained, and protected from weather
and sea. Additional earth-closets may be temporarily
installed, but for one month only^ or less.
The number of passengers to be carried in the after-cabins,
fore-cabins, state-rooms, etc., is determined by the number
ot, berths, or sofa% properly constructed for sleeping berths,
provided there are 72 cubic feet of space for each passenger
berthed in each state-room or cabin. The floor 0(f state-
looms 1% never to be measured, but so much of the floor
iDf the after-saloon as is not covered by tables, etc., may bei
indoded.
For the total number of cabdn passengers so accommodated
below there shall be reserved on deck, or provided on a bridge,
deo^ or other suitable place, promenade, or airing space at the
rate of 3 square feet per passenger, and this spaoe SAall not
BOARD OF TRADE REGULATIONS FOR SHIPS. 481
b© counted or included in the area available for deck or any
otheir passenffers.
To obtain the number of second-class or steerage
passengers, measure the unencumbered floor-space of the
dining saloon (it any), and the floor-space of shelters toi
deck passengers ; and divide the number of square feet by
tBuree. For compartments neither dining saloons nor deck
shelters, the number is that of the fixed beds or sofas
therein, Jbut there must be 72 cubid feet for each persion.
The sofias, ^to.^ in the saloons already measured muiit not
be included in the above number. If there are three clasaes
of passengers, airing space at 3 square feet per perskon
must also be reserved for the second-class passengers.
In general the main deck, the deck beneath^ and the raised
quarter-deck (if ^ny) may be measured ; also the poop or
bridge house, promenade deck, etc., over up to one-quarter
the length, provided the stability is satisfactory and that there
are bulwarks or rails 3) ft. high (with weather cloths) fitted
as specified above.
The main deck shouM, if necessary, be protected with
close Jsulwarks 4 feet high.
For yoyages not exceeding ten hours the whole of the
clear upper surface of the psomenade deck, poop^ etc., may be
included. 'if
The number of deck passengers is obtained by dividing
the clear area in square feet by nine. In measuring^
take the breadths from the point of waterway, rail
or covering board which is the most inboard ; deduct all'
incumbrances, sponsona (in paddle steamers) and houses'
over, jGtreas between rail and deck-house less than 2 ft. 6 in.
wide, forecastle deck (for the foremost one^eighth length if
joined to deck amidships), lower hold or cargo space, portions
of deck overhanging side or occasionally used for navigation,
decks carried pn stanchions or extensions of frames no(fe
plated. In saloon steamers the tops of saloons or bridge!
decks, if sufficiently strong and not carried on stanchions^'
may be included. Not more than three decks in all to h6
measured except in special cases.
The total number of passengers, other than saloon or first-
class, must not exceed six times the number (at 9 square feet
per person) that can be sheltered.
The total number of ^ passengers must never exceed the
gross tonnage of the vessel.
In well-decked vessels, the space between topgallant
forecastle and raised quarter-deck, etc.^ must not be included
unless sufficiently high, and having freeing ports on each side
with areas from 9i to 12i square feet when from 30 to
60 feet Jong, and 1 square foot extra for each additional 6 f eejtj
length fyf bulwarks.
li
482 BOABD OF TRADE REGULATIONS VOR SHIPS.
Cattle on the open deck most be separated from paaaengen
by partitions, not necessarily close, with efficient wash-boaSrds.
If under cover they must be separated by a movable close
bulkhead from deck to deck. If below, the compartmemts
and their ventilation most be completely sepaxated fltoim
those for the passengers. Deduct one passenger for each
square yard of passenger space oooupied by eattle or cargo.
Passengers' in Excursion Steamers.
For steamers used in excursions the rules for calculating
the number of passengers are the same as in sea-going home-
trade steamerS) except that if application is made for aji
excursion certificate &r short distances along the coast during
daylight, the number, originally calculated at 9 superficial
feet to each passenger, shonld it exceed the gross tonnage
of the vessel, need not be diminished so as to bring it down
to that number.
Windows in saloon houses should have efficient portable
shntters.
Passengers in Steamers plying in partially smooth water,
Qiie measurements are to be made in the same manner as in
home-trade sea-going steamers, except that one saloon only
is to be included.
There will be no distinction between lore- and after-cabin
passengers.
Divide the number of superficial feet on deck, obtained as
before, by six, and the clear space in the saloon by nine,*
and the sum of these quotients will be the number of
passengers allowed.
In the last-mentioned class of steamers one and a half
passengers per square yard of the space measured for
passengers which is occupied by cattle, cargo, etc., to be
deducted.
Between October 31 and April 1 the number of passengers
which, according to the preceding rules, is allowed to be
earried other than in cabins Or saloons during summer is to
be reduced by one- third.
These vessels are to be provided with a suitable anchor and
cable, and a compass properly adjusted, and suitable life-
saving appliances.
' Passengers in Steamers plying in smooth water.
Divide the number of superficial feet on deck, obtained as
before, by three,* and the clear space in the saloon or on
bridge decks, etc., by nine, and the sum of these quotients
is the number of passengers allowed.
* In new vessels the divisor may be nine for small rooms on upper deck
'ip to 90 square feet area, or six up to 270 square feet in smooth water.
STRENGTH OF BULKHEADS.
483
Thmee passengers are to be dedncted for every square yard
of space measured for passengers occupied by cattle, c^rgo
etc.
No reduction to be made in winter months.
These vessels are to have a suitable anchor and cable.
Open Boats or Launches. <
The number of passengers must not exceed the seating
accommodation equal to the total length of fixed seats,
inoluding thwarts, in feet divided by 1.5. The stabilitQr
must be tested, unless obviously ample ; and, if satisfactory,
additional passengers standing may be allowed. The height
of gunwale should be in accordance with the table, page 479.
STBEKOTH OF BULKHEADS
{Recommended in the First Report of the Committee on the
Subdivision of Shi^s),
{See tables on pp. 484, 485.) *
Attachments fob Stiffenkus (dimensions are in inches). |
Type and Depth of Btiflener,
•
Bracket Attachments.
Lug
Attaohments.
Thickness
of Bracket.
Width
of Flange.
Rivets in
each Arm.
Rivets in
Lugs.
No.
Diom.
No.
2
3
4
6
6
8
8
10
Diam
Angles up to 6" .
Bulb angles 7" . .
ft"
»» »»•'••
1^'
Channels* 12" x 31"* !
„ 16" X 4" .
Plates 16" Vith 3" ang
91"
les
•34
•44
•44
•44
•44
•60
•60
2i
3
6
7
0
10
14*
16*
i
i
i
1
i
i
i
1
i
S
i
i
i
i
i
Note. — Distance from heel of boundary bar to ends of bracket
arms = three times depth of stiffener ; if more than 24", bracket,
should be flanged. Either bracket or lug attachments in acc(nd-'
aiioe with the table can be used.
Thickness of Bulkhead Plating.
D « Depth at middle liiie fr<Mn bulkhead deck to lower edge of
plate in feet.
t *= Thickness in inches.
*.In two rows.
484
STRSNOTH OT BULKHEADS.
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STRENGTH OF BULKHEADS.
485
486 STRENGTH OF BULKHEADS.
t (Biitteutn spaeed 2(f) (stiflenera spaced 86^
np to op to
•28 12 7
•34 24 17-5
•40 36 28
•46 48 88-5
•52 60 49
•68^ — 69-5
General Notes on W.T. Bulkheads, etc.
(Becommended in the First Report of the Committee on the
Subdivision of Ships.)
The hulkhead deck is the uppennost oontinuous dock to
whioh all trans versid bulkheads are carried.
No W.T. compartment shall exceed 92 feet in length
nor be regarded as part of the W.T. subdivision (exoept in
the p6aks of ships less than 200 feet in length) if lesa than
10 feet in> length.
Side scuttles below » deck which is less than 7 feetr above
the L.W.L. shall be permanently fixed, except special shuttles
provided with metal shutters, and kept closed and looked
at sea.
The minimum distance of an inner skin from the outer skin
should be 2 feet plus 2 per cent of the moulded beam.
The lowejt strake of a bulkhead at the end of a stokehold
or bunker space should be 36 in. high and 0.1 in. thicker
than ffiven by the table. In other cases the lowest strake
should be '04 m. thicker ; limber plates being '1 in. thicker.
Boundary angles should be *1 in. thicker than the bu&head*
The en(k of the stiffenera should be connected by either
brackets or lugs to efficient horizontal plating ; the- lower
brackets should extend over the adjacent floor (which should
be solid) ; the upper brackets should be connected to angles
which extend over the adjacent beam space.
The rivets In seams, end connexions;, and boundaty bcu«
should be spaced 4|^ diameters, exeept in the shell flange of
boundaries where 5 diameter spacing is permissible. The
bouiidiiries should be double riveted when more than t4 feet
below bulkhead deck ; over 35 feet, the vertical butts slionld
also be double riveted. The stiff eners rivets should be spaced
7 diameters, except for 15 per cent of the length 9^ eaci
end^ when spacing should be 4 diameters if lu^ attachments
are need. Collision bulkheads should have stiffeners 24 in.
apart ; but the scantlings should be based on 30 in. spaeing.
W.T. decks and trunks should have the same strength
as required for bulkheads in the same position.
Bulkheads should be hoee-tested at 30 Ib./in.^ pressure ;
fore and after peaks, inner skins and double bottoms sh<mld
be filled with a head up to the bulkhead deck.
REGULATIONS FOR PREVENTING COLLISIONS AT SEA. 487
The double bottom should extend from machinery space to
f orepeak bulkhead in ships 200 feet to 260 feet long ; outside
machinery space to fore and after peaks in ships 250 feet to
300 feet long ; also amidships in ships over 300 feet long, when it
should extend to the bilges.
Bilge Sxiction Pipes. — Hiese should have a minimum
diameter in inches of 1 + ^ V (l x (b+d)) for main suctions, and
1 + -/ (Z X (b + d) -r 1500) for branches to cargo and machinery
spaces, where L, b, D are the principal dimensions, and Z the
length of compartment in feet. In no case should these be less
than 2Jin. and 2 in. respectively.
INTEBlfATIOirAIi BEGULATIOKS FOB FUEVElfTINa'
COLLISIONS AT SEA.
Lights.
To be carried from sunset to sunrise. ' Visible * applies to
a dark clear night.
1 . (a) A steam vease^l under way carries on the foremast (or
equivalent position)^, at a height abovQ the hull equal to the
breadth of the vessel (hut not less than 20 feet or over
40 feet), a white light, visible 5 miles, showing from. ri^ht[
ahead to 2 points abaft the beam on each side.
(fi) Also a green light on the starboard side and a wh|t^
light on the port side, each visible 2 miles and showing from
right ahead to 2 points abaft the beam. A screen projecting
at least 3 feet forward to be fitted to previaut those lights f ron^
Grossing the bow.*
(c) A white light may be carried in addition to the first*
described. Of the^ one to be at least 15 feet lower than
the other, and at a greater distance before It.
2. A steam vessel towing shall carry two white lights
similar to 1 (a), at least 6 feet apart in the same vertical.
If the length of tow from atern to stern exceed 600 feet, a third
light above or below shall be carried. A small white
light may be placed aft, bat it shall nob be visible beforo
the beam.
3. (a) A vessel not under control shall carry in lieu of
1, (a) two red lights (two 2 feet black balls by day) at leasl^
6 feet apart in the same vertical,, and visible 2 miles all round
the horizon.
(h) A vessel employed on telegraph cable work shall carry
in lieu three suioh lightaj, 6 feet apart, the central white an^f-
ihe others red. By daj three 2 feet shapes, the central
diamond and white and the others globular and red.
(c) In both the above, the side lights should be carried
(mly when under way.
4. A sailing vessel under way or any vessel that is being
towed shall carry the side lights only.
* The edge of the screen should be in a fore and aft line with the inner
edge of wick, the athwartship width of which should be — ^maximum 2*,
minimum 1" (paraffin) or If (colza).
488 REGULATIONS FOR PREVENTING COLLISIONS AT SEA.
5. In small vessels under way during bad weather, the
side lights need not be fixed, bat may be exhibited on
approach of other vessels.
6. Steam vessels of gross tonnage less than forty, and
sailing or rowing boats less than twenty, need not carry
the above lights, bat must have in lieu : —
(a) Steam Vessels,— Nine feet above the gunwale (or less
in small steamboats) in front of the funnel a white light
according to 1 (a)', but visible 2 miles. Also green and
red sidelights according to 1 (6) ; or a combined red and
green lantern at least 3 feet below the white light, visible
1 mile.
(6) Vessels under Oars or Sails.— A portable combined
red and green lantern as above. Rowing boats under oars or
sail shall have a portable white light.
7. Vessels on pilotage duty shall show only a white all-
roand masthead light ; together with a flare-up light showing
every fifteen minutes or less. The red and green side-ligh^
shall be shown only On the approach of other vessels.
A vessel exclusively employed by licensed pilots shall show
her side-lights, and in addition an all-round red light, visible
2 miles, 8 feet below her masthead light. At anchor the side-
lights shall be omitted.
8. (a) Open boats, when fishing, shall carry one all-round
white light ; if outlying tackle extends more than 150 feet
horizontally, on approach of other vessels a second white light
3 feet below the first and 5 feet away in the direction of the
tackle shall be shown.
(6) Vessels, other than open boats, when fishing with drift
nets or lines shall carry two white lights visible 3 miles. They
shall be from 6 test to 15 feet apart vertically, and 5 feet
to 10 feet horizontally, the lower being in the direction of
the nets.
(c) Trawlers and vessels with dredge nets, if steam, shall
carry m lieu of 1 (a) a tricoloured lantern showing white
from right ahead to two points on each bow ; also green on
starboard side and red on port side to two points abaft the
beam ; from 6 to 12 feet below this a white all-round light.
If sailing, they shall carry a white all-round light, and also
show on approach of other vessels a white flare-up light.
All these lights to be visible 2 miles.
(^d) All fishing vessels or boats when under way shall
exhibit the usual lights in lieu of the special ones above
described. At anchor the light specified in (10) should be
shown, and, in addition, if attached to a net or similar gear,
an addituonal white light as described in 8 (a) shall be
exhibited on approach' of other vessels. They may also use
a fiare-up light and use working lights as desired. In day-
time, all vessels fishing with nets^ lines, or trawls shall
RFfilTI.ATJONS FOB. PREVRNTINO COLLISIONS AT SEA. 489
display a basket or oimilar signal whether at anchor or
under wayr
9. A vessel overtaken by another shall show a white or
flare-up light from the stern. If fixed it should be visible
1 mile, showing from right aft through six points on each side,
and at about ike same level as the side-lights.
10. At anchor or aground a vessel under 150 feet in
length shall show a white all-round light visible 1 mile, not
higher than 20 feet above the hull. Over 150 feet this light
shall be from 20 to 40 feet high, and a similar light shall
be placed at the stern at least 15 feet b^ow the forward
Ught. ' \
11. A steam vessel under sail with funnel up shall carry
forward in daytime a black ball 2 feet diameter
Fog Signals.
These consbt of a whistle or siren in steam vessels, and
a mechanical fog-horn in vessels sailing or towed ; also a bell
in each case. In tog, mist, falling snow, or heavy rainstorms,
day or night, the Allowing signals shall be given : —
(a) Steam vessel und«r way — a prolonged blast erory
two minutes. '
(6) As above, but stopped — ^two prolonged blasts every
two minutes, with one second interval.
(o) Sailing vessel under way — every minute, on starboard
tack one blast, on port tack two blasts, with the wind abaft
the beam three blasts in succession.
(d) A vessel at anchor — every minute, ring bell rapidly
daring five seconds. ^
(e) A vessel towing or not under command (in lieu of
above) — every r'two minutes sound one prolonged blast;
followed bv two short blasts.
(/) Fishing vessels with lines out — every minute one blast,
followed by ringing the bell.
Sailing vessels and boats of less than 20 tons gross tonnage
are exempted, but they roust make some efficient sound signal
every minute.
Stkeuinq and Sailing Rules.
1. When two sailing vessels are approaching-^
(a) The one running free shall give way to the one close-
hauled.
(6) The one close-hauled on port tack shall give way to
the one close-hauled on the starboard tack.
(0) If both are free, the one having the wind on the port
aide shall ffive way. '
(d) If both are free, the one which is to windward shall
give way to the one which is leeward.
(^) One which has the wind aft shall give way to any other.
490 TONNAOF.
2. Wh«n two steam vessels are iiK^eiiiigf oearly and on,
each shall alter her course to starboard.
8. When two steam vessels are crossing, the one which fans
the othev on her starboard side shall give way.
4. A steam vessel shall ^ve way to a sailing vessel.
6. Any vessel overtaking another, i.e. coming ap from
a diroction which is at any moment more than two points abaft
the beam of the other, shall give way to the other.
6. In narrow channels each vessel shall keep to her atar-
boazd sido as far as practicable.
7. Sailing vessels shall give way to vessels engaged in
fishing.
8« A steam vessel shall indicate her course to another vessel
in sight by one short blast with whistle or syren on taming
jjto starboard, two short blasts on turning to port, three
short blasts on putting engines astern.
9. A vessel in distress shall signal together or separately
as follows :— •
By day (a) a gun fired every minute, (5) the code signal
NC, (&) a square flag having a ball beneath it, (d) a oon-
tinnous sounding with a fog signal. By night (a) and (d)
as before ; but (6) flares on the vessel, (c) rockets or shells
throwing stars at short intervali.
TOKNAOS.
BfiGISXEB TONNAOS.
The gross tonnage of a ship expresses her internal cubical
capacity in tons of 100 cuino feet each. It is calculated as
indicated below ; but the gross under-deck tonnage (i.e.
exclusiive of that dne to erections) may be found approxi-
mately by the following formula :—*
ij = i£e length at load-line fnmi front of stem to back
of sternpost.
B = the breadth extreme to ouiatde of plating.
Das the depth from top of upper deck amidships to top
of keeL _.
Gross tonnage under deck = — yqq — ^*
Value of C.
Passenger steamers of high speed and sailing
ships . . '6 to 65
ipasseuger and cargo steamers . « . . *7 to *72
Cargo steamers and oil- tank steamers • . '72 to '8
To calculate the Choss Tonnage.
The tonnage deck is the upper deck in all vessels under
three decks, in all other vessels the second deck from below.
Measurements to be expressed in feet and dpclmals
of a foot.
TONNAGE. 401
The lei^ik fof register tonnag^e is takeft from insufe of
plaak at aiem to inside of midship stecn timber, or plaal:
there> as the case m&j he, and is taken on the tonnagie deck ;
the Itfitgth flo taken (having made deductions fw the sake
of stem and stern, if any, in the thichnau of the deck, aad
one-third of the round of the beam) is iR>> be dabrided iJiitt
the presocibed number of equal part8> aocording to the lengthy
as ix>llows ; — . ^
Kot exceeding 50 feet and under . ^ . .4
Exceeding 50 feet and not exceeding 120* feel; . 6
Exceeding 120 feet and not exceeding 180 feet .. ' 8
Exceeding 180 feet and not exceeding 225 feet . 10
Exceeding 225 feet 12
la the case of a break in the double bottom- for wateac
ballast^ take the length in parts bet<ween the bnee^Sy qsiftg
the aibove rule.
Transverae sections are then measured at each of the
points of division, as follows :—
The total depths of the transverse sections are. measured
from the under side of the tonnage deck to the upptegr side
of floor timber (or inner bottom plating) at the inside- of
the limber strake, after deducting average thickness of
ceiling and one-third of the round of the beam. The depths
so taken are to be divided into- five equal parts^ if midship
depth does not exceed 16 feet ; otherwise into seven
equal parts.
The breadtiis are measured horizontally at the points of
division,, aad also at the upper and lower points of each
depth, each measurement extending to the average thickness
of that part of the celling which is between ii!ie points of
measurement.
The areas of the transverse sections are theai computed
down to the lowest point of division by Simpson's first xida
(p. 44) ; the area below is then calculated by subdividing
the lowest interval into f oi^r equal parts and applying the
same rule to the additional horizontal breadths thus obtained;
the sum of the two parts is the whole area of the section. The
capacity of the ship is computed by the same rule (Bul^
II, p. 54) — ^that is, the areas are treated as the ordinates of
^ new curve of the same length as the vessel ; anil the area
of that new curve, found by Simpson's first rule, will be
the capacity of the vessel in cubic feet, which being dividqi
by 100 gives the gross tonnage imder tounage deck.
If the ship has a deck or decks above the tonnage deck,
the volume of each 'tween deck space is computed by a similar
method using the same number of ordinates in the lengthy the
length being measured at mid-height.
492 TONKAOB.
In thiiM where ibe nnder-deck tonaaage eannot be obtained
by direct measurement it ehoold be estimated as follows :
Measure extreme length on highest deck, extreme breadth,
and corresponding girth from height of npper-deok (as
measored by a chain nnder the keel). Add half the girth to
hsjf the breadth ; equare the sum, and multiply it by the
leivth. The product multiplied by *0017 (for wood ships)
ana '0018 (for iron ships) shall be deemed the tonnage of the
ship, sabjeot to the osnal deductions and additions.
The capacity of the poop, deck-house, forecastle, break, or
any other permanent closed-in space available for cargo or
stores, or for the accommodation of passengers and crew, shall
be similarly obtained, and included in the rross tonnage.
?nie foUowioff spaces, however, are exempted from the above
rule : (1) Shelter-deck spaces, with permanent middle-line
deck openings at least 4 feet long and of the width of the
after cargo hatch, (2) shelters for deck passengers on short
voyages, (3) closed- in spaces solely for machinery, (4) wheel-
house, (6) cook-house and bakeries, (6) condenser space,
(7) w.c. s for officers and crew.
To ealeulate the Begister or Net Tonnage.
The deductions allowable from the gross tonnage are as
follows, no deduction beii^ permitted for any 8j)aoe that has
not already been measiued and included m the gross
tonnage : —
(a) Propellins space. This may include spaces actually
occupied by engines and boilers and closed-in spaces over for
admitting light and air. Also shaft trunks in screw ships.
Bxclude store-rooms and cabins.
Divide the volume in cubic feet by 100. '(X) If in screw
steamers this be over 13o/o and under 20o/o of the gross tonnage,
deduct 82o/o of the gross tonnage ; otherwlBe deduct tonnage
of space multiplied by 1*75. (2") If in paddle steamers this
be over 20o/o and nnder 30^/0 of the gross tonnage, deduct 37 <^'o
of the gross tonnage ; otherwise deduct tonnage of space
multiplied by 1*5. In all new ships, except tugs, the maximum
deduction for machinery is limited to 55o/o of the gross
tonnage diminished by the further allowance detailed below
(crew space, etc.).
(6) Master's and crew's spaces. (In warships only a small
proportion of this is deducted.)
(c) Spaces for working helm, capstan, anchor gear,. or for
keeping charts, signals, and other gear for navigation, and
boatswain's stores.
id) Space for donkey-engine and boiler, if connected to
main pumps»
(jb) WaW ballast space (other than double bottom).
if) SaU-ioom, limited to 2}o/o of the gross tonnage, in
ships wholly propelled by sails.
TONNAGE. 4^
Note, — ^All such spaces must be plainly marked, and
exclusively resenred for the object indicated. Double-bottom
spaces available for water-ballast only (not for fuel, stores,
or cargo) are liot included in the gross tonnage, and no
deduction is necessary. In open boats the volume is to ba
meaAured to the upper edge of the uppeir strake at each
aeotloB.
Buez Canal Tonnage.
This is determined in a manner similar to that used for
Register Tonnage in Great Britain (see above) ; but there
are the following differences : —
1. All 'permanently enclosed spaces are included in the
flro88 tonnage. Unenclosed shelters are excluded ; also the
fore end of the forecastle extended from the stem over
a length equal to ^ the length of ship ; similarly in the poop
from riffht aft over ^ length of snip ; also such length of
the brioge as is equal to the length of machinery space deck
openings. Shelter-deck space is included, except immediately
opposite any openings in the side.
2. For tne net tonnage ^e following deductions are made
j(mazunnm 5o/o of gross tonnage). Spaces (including mess-
rooms, cook-houses, bathrooms, and latrines) for ship's
officers and crew ; spaces for working helm, capstan, and
anchor gear, and for keeping gear for navigation. Kd
deduction is made for accommodation wholly or partly for
passengers, captain, purser, stewards, etc., or for peak
ballast tanJcs.
In steamships the following additional deductions arc made
(limited to 60<>/o of the gross tonnage, except in tugs). Spaces
occupied by enj§pines, boiler, coal-bunkers, shaft trunks, enffind
and Doiler casings between decks (German rule). For snips
with moveable coal-bunkers (or alternatively in any case) the
deduotion allowed is equal to the machinery space exclusive
of bunkers x 1} in paddle steamers or If in screw steamers.
(Danube rule.)
Tannaige and Displaoement,
The net tonnage (Suez Canal) of steamers is therefore at
least 450/p of the gross tonnage, and in high-speed ships is
esiactly that ratio. The net tonnages under British and Suez
Canal rules are now approximately equal. The ratio of the
net tonnage to the displacement in tons b approximately :
battleships 35 ^/o, light and heavy cruisers 30o/o, destroyers
400/0, fast passenger steamers 30o/o, coasters 25<yo, sailing
ships 40o/o. The gross tonnage in many vessels is about one-
half the load displacement, rather less in large ships.
Pananui Canal,
The tonnage for vessels passing through this canal is
estimated on a basis generally similar to that for Suez Canal ;
the following are the principal differences : —
491 TONNAGE.
Dottble-bottom space* for oil fuel and feed>w&ier are
inoloded in the gro« tonnage, but dednoied for the nefc tonnage
independentiy of the aUowanoe for propelline powar.
Erections are included as in Sues regokiiaons, bat the
deductions allowed are slightly fewer.
Fro^lling^wer space is allowed for as in Snea
regulations. Deductions are allowed for master's and ocew'a
accommodation, navigating spaces, and peak ballast-fcanks as
in British roles ; there is no percMttage limitetsDn.
Habkinq of Ship.
Every British ship to be permanently marked as 'follows: —
(a) Her name on each bow, and her name and port of
registry on the stern ; letters <{o be at least 4 inches long,
and either light on a dark ground or the converse.
(6) Her official number and her registered tonnage to be
ont on her main beam. '
(c) On each side of the stem and storn'pK>3t a scale of
feet denoting the draught of water. Lefa'ters to be 6 inches
hj^h, the lower line denoting the draught ; to be cut in and
painted white or yellow on a dark ground.
Bead-wgioht Oaaooes.
To estimate approximately the dead-weight cargo which
a ship can safely carry on an average length of voyage.
Bole.— Deduct the tonnage of the space for passenger
accommodation from the net register tonnage, and multiply
by the factor given below.
Type of Vessel. F^K^r (Sir W, H, White),
Iron and steel sailing ships . • 1*4
Cargo steamers .... IJ
Passenger steamers . . . 1| down to about 1
(fastest vessels).
BmLDER'8 Tonnage, ob Old Measurement Tostkaqe
(obsolete) .
To compute the Builder's Tonnage.
Rule.— Measure the length of the vessel along the rabbet of
the keel from the back of the main stem-post to a perpen-
dicular line let faJl from the fore-part of the main stem under
the bowsprit ; measure also the extreme breadth to the outside
planking, exclusive of doubling planks. Three-fifths of thai
breadth is to be subtracted from the length ; the remaindetr
is called the length of keel for tonnage. Multiply the length
of keel for tonnage by the breadth, that product by the half-
breadth, and- divide by 94 ; the quotient will be the tonnage.
If L = length, B = breadth, then
^ ^ ,- (l-#b) X b x4b
Tonnage (B.O.M.) =^^ ^-~ ^.
LIFE-SAVING APPLIANCES. 495
Measurement of Yachts for Tonnage.
(For list of measiufement formulae, see paper by Mr. B. £.
Froude, Trans. l.N.A.y 1906.)
1. Internatf<*nal Conference Mule. — Seep. 520.
T X a/a
2. yew York Tacht Club Sule.-Rskting = 3 .
6 • 5 ^ W
L s= mean of lengths on water-plane and over all, botfi
in feet,
s e= sail area in square feet,
w = displacement in racing trim measured in cubic feet.
LIPE-SIVIKO AFPX.IA9CE6.
Board of Trade Rules (1914).
Classes, — Ships are divided into ' Foreiga-^oing * and
Home-trade (including Channel Islands and as far as Brest and
River Elbe). Each is divided into a number of classes.
Toreign-going ,
Class I: Passenger Steamers and Emigrant Ships. — ^Total
lifeboat capacity (subject to the alternatives referred to,
below, V. " General ") to equal total number of persons carried
or certified. The number of davits to be as in A, Table 1,
p. 501 ; but this number need not exceed the number of
boats required. Each davit set to have a Class I lifeboat
attached, of which at least the number stated in B, Table 1,
Tuust he open boats. The remaining boats may be open or
pontoon boats of Classes 1 or 2.
Class II : Steamers not Certified for Passengers. — Lifeboats
on each side of ship and attached to davits to be suffioienti
to aoeommodate all on board ; if several boats are required,
the excess above two may be of Class 3 in lieu of ClasBCB 1
or 2.
Class 111 : Sailing Ships with more t/ian Twelve Passengers.
— ^Lifeboats, attached to davits where practicable, to accom-
modate all on board.
Class IV : Sailing Ships with not more than Twelve
Passengers. — Lifeboats of Class 1 to accommodate all on board.
If one only is required> a Class 3 boat in addition to be
carried ; if several, a Class 3 boat can b3 carried in lieu.
Two Ijoats (one on each side) must be attached to davits.
Home Trade.
Class I : Passenger-carrying Steamships. — ^Number of sets
of davits to be as a, Tabde I, bat not more than number ol
boats required ; each to have a lifeboat attached. The nomber
of open Doats to be as c, Table 1 ; the remaining boats may be
open or pontoon. In new ahipst, if total lifeboat capacity iii
lr>sfl than the number of persons carried, it must be incrossed to
496 LIPE-SAVING APPLIANCES.
that in 1, Tab!e 2, p. 502; any defect then remaining to be made
up by approved baoyant apparatus. For daylight voyages,
Maren 20 to September 80, if the ordinary accommodation is
allowed to be increased, the total capacity of boats and buoyant
apparatus to be at least 80 per cent of the number of persona
on board.
Class II : Steamers not carrying Passengers. — If over
100 feet length to carry a boat in davits on eacli side ; one
may be Gla^ 3. If 100 feet or less to carry one or morsi
boats of Glass 1 which can be readily lowered on cither side.
In any case total boat capacity available eaoh side should
equal number of persons carried.
Class III: Passenger Sailing Ships. — ^Lifeboats, attached to
that in A, Table 2, p. 502 ; any defect then remaining to be made
davits where practicable, to be capable of carrying all on board.
Class IV: Sailing Ships not carrying Passengers. — Life*
boats, capable of being lowered on either side, to accommodate
all on boa^. In ships 100 feet or more in length, one boat
to be Glass I.
Clftss V : Passenger Steamers either oQostal or plying be"
tiffeen Great Britajin, Ireland, and Isle of Man. — ^In general
as Glass I. In certain daylight voyages between June 1
and August 31, the number of davits and capacity of lifeboats
may be in accordance with B and c. Table 2.
Class VI : Passenger Steameps making short Sea Excursions
in Daylight between April 1 and October 31. — ^The number
of davit sets, each having a Glass 1 lifeboat of reasonable
capacity to b« as d. Table 2 ; but total boat accommodation
need not exceed the number of persons carried. If necessary
the total accommodation to be increased by rafts or buoyant
apparatus to 70 per cent of the number of persons carried.
Class VII : Passenger Steamers in Partially Smooth Water,
— As Glass VI, except that total accommodation need not exceed
60 per cent of the number carried.
Class VIII : Passenger Steamers in Smooth Water in
Estuaries and Lakes. — ^If length is 70 or under 150 feet, one
boat is required ; if 150 feet or over, two boats. These to be
carried in davits. If necessary additional buoyant apparatus
to bring total accommodation (including that of boats) up
to 40 per cent of the number carried is to be provided. These
regulations and those for Glass IX may be waived in special
eases.
Class IX : Passenger Steamers in Smooth Water on Rivers of
Canals. — ^As Glass VIII, except that one boat is sufficient for
any length over 70 feet.
Class X: Steam Launches, and Motor-boats making Short
Sea Trips,— li over 60 feet in length, as Glass VIII.
Class XI -: Sailing Boats making short Sea Trips with more
ffmn Twelve Passenger i.~~li over 60 feet in lemrth, as
Clast III. *
LIFE-SAVING APPLIANCES .
497
Class XII : Tugs, Dredgers, Barges, etc., which proceed to
^tt^,. — As Class II.
Class XIII : Vessels as Class XII which do not proceed tp
Sea ;^-To carry a boat to accommodate all on board.
Supply of Life-btioys and Life-jackets,
The number of life-buoys to be as follows : —
Clasfl of Ship.
No.ol
Bnoys.
Class of Ship,
No. of
9aoyB.
Foreign, I
12
Home, I, V
10
(under 400' length)
Foreign, I
18
.. u, IV, xn
4
(under 600' length)
(100' and over)
Foreign, I
24
„ n, XV. xn
2
(under 800' length)
(u»der 100')
Foreign, I (above do.)
30
„ 111, IX
4
M n . .
6
„ VI. vn
(200' and over)
8
.• HI . .
6
.. vi, vn
(uS4er 200')
4
M IV . .
4
,. vm
(150' and over)
6
„ viu
4
(under 160')
„ X and XI (60'
2
and under), XIIX
, , X and XI (over
4 .
60')
One life-jaoket is to be oarried per person in all cases ; and in
addition a sofficient number suitable for children in Glasses I and
ni foreign, and I, HI, V, VI, VH, X, XI, home.
General.
Children under one year are not included in the number of
persons oarried ; two children under 12 count as one person.
Daylight extends from one hour before sunrise to one hour
after sunset.
Buoyant apparatus or oth^r approved appliances majjr be
accepted in lieu of lifeboats, except that in foreign-
going pawepger steamers ihe total lifeboat fsapacitv shall be
at least as c^ Table 1, and at least 75 per cent of the tot^I
number of persons on board.
The weight of an adult person including life-jacket is
assun^ed to he 165 lb.
Ships carrying twelve passengers or lessi and otherwise
suitable for Claw II, foreign-going or home trade, shall b9
subject to these rules respectively.
Kk
498 LIFE-SAVING APPLIANCES.
Boats,
Class 1. — ^A. Open lifeboats with internal buoyancy. The
buoyancy to be provided by W.T. air cases, whose volume is
one-tenth the cubic capacity of boat ; in meljal boats
capacity to be increased so that buoyancy equals thatj of
a wood boat.
6. Open lifeboats with internal and external buoyancy.
The capacity of internal air cases to be 7}o/o that of boat
(with an addition as above in metal boats). External
buoyancy, if of cork, to be 3*3 o/o of the boat capacity ;
if of other (material to be equivalent as regards buoyancy*
and stability. ( ' (
C. Pontoon lifeboats with well deck and fixed W.T.
bulkheads. Area of well deck to be 30 o/o deck area. Height
of well deck above water to be ^ <Vo length of boat minimum,
increasing to Ijo/o length at ends of the well. Reserve
buoyancy of boat to be 35 o/o.
Class 2. — ^A. Open lifeboats having upper part of sides
collapsible. To have internal buoyancy of W.T. air cases
and external buoyancy of cork, having capacities in cubic
feet per person of 1*5 and 0*2 respectively. The minimum
freeboard when loaded in fresh water measured to top of
solid hull is to be 8 inches for a length of boat of 26 feet,
9 inches for 28 feet, 10 inches for 30 feet, and so
intermediately.
B. Fantoon lifeboats with well deck and collapable
bulwarks. As 10.
0. Pontoon lifeboats with flush deck and collapsible
bulwarks. The minimum loaded freeboard in fresh water
depends on the depth amidships ; both are measured from top
io|f ddck at side^, and the latt«r down to underside of garboard
strake. The tabular freeboard applies to a mean sheer of
30/0 length ; for boats with a smaller sheer, add to the
freeboard one-seventh of the difference between the actual and
tabular mean sheers. For intermediate depths freeboard shonld
be interpolaCed.
Depth in inches . . 12 18 24 30
Freeboard in inches . 2{ 32 di 6i
Class 3. — Open boats without intemsJ or external
buoyancy.
Motor-boats. — One of nino or less, or two of ten or more,
lifeboats may be a motor-boat : in pertain oases a greater
number may be permitted, Tqa^ mugt oompiy wi& tije
reMquirements of Class 1 lllebaats and be kept well provided
with fuel ; in fixing the buoyancy ^he extra weight alt motor,
etc., must be allowed for.
Construction of Boats, — 'Boats to be amply stable and
strong ; in new foreign-going ships they must be capable
LIFE-SAVING APPUANGES. 499
of being lowered with fall complement and equipment.
Thwarts and seats to be low, and bottom boards not more
than 2 ft. 9 in. below them. Internal buojanoy chambers
to be of copper or yellow metal at least 18 V)z. per
square feet ; to be placed at sides or ends, but not ati
bottom. External buoyancy to be provided by solid material.
The mean sheer of Class 1 open lifeboats to be 4o/o or more
of the length.
Pontoon lifeboats, if wood, to have two thicknesses fieparaied
by textile material in bottom and deck ; if met|^, to have
W.T. compartments with access and means of pumping. To
test the means of clearing water from the deick, boat to be
loaded with weight of iron equal to complement and equip-
ment ; the time for clearing two tons of water to vary
directly as the length, and in a 28-foot boat to be 60 seoonds
for Classes 1 C and 2 B, and 20 seconds for Class 2 C.
The buoyancy ma«t not depend on the adjustment of. any
principal part of the hull. All boats to be fitted to use
a steering oar ; and to be marked with their dimensions And
complement.
Complement of Boats. — ^Pjrovided the freeboard is satis-
factory, . and that the number of persons can be carried
without inconvenience to the ioarsmen, the complement is
derived from either the Cubic capooiiy or the surface of the
boat as follows : —
Classes 1 A and 3, divide capacity by 10 cubic feet ;
Glass 1 B, divide by 9 cubic feet ; Classes 2 A and 2 C, divide
surface by 3} square feet ; Classes 1 C and 2 B, divide surface
by 3^ square feet (or exceptionally 3 when seating accommo-
dation permits). In all cases the capacity must be at least
125 cubic feet.
The complement may be limited in either very fine-ended
or very full boats.
(Jubio Cctpacity, — This is obtained by obtaining the
sectional areas inside plankinff and up to level of gunwale.
The volume and the areas are Both obtained by Simpson's first
rule (p. 43), using four intervals. Certain corrections are
introduced (a) for excessive sheer, (b) if depth amidships is
more than 45 per cent the breadtn or more than 4 feet.'
Alternatively capacity may be taken as *6 x length x breadth
(external) x internal depth. The depth taken must not exceed
'45 breadth. In a motor-boat deduct space occupied by
motor, etc. . ^
Surface, — ^This is measured ooitside the planking by taking
the horizontal breadths at amidships andf at '\,l and %t
(2 B3 length outside) from amidships. If these breadths are.
(in order, starting from one end) a, d, c^ d^ e, then area of
Burface = Jq (2a + 1 • 56 + 4c + 1 • 5<i + 2^) . This is applicable to
pontoon boats and Class 2 open boats.
500 LTFE-SAVINO APPLIANCES.
JAfe-rafU,
To be Tev^rsibld and fitted with bulwarks on botli' mdes.
To be capable of being easily bandied without mechanical
appliances. To have 8 cubic feet of air oases for each person.
TO have 4 square feet deck area for each' person ; deck
when loaded to be 6 in. above water. ,
Lifeboats may be placed in tiers or inside one another,
provided they can be readily lowered. Supports to be
provided between two boats stowed together. All gear to
be readily available in an emergency.
AppUanem for Lotoerinff.
Davits are not to be fitted at bows or too near propellers.
In new foreign-going passenger steamers the boats must be
lowered safely with full c(»nplement when there ia a list
bf 15^ ; gear capable of turning out davits against this
list to be provided. Life-lines to be fitted to davit spans ;
these and the falls to be long enough for use whien vessel
is light. The boats to be capable of being speedily detaehed
from the falls ; those under davits to be rSeAy for service.
plugs
tUler
Boats. — ^To consist of oars (two spare and one steering),
igs, thole-pins, sea-anchor, bailer and bucket, rudder and
killer or yoke, painter, boat-hook, f reah-water keg and
dipper Tone ^na^ for each person), two hatchets, a line
bedcettea outside the boat, and lantern to burn eight houra.
Boats for foreign-going ships and I to V home, also to
have in general — ^mast, sail, and gear (not home trade, nor
motor'^boats), compass, air-tight case (foreign-going only)
with 21b. biscuit per person, one gallon oil, and one dozen
red lights and box of matches in W.T. tin.
Life-rafta, — In foreign-going ships to carjT ^^' oars,
steering oar, sea-anchor and painter, foesh- water "keg as above,
line becketted outside, life-buoy light, one gallon oil, biscuits,
and lights as above.
Buoyant Apparatus. — ^The number of persons supported is
taken as -^ part of the number of pounds of iron supported
in fresh water.
Life- jackets, — To be capable of floating twenty-four hours
in fresh water, with 151b. of iron suspended from it. l%e
buoyancy must not depend on air compartmehhs*.
Life- buoys. — ^To be of solid cork or equivalent material,
and to support 82 lb. of iron twenty-four hours in fresh water.
To be fitted with beckets. One Duoy on each side to have
a lifeline 15 fathoms long. Half (at least six in passenger
steamers) to be fitted with self-igniting lights.
LIFE-SAVING APPLUNCBi.
60J
Table L
Particulars regarding Class I (JForeign) and
Classes I and V (Home).
Registered Length
of the Ship.
Feet.
Under 120
140
160
175
190
205
220
230
245
255
270
285
300
315
83e
350
370
390
410
435
460
490
520
550
580
i610
640
670
700
730
760
790
820
855
890
925
960
995
1,030
If
>9
t>
))
if
ff
»
ii
if
»
}f
»
7f
if
if
if
if
iP
if
it
a
if
if
a
ff
if
if
»
if
if
»
if
»
If
if
if
(A)
Minimmn
namber of sets
of davits.
Minimum
namber of open
boats. Class I.
2
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
12
12
14
14
16
16
18
18
20
20
22
22
24
24
26
26
28
28
30
30
2
2
2
3
3
4
4
4
4
5
5
5
5
6
6
7
7
7
7
9
9
10
10
12
12
13
13
14
14
15
15
17
17
IS
18
19
19
20
20
(O)
Minimum aggre-
gate cubic capacity
of lifeboats in feet
(Foreign only).
980
1,220
1,550
1,880
2^^0
2,740
3,880
3,900
4M0
<5,100
^,640
6,190
7,550
8^90
9,000
9,6a0
10,650
11,700
13,060
14,430
15,920
17,310
18,720
20,350
21,900
23,700
25,350
27,050
28,560
30,180
82,100
34,350
36,450
38,750
41,000
43,880
46,350
48,750
502 KEGULATtONS FOK EMIGRANT SHIPS.
Table U.
Particulars REGARbiNG Home Trade Ships op various
Classes.
Length of
vessel in feet.
1
Affgreccate lifeboat
capacity in cubic feet in
^ps launched after
March 1. 1913.
NumlMr of seta of dayits.
(A)
Classes
I and V.
(B) .
ClassV
(daylight
excursions).
300
400
500
600
700
800
950
1,080
1,250
1,430
1,700
1,900
2,150
2,400
as
required
(C)
Glass V
(daylight
excursions).
(D)
Classes VI and VII.
Under 120
» 'i«o
„ 180»
„ IMi
» 2101
„ 226»
„ 240»
. » 2S«
„ 270
, ,, 2M
„ 800
„ 880
„ 360
„ 870
„ 410
„ 480
400
600
850
1,150
1,300
1,450
1,600
1,850
2,350
8,000
3,750
4,400
5,100
6,000
2
2
2
2
3
3
4
4
5
5
6
6
7
8
as
required
•
2
2
2
2
2 fonder 2000
3
3
3
4
4
4 (nsder 2800
5
5 (under 3200
•
as
required
*
BOARS OE TRADE REOULATIOKS EOR EMIGRAKT SHIPS.
(For regulations concerning Ventilation see p.=400.)
An ' emigrant ship ' ia one which carries from the British
Isles to any port outside Europe and the Mediterranean Sea
more than fifty steerage passengers, or a greater proportion
6f them than one adult to every 33 tons of the registocred
tonnage of a sailing ship, or of every 20 tons for a steamship.
An a^ult is a person of 12 years or more ; two younger
ehildren are counted as one adult. ' Passenger ' refers to
'.steerage passenger ' ^ in the rcg^ations below, which apply
Only to steerage accommodation.
^ Up to 5 feet less for column A. ^ Up to 5 feet more for column A.
8 i.e. other than cabin passengers who must each have at least 86 sq. ft
for their exclusive use.
REGULATIONS FOR EMIGRANT SHIPS. 608
Emigrant dhips are subject to the surveys usual to
passenger steamers ; and in addition to the following
regulations : —
Deoka, — If of wood to be properly fastened and caulked.
If of steel to be sheathed with wood or approved composition.
Height between decks in steerage compartments to be at
least 6 feet.
Smh, — ^To be not more than 6 inches apart between centres
unless fitted with netting.
Berths, — ^Two tiers only to be fitted on one deck. The
lower to be at least 12 inches clear above the deck ; the
interval between each tier and between the upper tier and
deck to be at least 2 ft. 6 in.
Each berth to be at least 6 feet lons^ and 1 ft. 10 in. broad
(for adults). To be sufficient in namoer for all passengers.
To be separated by gas-tight partitions from w.o.'s and
urinals.
All male adult passengers to be separated by substantial
bulkheads from aU other passengers. Not more than one
adult passenger, except husband and wife 'or females, to
occupy the same berth.
Doora. — ^To be equal in width to the ladders or stairways
to which ihey give access.
Over each hatchway a booby hatoh admitting light and air
but affording protection from the wet to be placed.
Short passages between cabins to be as wido as the bunks.
Stairways, — The ' weather deck ' is the highest complete
deck, except in compartments w'hose only egress is on the
deck of a poop, bridge, forecastle, etc. — ^in which case the deck
over is termed the weather-deck.
Separate stairways to be provided to each passage com-
partment ; their aggregate width, being at least 2 inches for
every five adults accommodated. In stairways for access
to weather decks from two compartments take the total
accommodation in both for applying this rule ; for three
compartments take the accommodation of the two largest
added to one-half that for the third.
When the stairways leading to weather-deck are enolosed
by a poop, bridge, or similat space, the width of the doors in
the end bulkheads plus that of the stairways leading to the
weather-deck from poop, etc., is to comply with the rula.
The stairways should lead to a weather-deck space always
accessible to steerage passengers. No ladder or stairways to
be less than 30 inches wide ; if more than 50 inches wide
intermediate rails to be fitted from 30 inches to 50 inches
apart. The width is always the inside clear width on treads
or between rails.
There must be 6 feet clear space vertically above each
stair. All stairways to have efficient handrails, not ropes,
504 BEGTJLATIONS FOR EMIGRANT SHIPS.
on each side. To he well lighted by d*y and night. Ladders
to be generally pitohed fore and aft ; angle to vertical
aboat 37** ; those for women to be lined on the back. Those
passing through an open sleeping spaoe to be enclosed by
olose boarding*
Ligktinff. -^^ood natural lighting to be provided in all
steerage spaces. Side-lights to be at least 9 inches diameter;
to have brass or gan-metal frames, and to be fitted with
dead-lights. When electricity is the sole means of lighting,
the generators must be situated well above the water-line.
Wnier-elotets. — ^Four to every 100 passengers up to 300,
and two for each additional 100 to oe provided. To be
placed on a passenger deck oth^ than the lowest. Separate
closets to bo apportioned to, and marked for, males and
females. Two additional urinals (or tip-np w.c/s) to be
provided for every hundred male paSBehgers up to 300, and
one for each additional hundred.
Ht>spitah,*SpwcB at the mte of 18 square feet clear for
every fifty passengers to bia dlvfdod oS for hospital accom-
modation. Hospitoi i^iace to be at least 190 square feet ;
and one hospital to be set apa^rt for inf4»ctloas diseases.
They should not contain more fltibed beds than one to each
15 square feet : one lying-in or doable berth to be provided.
jiupBn9ary,—K eeparste dispensary is desirable on emigrant
ships.
Nnmber ef Pa88enffer9,-^-Ncia.e to be carried on more than
one deck below the statatory load-line. This is termed
' the lowest passenger deck *; * passenger deek * includes every
deck above this which is appropriated to passengers.
The lowest passenger deck to be efficiently lighted by side
scuttles or otherwise. The nn!mt>er of passengers carried on
it is not to exceed one adult to evety 18 square feet clear
appropriated. If the 'tween deck heig4it is less than 7 feet,
or if tiie apertures, other than side scuttles, for light and
air are less than 3 square feet to every 10 ) passengers, the
number is limited to one adult to every 2j sqnare feet.
On a passenger deck the number is limited to one adult for
eveiy 16 square feet clear appropriated ; 18 square feet are
necessary if 'tween deck height is less than 7 feet.
In addition, promenade space, not otherwise reserved, to
be provided on a deck so open 83 nbt to be included in the
tonnage, at the rate of 6 sqnare feet to each adult.
In measuring the passenger and lowest passenger decks,
passengers' light luggage space, and l^at occupied by pnbliu
rooms, lavatories, and bathreoms exclusively reserved for the
steerage may be included ; hospital Space to be excluded.
When separate mess-rooms are provided, the sleeping- space
appropriated per passenger to be at least 15 square feet on
the lowest passenger dedc and 12 square feet on passenger
KEGULATIOXS FOR EMIGRAlfT SHIPS. 505
decia. Hatchwajs below masts, ventilaiors, and otii«r obstrao-
tions to be deducted.
Water, — ^Four quarts daily for each adult, plus 10 gallons
a day for each 100 adults for cooking purposes (exclusive of
that for cabin passengers and crew). Where efficient dis-
tilling apparatus is fitted, only one-half the normal quantity
for the Toyage need be carried. If carried in double bottoms
it must be distributed in four compartments.
Provisions. — Weekly scale per adult : beef or pork, 36 oz.;
preserved meat, 16 ; suet, 6 ; butter, 4 ; bread or biscuit, 40;
wheaten flour, 56 ; oatmeal, rice, and peas (any two), 32 ;
potatoes, 32; raisins, 6; tea, 2; sugar, 16; salt^ 2; mustard, |;
pepper, ^ ; dried vegetables, 8 oz. ; 1 gill of vinegar., To&I
weight, 16lb« 6oz.
Certain substitutes are allowable, including : 8 oz. fresh or
tinned vegetables in lieu of 1 oz. dried vegetables ; 1| Ibw
fresh meat = 1 lb. salt meat = } lb. preserved meat; ^oz.
tea ^ i oz. coffee or cocoa ; 1 lb. flour = 1 lb. biscuit «i 1 lb,
rice ; i lb. butter = 1 lb. jam or marmalade.
Cargo. — Not to be dangerous to health or lives of passengers,
or to safety ef ship. Iiron or steel rails, or similar dead-weight
cargo should not exceed one-third dead-weight capacity
of ship.
Cattle. — ^Not more than twelve dogs and no pigs oir wtSe
goats to be carried without special permission. ' Cattle ' in-
cludes deer, horses, and asses ; four sheep or female goats ar^
equivalent to one head of cattle. Not more than one head of
cattle to be carried for every 200 tons gross tonnage ; not more
than ten head in all. No cattle to be carried below, or
immediately above, any deok ot in any co^npartment in which
emigrants are berthed ; when in adjoini^ compartments an
efficient iron or steel bulkhead lined with wood or felt on
the passenger side to be fitted. Cattle carried on the weather
deck used as a promenade space shall be separated by a deck-
house or bulkhead from the passengers ; promenade space
within 50 feet of the cattle to be reckoned at 8 square feet
per adult.
Miscellaneous, — The above r^^lations also apply to ships
bringing steerage passengers to the British Isles from any port
out of Europe and the Mediterranean Sea.
506 LLOTD*S BULBS FOR SHAFTS.
LL0TD*8 BVLE8 FOB DBTESMHriHe 8IZE8 OF
SHAFTS.
For compound engines with two cranks at right angles —
Diameter of intermediate shaft in inches
a (-O^A + OOGD + 028) X -yp.
For triple expansion engines with 3 cranks at equal angles —
Diameter of intermediate shaft in inches
= ("OSSa + -0098 + 0020 + -01658) x .^P.
For quadruple expansion engines with 2 cranks at right
angles —
Diameter of intermediate shaft in inches
=r(034A 4.011B + '0040+ •0014P + -0163) X ^p.
For quadruple expansion engines with 3 cranks —
Diameter of intermediate shaft in inches
= (•028a + -0148 + 0060 + OOITd + 0158) x ^p.
For quadruple expansion engines with 4 cranks —
Diameter of intermediate shaft in inches
s (-OSSa + OlB + -0040 + OOISd + 01558) x ^P.
where A = diameter of high pressure cylinder in inches.
B = diameter of first intermediate cylinder in inches.
c « diameter of second intermediate cylinder in inches.
Da diameter of low pressure cylinder in inches.
8 = stroke of pistons in inches.
P = boiler pressure above atmosphere in lbs. per sq. inch.
The diameter of crank shafts to be at least f^ths of that of the
Intermediate shaft.
The diameter of the screw shaft is —
•G3t+ •03p, but is in no case to be less than 1'07t
where P is the diameter of the propeller, and
T the diameter of the intermediate shaft, both in inches.
Lloyd's rules for ships. 507
LLOYD'S BITLES POB 8HIPS.
Tests of Materials. — See p. 284.
Numbers and Tables. — ^Lengtib (l) is measured from fore-part
of stem to after-part of stern-post on the range of the
upper deck beams ^second deck in awning and shelter
deck vessels).
Breadth (b) is the greatest laoulded breadth of the vessel.
Depth (d) is that at middle of length from top of keel to
top of beam at side of uppermost continuous deck.
In awning and shelter deck ships to the second deck
or to 8 feet below the awning or shelter deck, whichever
is the greater.
Depth (d) is that at the middle of length from top of
ordinary floor at centre (or of double bdttom at side)
to top of the lowest tier of beams at side, whether widely
spaced or not.
B + D is called the transverse number ; with * d^ * it regulates
the dimensions of frames, floors, and web frames.
liX(B+D) is called the longitudinal number and regulates
the scantlings of structure contributing to longitudinal
strength.
Note. — ^When obtaining the proportion of length to depUi foi:
scantlings of topsides, the depth is to be taken to the highest
continuous deck, whether upper, awning> shelter, or long bridge.
Except when otherwise stated, the scantlings in the tables are
those amidships, and can be reduced at the ends.
Keel, Stem, Stern-post, Keelsons, and Strmgers. — See tables.
Whj^n breadth is under 27 feet, one side keelson is to be fitted on
each side ; up to 50 feet, two side keelsons ; up to 54 feet, two
side and one bilge keelson. All keelsons and stringers to be weU
butted, the straps to the angles being 2 feet long. Keelsons
should preferably be oontiiraous through bulkheads.
Frames y Reverses, and Floors. See tables. The spacing
should never exceed 24 inches in the peaks, and 27 inches for
one-fifth lengtii abaft collision bulkhead. The height to which
the reverses are carried depends on *d* ; if this be from 7 to
9 feet tbey should terminate at bilge ; if from 18 to 27 feet at
upper deck, and so intermediately, for a single tier of beams.
With more than one tier the reverses should extend to upper deck
and deck below alternately.
The depth of floors, should be, at three-quarters the half -breadth,
at least one-half that at the middle line ; they should be carried
up the side to a height above top of keel equal to at least twice
the midship dep^. They should be '04 and ^lO in. thicker
respectively in the engine and boiler spaces.
Web Frames. — To avoid excessive frame dimensions, web
frames, six frame spaces apart, of the sizes given in the table,
with light intermediate frames may be adopted. Special side
stringers, not more than 8 feet from each other or from deck oi
508
LLOYD'S RULES ITOR SHIPS.
top of floor, of the deptii of the web frames are to be fitted.
When using the table, 'd* is here taken up to the loire^ laid deck.
The frames above the lowest deck to be of the sizes of the
intermediate fiames.
Double £o<ioffi. — This may be fitted in lieu of ordinary
frames (see tables). The margin plate is ooatinooos, and the
transverse frames also oontinaous from centre to margih plate.
The floors may be fitted at evexy fmme, or at efvery otfauer frame ;
the intermediate frames in the latter oase consist of the top and
bottom girders, vertical stiffenecs to the side giiden, and plate
brackets at centre and against mai^io plate. Brackets outside
the margin plate at each frame extend to a height above top of
maigin plate of 2" lor longitadinal number 7,^00* 2£f' for 20,000,
and 39" for 80,000, and so intennediately. The thickness of the
central girder is increased about -lO" to 'Oi" in boiler rooms.
The number of side girders is as follows, taking the alternative
giving the greater number : —
Floors at every Fnune.
^MUihoI
Ship.
Breadth of
Imier
Bottom
Ajnidfihips.
Floors at alternate Fxames.
Breadth of
Ship.
TTwlerSO'
60' to 08'
62' to 74'
74' to 86'
Under 86'
36' to 46'
48' to 60'
60' to 72'
Breadth ol
Inner
Bottom
AmMihipa
1
2
3
4
Under 34'
34' to 50'
Under 28*
28' to 36'
^^5 I
1
2
BtUkheads,—Bee table. Steamers axe to have the four W.T.
bulkheads demanded by the Board of Trade (see p. 473} ; in
addition up to 335 feet in length, another to be fitted ia the
forehold ; up to 405 feet, another in the afterhold ; up to
476 feet, seven in all ; pp io 540 feet, eight in aH ; np to 610 feet,
nine in all ; up to 680 feet, ten in all.
Beams. — See table. The round up in all weather dec^ except
where the longitudinal number is greater than 80,000, and haii
the deck is covered by erections, should be i inch per loot of
beam. They should be fitted at every frame (a) at all W.T.
flats, (5) at upper decks of single deok vessels exceeding 16 leet in
depth, (c) at unsheathed upper awning, shelter, oar bridge deds,
and at all such decks in vessels over 450 feet long. Elsewfaero,
when frame spacing is 27 inches or less, tiiey may be fitted at
liltemate frames. The knees to be from 2i to 3 timos tiie specified
depth of beam with one row of pillars. The depth across the
throat is to be -6 times that of the knee. In u^per decks of laige
LLOYD'S RULES FOR SHIPS. 609
vessels to be brackets varying from 83"x33"x 'SCT when d is
24', to 42"x42"x .64" when d is 27'. The rivets to vary from
4 - J" when knee depth is IT to 9 - i" when it is 86". All web
frames to have bracket knees of the same thickness and depth as
the frame ; these are doable riveted in each arm.
Pillars, — One row when beam is less than 44', two up to 60',
and three above. When widely spaced they should be in accord-
ance with the tables where s is their longitudinal spacing, B is
} breadth of ship with two rows and i breadth with three rows, H is
the sum of the heights of the several 'tween decks above the pillaiB
with an addition of five for the top deck, all in feet. A 'tween .
deck exclusively appropriated to passengers need count only as
5 feet. liongitudinal girders in accordance with the table are
also to be fitted. Alternative forms of pillars and girders of
equivalent strength may be used.
Plating. See table. Full thickness to be maintained for
i length. The butts of adjoining strakes to be two frame spaces
apart ; those of alternate strakes to be shifted one frame space.
Unless special arrangements of butt riveting be devised, thQ
breadth of strakes should not exceed that given below : —
MookM depth Din feet under SO 90-94 94-98 98 and above.
Breadth in inches . 54 60 66 72
The thickness of bottom covered by a double bottom with
a floor plate at every frame may be leduoed by 'OS" if ^52^', by
•04" if from -54" to -64", and by -02" if -66". In way of W.T.
bulkheads, wide or diamond-shaped liners extending from frame
before to frame abaft bulkhead are to be fitted to the outer strakes.
Decks. — See tables. Pitchpine planks for weather decks to
be- four to six months old, and the breadth of plank should be
5" or less. The margin planks of weather decks should be of
teak or greenheart. A single nut and screw bolt per beam is
sufficient up to. 6" width of plank ; from 6" to 8", a bolt and one
short screw bolt ; above B^' two nut and screw bolts. Bolts to be
i" diameter up to 3i" piae or 2|" teak, and |" for greater thick-
nesses. If a wood &t be laid for a steel top deck, its thickness
shield: be 3" if pine, 2)" if teak, or more ; for Uie second deck it:
skoald be 2^". Steel decks are to be caulked unlets sheathed
-with a caulked wood deck.
Eiveting.—See pp. 289-92.
Steering Chains,'— li D be diameter of rudder head, R radius
of quadrant or length of tiller, d- diameter of steering chain, all in-
inches 4= -3^ V (o'/»). The dian^eters of the leading l?lock
sheaves should 'he at lettst 16(2, -and the pins of the sheaves 2el.
Deck Coamings. -^The minimum height of these above weather
decks should be 18" on awning, shelter, or bridge decks, 24" on
npper or raised quarter decks, 80" on upper decks in wells or
under tonnage openings of shelter decks*
510
IXOYb's RULES FOR SHIPS.
Keel, Stem, Stern-post,
(All dimensions
■§1:
dfex
5 •^
Eeelfl.
(For fiat
keels see
with
bottom
plating.)
Stems.
Stem-poBt
without
Apertures.
Stern Frames
wiih
▲perLures.
Keelson
Angles.
Propeller
Post.
Rudder
Post.
1
1
2,500
7,000
11,500
18,000
36,000
80,000
6 xl>
8 x2i
10 x2|
12 X 3
5jxli
6ixl|
7 X24
9 x2i
lOJ X 2f
12 x3i
eJxij
7 x2i
9 x2§
10ix3i
12 X 4
6ix2j
6|x4
7 x5
9 x5j
lOjxS
12J X 11
5 x2i
5Jx4
6}x5
8 x5i
9 x8
11 xll
3x3 x-26
3i x 3 X -32
4| X 3i X -36
6 x3|x*44
7 X 3^ X -50
■SfeO
'§■§ +
cSz5x
5 -
5 <« s
tit
in
2,600
7,000
11,500
18,000
36,000 * •
80,000
32
38
41
43
48
58
Outside Platino, Loweb
(All dimensions, except
Outside Plating.
IS
•42
•56
•68
•80
1-04
2-12
o ^•d
•32
•44
•50
•56
•70
Thickness,
Outside Plating
Below
upper
turn
bilge.
•26
•38
•46
•52
•66
98
From
sheer-
strake
to bilge
2nd Deck ; Upper Deck
in Awning and Shelter
Deck Vessels.
Stringer.
•26
•36
•46
•52
•64
•90
41 x -36
43 X -40
48 -48
58 -62
Tie plates
(t) or Deck
Plating (d).
10 X -361
12 y •40T
•38D
56D
J
LLOYD'S RULES FOR SHIPS.
511
Keelsons, and Strinqebs*,
in inclies.)
Middle Line Centre
through plate
keelson.
(b= double bulb
angles ; p=/plate,
4 angles, and rider.)
Plat Keel
Plate Angles.
Side Keelsons.
(b= double bulb
angles ; p= plate.
4 angles, and rider.)
Side Stringers.
One each side when
d is less than 14 ;
two from 14 to 21;
three when greater
than 2i:
Double
Angles.
Thick-
ness
Centre
Plate.
Double
Angles.
Inter-
costal
Plates.
Angles. .
Inter-
costal
Plates.
3 x3 x'26
3^x3 X 32
6 x3 X.40B
IIP
18 P
•30
•34
•40
•46
•62
3 x3 x26
3Jx3ix34
3ix3i^x.48
4 x4 x^52
4ix4ix-60
3 x3 x-26
3^x3 x-32
4ix3|x.36
7 x3|x-42b
18x.64p
•26
•30
•36
•40
•44
3 x3 x-26
3^x3 X'32
4ix3ix.36
6 x3|x-44
7 x3|x-50
8 x4 x-66
•24
•32
•36
•40
•44
•50
Decks, and Shobt Bbidges^
[engths of bridges, in inches.)
M Deck; 2nd Deck
in Awning and
Shelter Deck
Vessels.
Stringer.
13 X -38
18 X -44
IS X -54
Tie plates
(t) or
Dsok
Plating
(d).
Decks below the
precediniT'
Stringer.
12 X •38t
16 ^ 441
•44d
Tie plates
CV or
Deck
Plating
(»).
58 X -54
27 X •52t
'Short* Bridges.
Maxi-
mum
Length,
feet.
25
45
55
65
85
105
Side
Plating.
•22
•26
•30
-36
-42
•62
Stringer
Plate.
19 X -22
30 X -26
37 X -30
42 X -36
40 X ^42
51 X -62
Tie Plates
(t) or
Deck
Plating
X -221
X '261
X "301
X •36T
•30d
•46D
512
LLOTDS RULES FOB SHIPa.
i
24
40
62
80
L04
128
Fkfton.
20
n
23i
m
29i
33
I
d
8
H
&
Thickoess
in inohes.
8i
141
32
b0
d
fM
I
9
22
•32
'42
48
•22
28
38
wiUi singl* nveisM.
All dimenfliona in inchea.
"Hie three sizes givea are—
(a) that of frame,
(6) that of reverse,
(o) depth of framing.
36'
d=B
C^ CI
•
V y
«ei CI
X X
00 00
• •
X X
eoei
X X
ec
eo C4
X X
CO CO
X X
•^ CO
W5
00 00
CO CO
• •
^ X
ob co<o
X X-
co eo
30 00
00 CO ^^
X X
;o 00
=15
op
CO C'J
• •
X X
CO CO
X X
^ CO
00
X
CO
X
00
eo
X
CO CO
X
-"l**
-"l**
00
X
X
X
CO to
»o
CO "^
X X
1^ op
= 21
= 27
X X
CO CO
X X
kO to
00
00 00
y y
X. X
CO
»o
CO
M5
y X
Floqbs^ F&a&ies,
Frames formed
or chamiela
A sismifies a single
angle bar.
B signifies a single
angle bulb.
c signifies a single
channel bar.
B and c together are
alternate, accord-
ing to choice.
4=9
eo
X
X ^
aoee
3)x3i
X -26
=15
4ix3
x-84
A
6
x8
X -38
B
7|x3i
x-44
B
11x3)
X -480
11x3}
X •58b
11x4
X 540
4x4
X *54b
5x3
X -86
B
7^x3
x-44
B
8x3J
X -440
9x3J
X-50J
9x3i
X-46C
3}x3i
X-46B
LT.OYD S RULES FOR SHIPS.
513
lAND Web Frames.
qf uinyrle bars
and reverses.
t, Cwith-c) ftignifies
I a reverse angle,
e X tending to
' lowest deck, fitted
' to a channel bar.
^11 dimensions in
InoheA.
=21
=27
Web Frames. Intermediate Frames, and
Stringers.
Size of Web Frame
ia inches.
d=15
=19
=28
=27
u
o
QQ PS
eo
Intermediate
Frames
(inches).
J
5 "S
bo ti
•8 ^
a
e9
hi
. 85
P=4 P?
V
!
5 f
■-.4 f-H
«
s
eo
O
OB
go
'St'feB
QQqq
9x3
x-50
B
11^x3^
x'52c
11^x3^
X'62b
15X-36
18X-38
21x-4a
•32
CO CO
• •
>c X
00 CO
X X
^ CO
00
CO
X
9ix3J
X -480
10 X 3i
X *56b
11 x3J
X •50c
X -SOe
10ix3^
X-48C
3^x3^
X-48R
19X-40
22X-42
25x46
30X-52
•36
©O CO
. •
> X
CO CO
X X
«0 CO
X
CO
X
28X-48
30X-52
34X-54
39x'60
•44
• •
X X
CO CO
X X
o
37X-58
39x 62
42x^62
Ll
•52
X X
OO CO
X X
Oi 00
X
o
co»o
• •
X '
CO w
X X
Hc« _
CO »0
• •
X X
HNHct
CO CO
X X
CO «*"
00
•
X
^ I
X
o
X
514
LLOTD 8 &ULES FOR SHIPS.
Topside and Deck Plating at Upper, Awnino, and Shelter
4
3ja
^55
3
+
(4
X
Hi
2.600
7,000
11,600
L8,000
16,000
56,000
n,oo>
s
eS
1-
JoQ
o
u
OQ
32
88
41
43
48
62
68
Si3
•28
•40
'48
•64
•66
•80
100
Length H- Depth up to 10.
is o
is ©
•28
•40
•46
•62
•64
•76
•90
Stringer
Angle.
8 XS X'28
3 x3 X'40
4 X4 x^4(;
4jX?|x'60
6 x6 x-63
6 x6 x"72
7 X7 X 83
Stringer
Plate.
18x'26
34x '38
48X44
42X'48
66 X '62
49 X '60
67 X '78
Deck
Plating.
T
6X 26
T
9x 38
T
13 X 44
•30
•40
•48
•68
Length -7 Depth OTer
u
II
SI
•
ii
•32
•28
•46
•40
•64
•46
•62
•62
•78
•70
•98
•88
D
D
•90
■90
Stringer
Angle.
3 X8 X'28
3 X3 X-40
4 X4 x'4S
4ix4ix-£0
6 X6 X'62
6 x6 x'"2
8 x8 X"83
D = Double.
Cellulab Double Bottoms. >
Depth of
Central Qirder.
-
Thioknesa.
Inner Bottom.
Floor plates
and Brackets.
Central
girder.
Side
girders.
Margin
Plate.
Breadth, Middle
Line Strake.
Thickness.
-^ d A
•f^ d v^
3 -
Middle
Line
Strake.
1
Kg In Holds
» generally.
In Engine-
room.
^3.2 =
Up to
7.600
30
•28
•36
•28
•30
30
•34
•32
•44
About
19.000
37
•34
•46
•
•34
•40
37
•44
•36
•42
•52
36,000
44
•40
•62
•40
•48
44
•62
•40
•50
•56
66,000
61
•48
•66
•48
•68
61
•60
•48
•58
•64
80,000
57
•54
•82
•64
•72
64
•70 -66
•66
•70
i
LLOYDS UULES VOll SHIPS.
516
Decks and **Long** .
Bridges. (All dimensionB in inches.)
11 and up to 12.
Length -^
Depth over IS and ap to lU
1
Upper Dec
Oiidcr 'Loni
Stringer
Plate.
Deck
Platiug.
Sheer
•crake.
Strake
below.
Stringer
Angle.
Stringer
Piute.
Deck
Piatiiiif.
•
Bridges.
Stringer
Plate.
Deck
Platinflr-
22X'28
T
6x *28
•40
•28
3 x3 x'36 26X 32
6x '32
•24
T
•24
38x'i0
Ox '40
•C6
'42
a|x3ixd8
42 X '44
11 x'^'44
'32
T
•32
52 X *46
13 X *4G
•G8
•52
4 x4 x'64
44 X '50
•30
•40
30
46X*52
•30
•80
•60
4ix4|x'GO
COX '56
•38
•44
30
59X-68
•42
IOC
•84
5 x6 x*70
G3X •68
•46
•48
•38
51X*70
•62
1)
•1X5
•yo
7 x7 x;80
63 X •sa
•56
•52
•44
59x*9S
'72
1*20
112
8 X8 x'U2
Gl Xl^Od
•78
•62
•54
T = Tie-plates.
(All dimensions in inches.)
Angle Bars.
(1) Top.
(2) Bottom of
Centzal Girder.
Connecting
Margin Plate
to Outer
Bottom.
3 X 3 X '30
8| X 3} X 40
4 X4 x'48
4 x4 x58
4 x4 x'72
Frames and
Beverses
on Floors,
Side Girder
Angles, and
Vertical Angles
on Central
and Margin
Plate.
3 X 8 X *28
8| X 3i X 81
8| X 3| X '42
8| X 3| X ^64
4 X 4 X •eo
Vertical
connecting
Floors and
Side Girders.
2i X 2i X '23
3 x8 x'31
3 X 3 X '40
d| X 8} X '48
3i X 3i X '54
Intermediate
Frames and
Reverses
where Floors
are on
alternate
Frames.
31 X 8 X '30
3 X 2« X ^26
5 X 3{ X '33
3|x8 x'31
i IKlliil^l 1
s
5
« S S
1 1 » 3 1 1
■s
« s s s
1 1 ^ 1 1 «
« 1 1 a
-=
1 *- ■ « o « ■
- « - * a
s
£89999
*f "b *« « « «
« ^ ^ ^ = ^
!!!!!!
« « a ^ « a
^ « » ^ * «
-B^saao)
laayocqnoa
! ! ! ! !
kh
-nosa
a .a a s a 8
i««^ie™i
s a s a g s. ,
a'- |s ' a s s «
LLOYD S RULES FOR SHIPS.
5r
OQ
I
O
i
.§
I
O
OB
i
g
e
"a
>4
«
«
n
- -.9
I §5 lis
S ilia
« am
p,«^.
•^ - « 2 S «»
gSgQgfpq
5
eB
A
.Q
d
.a
s
I
I
II
CO
eo
tc
a
«
S
c3
»«
O
1
-a
a
,4
.Q
§
"^
03
eo
ef9
o ,5
*999} a| sdfqs
«
X
X
X
eo
SS 9 8
X
to
X
00
X
Q
9
X
X
o
I I
X
to
X
X
as
X
X
s 9
X
X
00
S 8
X
X
Q
3
X
03
X
X
09
X
eo
X
^ ^
X
eo
X
X
03
X
X
X
X
rigl
09
X
s §
^x
00
X
X
00
X
S 8
eo
X
9
X
eo
X
• •
X X
C^ 00
X X
X
eo
X
00
X
X
I I
X
a?
I i
"«a
s s
X
eo
X
a* i-i
S S 9 $ s
X
X
I I
I 1
^ s s ^
518
LLOYD 8 aULlS 70S SHIPS.
Widely Bpaced Pillabs.
amber.
Length of Pillar in
feet.
xBxH
6 to 8.
12 to 14.
18 to 20.
100
ee de-
riptton,
?.60O.
If tnbular,
ontside
diameter
X
thickness.
If built of
4 angle ban,
riveted
together
back to back.
6x5x .50
7X7X.70
If
tubular.
If built of
4 angle bars.
If
tubular.
12
24
46
ir,
115
6>^-40
8X.4O
llx.50
Hx-eo
18X-64
7 X.40
Six. 44
12 x-SO
15 x.fiO
18 X -70
4 x4 x-40
4^x4ix.44
6 x6 x-60
6 x6 X.70
8 x8 x.70
8X.40
IOX.4O
MX. 54
16X.60
I8X.74
GiRDKRs AT Heads of Widely Spaced PHiiiABs.
limber.
xBxHig^'S
05S
"ioo~ §^
c
ee de- S g g
ription, &^ -S *
>. 509. A
200
350
580
1000
1800
•84
•40
Sizes of Double Channel Bars riveted to Girder Plate
below Beam. Where B is marked, a Rider Plate
about '15 thicker than Girder Plate is added.
Depth of Beam in inches.
9
12
7x3 x3 x-38;
8x3jx3ix.46
•44 Ilx8}x8ix.60
•50 —
•54 —
7x3^x83^x40 7x8 x3 x 88
9x3jx8jx.60 8x8ix8ix.46
12X8jxSj x.70 12x4 x4 X.64
7x8 x8 X.38
10xa|x8ix.6D
12x4 X4 X.72
LLOYD S RULES FOR SHIPS.
619
(AU dimensions, except in length, in inches.)
Length of Pillar in feet.
18 to 20.
If bnilt of
4 angle bars.
24 to 26.
If tubular.
4ix4^x.40
5 x5 X.50
6 x6 x-60
7 x7 x-70
8 x8 x-84
9X-44
lOx.fiO
18X.54
17 X-60
'If built of
4 angle bars.
28toa0.
If tubular.
6X6X-50
5X6X-60
6X6X.64
7X7X-74
10X.60
llx-SO
18x60
18X.60
U built of
4 angle bars.
6x6x.60
6X6X.60
7X7X.60
8X8X.74
Depth of Margin Plate in
Double Bottom. •
ft
Xi
I
Up to
45
About
68
74
92
110
Depth d from top of Margin
Plate to lowest tier
of Beams at side, in feet.
7-10
19
24
28
34
40
13-16
19-22
21
—
26
27
30
32
96
38
42
44
1
25-27
34
40
ThickAess of Deck Plcnking.
Longi-
tudinal
Number
IjX (b + d)
up to
2.400
8.700
6.200
'7.000
8,800
11,800
82.000
At Upper
Decks.
At Awning
or Shelter
Decks.
and on
Erections.
Pine.
2h
SI
8
8i
Si
Teak. Phie.
2i
2i
21
8
3
8i
2^
2i
22
Teak
21
2i
2i
520 YACHTS OF THE INTERNATIONAL RATING CLASSES.
LLOJL'B SUUM FOB TA0HT8 07 THB IMTnfcVATiOirAL
JtATDTG CLA88ES.
These reg^ulations are tihe ontcome of ihe reoommeiidation
of the InteniaiioBal Gonfereiiee <m Yaelit Messorenteitt, 1906,
which decided that there should be scantling restrictions for
tiacing yachts in all tiie countries representeii (Uie principal
JEnropean states). They have been prepared by the British
and German Lloyd*s and the Bureau Veritas in consultation ;
their aim is to arrange for such spantlings as will enable
a yacht to withstand the strains consequent on racing^ and
lafterwards to be converted into a serviceable cruiser. They
apply to wood, composite, and steel yachts rated as follows : —
Ratinff in metres 5 6 7 8 9101916 19 S3
Corresponding
No. of feet . 16-4 19-7 SS'O 39*3 29*6 83-8 89*4 49-3 63*3 75*4
The rating in metres or feet isJ(L + B + }o + 8<2 + }Vs-F);
where h =» length on water-line + (^) the difference between
the girth, covering board to covering board, at
the bow water-line ending, and twice the free-
board at that point, -|- (&) one-fifth of the
difference between the girth, covering-board
to covering-board, at the stem water-line
ending, and twice the free-board at that point.
B =:= the greatest beam, including wales, doubling
planks, and mouldings.
o ass the greatest chain girth between upper sides of
covering-boards round the keel, leas twice the
free-board at the same station ; alternatively
if the underside of keel be straight, the above
may be measured anywhere abaft '55 L.W.L.
length from the bow, provided that( the
maximum chain girth does not exceed it by
more than 3 per cent. If there be a hollow
in the fore and aft under water, o and d shall
be taken under an Imaginary line excluding
such hollow.
d = the difference between the chain giri^ a and the
skin girth between the same points measured
along the outline of the cross-section.
8 =s the sail area as measured under Y.B.A. rales.
F =s twice the free-board at ffirth station -{- free-board
at bow water-line enoune -|- free-board at stern
water-line ending, the whole divided by four.
All measurements to be taken without the crew on board.
A selection of the scantlings for wood and steel yachts is
given in the following tables ; those for intiermediatei
ratings can be obtained approximately by interpolation.
No rules were formulated regarding the masting and
Tigering.
YACHTS OF THE INTERNATIONAL RATING CLASSES. 621
Steel Yachts.
Th» vize of the stem may be reduced uniformly from
full size at heel to three^uarters area at head. The itern-
post similarly from the counter to the head. The scarves to
keel to have a lei^th nino times the bar keel thickness.
Reverse frames to be fitted to all floors ; in 12 and
15 metre yachts to extend alternately to bilge stringer ; in
19 and 23 metre yachts to extend saternately to oabin sole
beams and 4 feet above. *
Beams to be pillared at the middle in way of masts, wind-
lasses, deckhouses, and large openings, and about four frames
apart for yachts 15 metres and above. Hidf -beams to be
attached by double lugs to carlings or eoamings. The depibs
of the knees to be 6^' for If" beam, 9" for Si" beam, 12J"
for 5" beam, and so intermediately ; to be connected by fonr
rivets to frame ; depth at throat to be 60 per cent that
of knee.
Fine decks to have the grain vertical.' For 8 metre yachts.,
and less, the planking thickness may be reduced by iV' when
covered with canvas and painted.
'Butts of outside pla$ng to be planed ; they are to be
shifted as by Lloyd's xuJes for ships. Those or keel plate,
sheer strake, and dieok stringer to be generally double-
jriveted ; the remainder, except in the largest olasees, single-
riveted. The riv^tin^ to be spaced four diameters in outside
plating (at edges 4^ diameters), and seven diameters in frames,
beams, etc.; the spacing is somewhat (doser with the larger
sizes of rivets. See a]»> p. 290. The sizes of the riveting
to be in accordance with tiie following table :—
Tbiekness of plates or angles in inches —
•10 15 -20 .25 -30 -35 -50
aud under and under and nader and nnder and trader and under and tmler
'15 '20 '25 '30 '85 -50 '60
Diameter of rivets in inches —
At A J i i i
The rivet hole to be xV" larger than the rivet ; the diameter
at the top of countersink (in outside plating) being greater 4han
that of the hole by about half the rivet diameter.
Wood Yachts.
The table scantlings are, except for decks, those required for
oak. East India teak, greenheart, acacia, English elm, American
rock elm, and mahogany weighing at least 35 lb. per cubic foot.
The scantlings are to be increased by 5 per cent with pitehpine,
by 10 per cent for Oregon pine, larch, Kaurie pine, lighter
mahogany, Qr (various descriptions), and red pine, and by
20 per cent for spruce or yellow pine. It is recommended that
any steel used be galvanized.
The keel may be scarphed for 12 metre and larger yachts, the
length of scarph being 3' 6" to 5' 5" according to size. If
522
YACHTS OF THE INTERNATIONAL RATING
SCANTIilNaS OP
Portion of Strnoftnn iwll dimensions in inches).
Keel, stem, and stempost
Budder.
Diameter at head ....
Section at heel
Thicknefis of plates
Diameter of pintles
Framing.
Frames
BeventM
Frame spacing
Floor plates.
Depth at centre
Thickness ....
Web frames.
Number on each side
Size of plate ....
Beams.
At
alternate frames.
Through beams f or } l« .
Do. at ends. All half beams .
At
erery frame.
Through beams f or } l .
Do. at ends. All half beams .
Hollow pillars (iron or s^eel). Outside diameter and thiokness .
Outside plating
Keel plate
Plating generally . . ...
Upper deck sheerstrake
Uppsr deck
Stringer plates
For three-quarter L amidships ....
At ends
Tie plates on
upper deck beams
Number of pairs of diagonal tie plates .
Breadth and thickness
Upper deck and b
ilge stringer angles
Upper deck
planking.
Thickness of
planking.
Beams at every frame •
Beams at alternate frames
Diameter of screw fastenings . , . .
YACHTS OF THE INTBRNATION'AL RATING CLASSES. 528
Stett, Yachts.
International Bating Class in metres.
6
7
10
. 16
23
2fxA
3JxiV '
4xg
4Jxif
2
IJxll
•1«
IJ
5ixiiV
li
.08
11
Ixi
•10
li
lixlj
•14
3
2ixi
•24
2
lJxlX.12
1 xix.i2
12
1 ;
IJ X li X 14
ijxl xl2
14
- « . • '
lf5<ll)<16
11 X IJ X 14
17
•2 x2 X-2D
2 xljx 16
19
2J X 2} X 25
to 22
2i X 2j X 22:
21
6
•10
. —
8
•12
11
*14
13
18 to 16
16
•25 to -20
2i x If X 14
2. xl^xH
—
2
14 X -25
2 xlJx -12
2 xlJx -12
3 X 2 X 18
2i X IJ X 18
3i x 2i X -25
3i X 2J X -20
5 X 3 X 30
4i X 3 X -25
2 xljx -12
l|xlix.l2
2. xlix 14
2 xlix 12
2i X li X 18
2i X li X 16
3 x2 x20
3 xlix 18
4 x2ix-25
3ix2 x-22
—
—
lix.18 .
2ix.l8
24 X .14
10 •
17.x .10
26 X 18
•12
19x14 to -12
29 X 26
•14 and 16
alternately
22x-20tol6
.32 X 35
•18 and -20
alternately
24x-26to*20
36 X -50
to -45
•24 .
28x36 to -24
. Oix-lO
. 4 X -10
8 X 12
6i X 12
10 X 16
7 X 14
14 X -22
10x18
I9i X -26
15 X 22
1
2 X -10
1
2i x -12
2
3ixl6tol4
2
4x -22 to -18
2
6 X -26 to -22
IJ xljx. 10
li xlix 12
2 x2 xl6
to 14
2J X 2i X -22
to 18
2i X 2i x 26
to -22
•74
•90
•96
112
1*34
150
• 1-69
1'86
224
240
• 21
'25
•33
•36
•39
524 YACHTS OF THB INTB&NATIONAL RATING CLASSES.
SCAinXINOS OF
Portion of Stiootaze (all aimeiuioiu in inches).
Moulding ....
Seotionftl area in square inches
Siding and moulding of stem and stempost, siding of after deadwood,
diameter rudder head
Bent wood
frames only.
'Grown*
frame
timbers only.
Siding X Moulding .
Spacing, centre to centre
Siding
Moulding.
At heel •
At head.
• $
Spacing, centre to centre
Floors.
Wood floors on grown timbers— moulding x siding
Angle steel floors on
' grown ' frame timbers.
Angle steel floors on
bent wood frames.
Length of arms
Angle steel
Length of arms
Angle steel
Web frames.
Number each aide
Siae of plate
Size of face angle
Sectional area of upper deck shelf in square inches
Sectional area of bilge stringer in square inches
Thickness of outside planking . . . .
Beams.
Spacing, centre to centre
Through beam for i h
amidships.
Wrought-iron hanging
knees to de^ beams.
At middle Of beam
At end of beam
Number each side
Lemgth of arms
At throat » •
At point .
Thickness of upper deck planking
Diameter of
fastenings.
I
In keel, dead wood, stem, stempost , floors to grown frames
Grown frames to iron floors and to deadwood
Bent frames to floors and deadwood, deck shelves, etc.
Outside planking to grown frames, (1) bolts, (2) screws
Outside planking to bent frames, (1) bolts, (2) screws .
YACHTS OF THE INTBKNATIONAL EATING CLASSES. 625
IJV^ooD Yachts.
Intenuktionftl R«fciiig Class in metres. |
'A
7
10
15
98
8
18
4
32
5i
61
7i
113
.16 '
200
8
. 8*
H
6
9
JxJ
IJxlJ
6*
2ixl}
8
«•
—
1
n
2f
4
6|
n
i
3
21
4
3i
7
18
9
11
14
16
2ixl
3JxlJ
5x21
7^x4
Hi X 6i
16 *
lixl x'15
20
2 x IJ X 18
26
2 J X 2J X -24
34
8J X 2 J X -28
48
5 X 3 X -35
12
ixJx-U
UxUx.U
10
l|Klix'15
26
2 X 2 X -20
*■"
^^^"^
4
7x18
2x l|xl6
6
14 X -25
2ix2lx-22l
3i
'50
6
75
13
10
114
. 24
18
1*65
46
225
7
0.
13
19
27
1^x1
Ix 1
2xli
U X 1}
2}xlf
l|xl}
3fx2f
2f X 2}
5Jx4l
41x41
3
12
fxA
4
14
ixf
ixA
7
19
l|x|
IJxi
114
10
26
ajxi
l|xi
14
36
3xl|
24x|
'50
•75
1-65
2-25
i
A
A -26
A
A
A -39
A
A
A -47
A
10 15
•14 18
•20 -25
•24 -30
526 YACHTS OF Tli£ IKTERNATXONAL &AT1KG CLASSBB.
ft keelson be fitted its sectional area may be included in thai of
the keel.
The heeh of frames to be let into the keel. The web fsames
required in large yachts are to be fitted in way of mast, rigging,
and lead keel ; also for'd and aft in the largest yachts. Efficient
breasthooks and crutches to be fitted at the ends of yachts.
Beams to be dovetailed or dowielled to the shelf ; and as far as
possible to be fitted to the frames.
Bntts of outside planking to be 5 feet apart in adjacent i^id
4 feet in adjoining strakes. There should be three strakes
between butts on the same timber. For deck planking see under
*• Steel Yachts ".
Through bolt fastenings to be clenched on rings of the same
metal as &e bolts except in 5 metre yachts. All iron fastenings
to be galyanised. The number of fastenings attaching outside
planking to frames are : for planking 1" to 1)", one or two up to
5" width of planks, two from 6" to 7", three from T to l(f.
There should be at least as many fastenings (and at least two) at
the butts.
All bolts for attaching lead keels to be of copper or yellow-
metal. Their diameter is given by a table where it depends on
the ratio of depth to breadth at upper edge of k«el, and on the
product of its sectional area in square inches by the fore and aft
spacing of the bolts in feet. With xatio 1*5 to :2*0 the diameter
is A" for product under .5, 1" for 1-7 td 2-8, If" for 6-8 to 8-0,
2il^ for 12*0 to 18*6, and so intermediately. In all but the
smallest sizes (minimum A") deduct }" for each reduction of
•5 in the ratio. It is recommended that the bolts be fitted
alternately on opposite sides of the middle line.
Equipment of Yachts.
To be in accordance with the table. Anchor stocks to b«
one-quarter the weight of the anchor. Two end shackles should
be included in the weight of each cable.
MtNIHXTV BEQtnXVMENTS OP ANCHORS, CHAINS. JLND HAWSEBB FOB
Yachts oi* tbb Intsbnatiokaij Racins CIiAbseb.
3i
Is
1
5
7
10
15
23
Anobors.
Chain Cables.
HemporManillsl
Hawsers. |
■
i
a -
1
1
2
2
8
Weight in lb.,
ex Stock.
•
Diameter
(inohes).
Minimnm
weight Ub.)
5|
^1
15
20
85
45
75
Ciromn-
ferenoe>
1st.
2nd.
51
116
294
8rd.
Stad
Tiink.
Short
Link.
in.
2
2i
2^
3i
6
in.
2
Si
25
35
68
154
892
168
40
60
115
A
-A
873
970
8709
398
1083
4081
i
AKCHOES AND CABLES.
527
BOABD OF
Tbade Tests
FOR
Anchors (Extract).
Weight of
Anchor,
ex Stock.
Proof Strain.
Weight of
Anchor,
ex Stock.
Proof Strain.
Weight of
Anchor,
ex Stock.
Proof Strain.
cwt.
tons.
cwt.
qrs.
cwt.
tons.
cwt.
qrs.
cwt.
tons.
cwt. qrs.
200
96
16
0
63
43
12
2
26
26
12 2
190
94
5
0
61
43
0
0
26
24
16 0
180
91
16
0
50
42
7
2
24
23
17 2
170
89
5
0
49
41
15
0
23
23
2 2
160
86
16
0
48
41
2
2
22
22
7 2
150
84
2
2
47
40
10
0
21
21
12 2
140
81
0
0
46
89
17
2
20
20
16 0
130
77
17
2
45
39
6
0
19
19
17 2
120
74
16
0
41
38
12
2
18
19
0 0
110
71
0
0
43
37
17
2
17
18
6 0
100
67
6
0
42
37
2
2
16
17
7 2
90
63
6
0
41
36
10
0
16
16
10 0
80
68
10
0
40
36
16
0
14
16
12 2
75
66
6
0
39
35
2
2
13
14
16 0
70
53
16
0
38
34
10
0
12
13
17 2
671
62
7
2
37
33
16
0
11
12
17 2
65
51
0
0
36
33
2
2
10
12
0 0
62i
49
15
0
36
32
7
2
9
11
2 2
60
48
7
2
34
31
12
2
8
10
2 2
69
47
16
0
33
30
17
2
7
9
6 0
68
47
6
0
82
30
2
2
6
8
6 0
67
46
12
2
31
29
7
2
6
7
7 2
66
46
0
0
30
28
12
2
4
6
7 2
65
46
7
2
29
27
17
2
3
5
10 0
64
44
16
0
28
27
2
2
2
4
10 0
63
44
6
0
27
26
7
2
Note. — The strain is tensile, and is to be applied on the arm
or palm, at a spot which, measured from the extremity of the
bill, is one-third of the distance between it and the centre of the
crown. There must be no more than { in. permanent set
measured between fluke and shrckle pin.
528
ADMIRALTY TESTS. ETC.. FOR CABLE.
Admiraltt Tests and Weiohts or Stud-linked Chain Cable.
o
a 4
S
in.
4 '
8i
8i
Si
8
2i
2i
m
2A
2i
21
21
n
2
n
li
li
li
If
li
IJ
1
a
i-
§
A
i
a a el's
51
55
59
68
69
73
75.
77
83
85
89
95
99
107
115>
123-
133
145
157
175
195
221
237
253
29i
321
355
395
445
509
595
Weight of
100 ftkthoms
m mm
Approximate weight of
of oable,
with the
neoessaty
Joining
shaokles, etc.
One
joining
shackle.
One
end
Unk.
One
inter-
mediate
link.
One
common
Unk.
owt. qr. lb.
768 0 0
lb.
5357
lb.
272
lb.
256
lb.
204-8
675 0 0
441
224
210-5
164*75
588 . 0 0
359
18225
171-5
134
507 0 0
287-5
1459
137
107-25
432 0 0
2261
11475
103
84-88
396 3 0
199
1011
95
74-3
363 0 0
174
8838
83
65
346 2 21
1624
82-5
77-4
60-7
315 0 21
140
71*5
66-3
526
300 0 0
130
66-4
625,
488
270 3 0
112
56-9
53-5
41-9
243 0 0
95
48-4
455
356
216 3 0
80
4075
38-3
30
192 0 0
67
34
32
25
168 3 0
55 26
28
26 33
20-6
147 0 0
449
2278
215
1676
126 3 0
36
18-25
17-2
13*4
108 0 0
28
14*34
13*5
105
90 3 0
2175
11
10-37
8-2
75 0 0
16'31
8*32
7-75
61
63 3 4
11-87
610
5 7
4-5
52 3 6
8-87
4-25
4
32
46 1 18
6-89
3-5
3-29
268
40 1 20
561
2-84
2-66
2-2
29 2 2
3*53
1*79
1*68
1-4
24 3 23
272
137
1*29
11
20 2 14
204
103
103
•8
16 2 23
149
•75
702
•58
13 0 22
104
•53
•47
•41
10 0 12
•7
•34
•33
•28
7 1 20
•44
•22
•21
•18
Proof
load to
be borne
without
injury.
tons.
201-6
189-8
176*4
161*6
1468
137*6
1293
1261
116-7
112^
lOli
91J
81i
72
63i
66i
40i
34
m
22f
18
15*
13J
lOJ
8i
7
5J
' H
The breaking loads of the several sizes of cables are 60 per cent above the proof
load : and these latter are eqixivalent to the following ttressas per circular i inch
of iron, viz. : 4 inch, 441 lb. ; 8^ inch. 604 lb. ; 8| inch, 636'6 lb. ; 8 inch, 667 lb. ;
2| inch, 682-7 lb.: 2| inch, 598-6 lb. ; 2^^ inch, 606-4 lb.; 2^ inch. 622 lb. ; Scinch
and under, 680 lb. The proof load for any cable under 2ft inch is {diameter^ x 18.
Note.— The Board of Trade proof tests are the same as abov/e ; the breaking
tests are also alike for l| inches and smaller sizes ; above they are slightly smaller,
viz.: li", 58-7; 2", 101-8; 2i",167-6; 3",2041; 8? , 246-9; 4",282-2; being 40 per cent
above proof for all sizes larger than 1| inches.
The breaking tests to bo anplied to 3 links in each 12^ fathoms (Admiralty), or
-Jb 15 fathoms (Board of Trade).
ADMIRALTY TESTS AXD WEIGHTS OF CHAIN. 52£
Admibalty Tests and
Weights of Bigging Chain and
Cat Chain.
II
Q o
in.
1-
Proof
Strain.
Weight per
Fathom.
Diameter
of Chain.
Breaking
Strain.
Proof
Strain.
Weight
per Fathom.
Rigging
Chain.
Cat
Chain.
Tons.
•43
Tons.
•19
lb.
2-0
in.
Tons.
12-65
Tons.
5-625
lb.
30-0
lb.
A
92
41
30
S
15-19
6-75
360
35-75
\
1-69
75
4-75
«
17-72
7-875
39 0
^
253
1125
6-75
i
20-53
9-125
48-0
49-5
i
3-66
1-625
9-5
u
23-625
10-5
530
^
506
2-25
13-25
1
27- 00
12 00
61-0
64-5
h
6-75
3-00
170
li
34-31
15-25
73. 0
79-75
A
8-44
375
21-0
u
42 19
18-75
920
96-0
8
10.41
4-625
250
If
50-91
22-625
108-0
116-0^
Note.— The above breaking strains are two and a quarter times the
proof strain ; a piece of seven links out of each fifty fathoms being tested,
The working strain for cranes, etc.. should not exceed two-ninths th«
breaking strain or one-half the proof strain.
For proportions of chains see p. 688.
The Board of Trade proof tests for short link chain cables are in accord-
ance with the above table, but the breaking strains (applied to three links
in each fifteen fathoms) are only double the proof tests.
Admiralty Tests and Weights of
Pitched
Chain.
Inside
Inside
Weight
Diameter
Length of
Width of
Proof Load.
per
01 iron.
Link.
Link.
Fathom.
Inch.
Inches.
Inches.
Tons.
lb. oz.
^
1
A
U
2 8>
i
i
A
1
3 13
A
¥
1
1
4 11
A
i
§
n
6 2
Chain
a
ft
\l
1^
6 11
for
§
3}
A
11
8 3 '
Pulley
A
lA
i
2i
10 14
Blocks.
i
If
A
H
13 4
A
1J«
fi
31
16 15
g
\l
« .
48
21 4J
f
1
lA
§9
4§
61
26 8)
36 8 -
Steering
i ■
2ft
lA
9it
49 OJ
Not6.— The breaking strain is double the proof strain, and is applied to
a piece of seven links out of each fifty fathoms.
Mm
Lloyd's
Requirements fob
,
■ ■ j * ■ ■
Bower ' Stream
AnchorB.* 'Anchor.
»
Kedge
Anchor.
stud-chain
Cable.
Equipment Number.
Weight
of
heaviest.
Com*
bined
weight.
•
■s
•
•*>
•s
i
1
•
S
m
inches.
1,600 to 2,100
cwt.
3i
cwt.
7
cwt.
i
cwt.
i
lath.
120
3,400 io 3,900
6i
13
2
1
165
a
5,300 to 5,900
12
34i
4
2
195
lA
7,700 to 8,400
19
54
6)
H
240
lA
11,100 to 12,300
271
79
8|
4)
270
m
16,800 to 18,600
36i
104
Hi
5)
270
lit
27,900 to 30,800
48
137
17
8)
300
i
21
2,400 to 3,000
3
7
i
i
120
«
4,800 to 5,400
6i
13
n
1
165
if
7,400 to 8,100
llj
33)
4i
2
195
ift 1
10,600 to 11,600
18i
53)
6
8
210
:A
15,200 to 16,700
26i
75
8)
4)
240
li*
22,700 to 25,000
36
103
12
5)
270
m
32,200 to 34,800
48
136)
16i
7
270
2A
43,200 to 46,000
61i
175)
22
10
800
2A
54,600 to 57,600
76
217
28
14
330
2ii
67,000 to 70,200
91
259
34
18
830
214 j
80,200 to 83,800
105)
301
40|
22
830
3A I
100,200 to 105,000
127
362
49)
27
330
a 1
«fi1^2^^ '^Y*"*?^^/*'® required, except for equipment numbers leas ths
aiJmg ships) or 6,000 (steam ships) when two only are necessary. If stockier
per cent, to weight and increase test accordingly. All weights given are r.
this Rhonia weigh at least I that of the wei^-ht in the table J
Anghobs and
Cables.
(See notes on p
. 532.)
stream, Chain, or
Steel Wire.
Towline : Hemp or
Steel Wire.
Hawsers and
Warps-t
i
i
Size
(chain).
!-Size
(steel
wire).
1
^3
Size
(hemp).
Size
(steel
wire).
1
Fathoms
45
Inches.
— — — —
Inches.
Fathoms
75
Inches.
5
Inches.
18
Inches.
8
Inches.
45
ft
2
75
6i
24
4
—
60
a
2i
75
8
2i
64
—
60
«
3
76
H
84
7
—
75
a
3i
90
lOi
84
9
54
75
ift
4
90
11
34
104
64
120
ij
«
120
H
47
13
9
45
i
—
75
5i
2
3
—
45
8
2i
75
7
24
5
—
60
H
8
75
84
21
6
—
60
if
84
90
9i
34
6
5
75
ift'
4
90
11
84
2 of 6
2 of 5
90
14
44
100
12
4
2 of 7
2 of 6
90
li
«
120
14
4i
2 of 8
2 of 7
120
ift
6
180
15
H
2 of 8
2 of 8
120
14
6
130
17
7
2 of 8
2 of 8
1 160
m
64
140
—
74
3 of 8
2 of 8
150
ii
7
140
—
74
3 of 8
3 of 8
150
1
2
74
160
—
8
3 of 8
3 of 8
+ Each hawHer or warp hnx a UmiuiIi of 90 luthoiiiB, except In bteauiships whose
sqnipment number lies between 40.400 and 57.600, when it is 100 fathoms, and above
>7.6.0it is lao fathoms.
)2
ANCHORS AND CABLES.
otei on Lloyd's IteqiHrements for Anchors and Ctthles
(p. 530).
For Board of Trade tests of anchors see p. 527; for cables
e p. 528 ; for Lloyd^s tests for steel wire rope see p. 577 ff.
Cast steel anchors shall stand being dropped twice through
I feet (15 feet for 15 cwt. and below) as follows :
) honizontaUy on an iron slab, (6) crown downwards on
wo iron blocks which reoeive it on the middle of each arm.
hey shall afterwards be well hammered with a 7 lb. sledge-
immer. A test piece, 1" diameter and 8" long, shall stand
mding cold throngh 90° with an inner radius of 1^". Each
ichor shall be properly annealed.
Supply op j
iNCHOI
IS AND Cables to Wabships.
I
Bower
Anchors.
Stream
and
Kedge
Anchors.
Stnd-link
Cable.
Flexibl.
SieelWin
Bop*.
Class of Ship.
No.
8
Cwt.
180
No.
Cwt.
42
5
476
58
Si
II
Si
it
Battleship . .
20,000
Battle cmiser .
28,000
8
160
46
5
500
3
^■^
—
2nd class oraiser
6,000
8
80
24
10
8
350
n
160
6
3rd class cruiser
3,000
3
54
14
8
3
850
u
150
6i
Sloop ....
1,000
3
28
9
5
8
3l2i
11
Gunboat . . .
700
8
14
2
6
4
225
li
■
T.B. Destroyer .
1,100
1
1
20
18
3
150
u
100
8i
The equipment number &s length x (greatest moulded
kreadth -f*.dopth from top of keel to top of upper deok beam
kt side amidships). In awning or shelter-deck vessels measure
Lepth to second deck or to 8 feet b^ow top dook^ whioh«ver
BRITISH STANDARD PIPES AND SCREWS.
538
is greater. For erections in steamships add product of height
ana length of erection multiplied hy one for raised quarter
deokj by f for awning or shelter deck, poop, bridge, or
(forecastie^ by Jl for erections not extending to the side. For
sailing ships with erections add -^g to the number.
Tlw second bower anchor may be 7i per cent lighter than
the heaviest, and the third (if any) 15 per cent lighter ;
or (in steamships) the three anchors may be of equal weight
provided that their collective weight complies with the table.
The weiffht of anchor stocks must be one-quarter that of
the anchor specified. The heads o^ stockless anchors
must be three-fifths the total weight.
For short Channel crossings only two bower (the second
15 per cent lighter) and one stream anchor need be carried.
Stockless stream and kedge anchors must be 25 per cent
greater in weight than that specified.
Cables may be of unstudded close-link chain, if their
proof strain be two-thirds that required for studded chain.
Flexible steel wire rope of six strands with twenty-four
wires in each strand, are admitted if the diameter of each
wire be one-fifty-sixth circumference of rope ; the breaking
jtest must be equal to that of the rope required in the table ;
the circumference will be about £" less for sizes 6}"
and above, and i" less for smaller sizes.
When any length of chain cable is worn so that the mean
diameter at its worn part bears the following ratio to ito
original diameter it is to be renewed.
Thirty-seconds of an Inch.
Original diameter
Beduced ,,
22
20
*
24
21
34
30
44
39
V
54
48
62
55
72
64
82
73
92
82
BBITISH STAHDABD PIPES AND SCREWS
Section of B.S. Whitworth Thread (p. 534).
Fio. 241.
584 BRITISH STANDARD PIPES AND SCREWS.*
British Standard Pipe Flanges t (Selection).
For working Steam pressures of 55 lb. per square inch and
Water pressure of 200 lb. per square inch.
S
1
in.
i
i
I
H
2
3
4
5
6
9
12
18
24
««
*4 •
o
«JS
Si
^
i^
«s
i&
in.
in.
3i
2i
4
2}
4i
Si
H
31
6
ii
74
6J
8i
7
10
81
11
n
14i
12i
18
16
2di
23
82i
29i
5l
4
4
4
4
4
4
4
8
8
8
12
12
16
in.
i
i
i
i
i
1
Thickness of Flanges.
e?
o
'#
Uu
i
i
i
I
1
11
1»
Jogo
JVb^.— Pipes of li, 2i, 8}, 7, 8, 10» 14, 16, 16, 20, and
21 inches internal diameter and tables for other working pressures
are given in the Ck>mmittee's Beport. The above scantlings are
intended to apply to pipes for working steam pressures up to 56 lb.
per square inch and water pressures up to 200 lb. per square inch.
Bolt holes to be i" larger in diameter than bolts, except for i"
and i" bolts where holes should be.^" larger.
British Standard Whitwobth Threads.
(See tables pp. 535-537.)
The angle of thread is 55^, and thread is rounded ar
shown in fig. 241, so that H = -9605p and D =s -6403?, P
being the pitch. The circular diameter in the table is twice
the minimum radius.
* All tables have been reprinted by permission of the Ensineerint
Standards Committee.
t See Beport No. 10 British Standard Tables of Pipe Flanges; pub-
lished by Crosby Lockwood & Son, price 29. 64.
BRITISH STANDARD SCREW THREADS.
685
British Standard Whitworth Screw Threads (with sizes of |
hexagonal nuts and bolt heads).* 1
1
1^
1
h
II
3S
•
S
1
S
o
O
nil
Nut.
^•4
•gs
i
1
* i^ d
ao O H
S-5
fn.
per in.
in.
In.
in.
sq. in.
in.
in.
in.
i
20
0500
0320
•1860
•0272
•525
• 6062
•2187
A
18
0556
0356
•2414
• 0458
•6014
:6944
•2734
f
16
0625
0400
2950
•0683
•7094
• 8191
-3281
A
14
0714
0457
•3460
•0940
•8204
'9473
•3828
i
12
0833
0534
8038
•1215
•9191
1-0612
•4875
A
12
0833
0534
4558
•1632
Oil
1'1674
•4921
S
11
0909
0582
•5086
•2032
101
1-2713
•5468
«
11 .
0909
0582
'5711
•2562
2011
1-3869
•6015
1
10
1000
0640
6219
•3038
3012
1-5024
-6562
4«
10
1000
0640
6844
•3679
39
1-6050
•7109
3
9
•1111
0711
7327
• 4216
4788
1*7075
•7656
1
8
1250
[0800
8399
•5540
6701
1-9284
•875
ij
7
1429
0915
9420
•6969
■X
8605
2' 1483
•
•9843
U
7
1429
•0915
0670
•8942
2'
0483
2- 3651
•0937
If
r>
1667
1067
1616
1-0597
2
2146
2- 5572
•2031
14
6
1667
1067
2866
1-3001
2
4134
2- 7867
•3125
l|
5
•2000
•1281
3689
1-471S
2
5763
2^9748
'4218
1!
5
•2000
1281
4939
1-7528
2
7578
«• 1844
•5812
2
45
•2222
•1423
7154
2-3111
3
1491
8- 6362
•75
2i
4
•2500
1601
9298
2- 9249
3
546
4- 0945
•9687
2i
4 *
•2500
1601
2-
1798
3- 7318
3
894
4- 4964
2
•1875
21
3-5
•2857
1830
2-
3841
4- 4641
4-
181
4- 8278
2'
4062
8
3-6
•2857
•1830
2-
6341
5- 4496
4'
531
52319
2
625
3i
325
•3077
1970
2-
8560
6* 4063
4-
85
5-6003
2-
843
Si
3*25
•3077
1970
3-
1060
7-5769
5
175
5-9755
3
062
81
3
•3333
2134
3'
3231
8- 6732
5-
55
6- 4085
3'
281'
4
3
•3333
•2134
3-
5731
10- 0272
5
95
6-8704
3-
5
a
2-875
•3478
•2227
4-
0546
129118
6
825
7- 8808
3'
937
6
275
•3636
•2328
4-
'5343
16- 1477
7"
8
9- 0066
4-
375
5i
2*625
•3810
'2439
5
0121
19- 7301
8-85
10-2190
4-
809
6
2-6
'4000
•2561
5-4877
23-6521
10
11-5470
5-25
Note. — The thickness of nut is always equal to the fu)} diameter
of bolt.
* See Report No. 90 Britaeh Stoadaxd Soi»w Threads and Bepoiri No* 28 British
Standard Nnte. Bolt^beads, and Bpannera, pnblished by Messrs. Crosby Lockwood
and Son, price 28. 6d. each.
^6
BRITISH STANDARD FINE SCREW THREADS.
BBiTisa Standakd Fikb Screw Thbeads.* |
Full
No. of
Threads.
St
Pitoh. I
of
andard
>epth
Fhread.
Core
Diameter.
Croflfl
Sectional
Area at
bottom of
Thread.
in.
i
per in.
26
•0400
0256
•1988
•0310
^
22
•0455
0291
•2543
0608
1
20
•0500
0320
•3110
•076*^
A
18
•0556
0356
•3664
•1054
i
16
•0625
0400
•4200
•1386
A
16
•0625
0400
•4825
•1828
§
14
•0714
•0467
•6335
•2236
ii
u
•0714
0457
•5960
•2790
1
12
•0833
0534
•6433
•3250
il
12
•0833
0534
•7058
•3913
i
11
•0909
0682
•7686
•4520
1
10
•1030
0640
•8719
•6971
IJ
9
-1111
•0711
•9827
•7586
li
9
•1111
•0711
11077
l-ggoT
18
8
■1250
0800
12149
11593
14
8
•1250
0600
1-8399
1-4100
IS
8
•1250
0800
14649
16854
If
7
•1429
•0915
15«70
2-9286
2
7
'1429
•0916
,1-8176
3-6930
2J
6
•1667
•1067
2*0366
3 •2676
24
6
^657
•1067
2-2866
4 1066
2i
6
•1667
•1067
2 5366
5 0536
3
5
•2030
•1281
2^7439
5 9133
SJ
6
•2030
•1281
2-9939
7-0399
34
4-5
•2222
•1423
3-2154
S^1201
81
4-5
•2222
•1423
3-4654
9*4319
4
45
•2222
•1423
3-7154
10 •8418
44
4
'2500
•1601
4 •179a
13-7216
6
^
•2500
•1601
4-6798
17-2006
S4
3-5
•2857
•1830
5-1341
20-7023
6
3-6
•2857
•1830
5-6341
24-9310
* See Report No. 20 British Standard Screw Threads, pnblished by
essre. Crosby Lockwood & Son, pri^e 28. 6d.
BRITISH STANDARD PIPE THREADS.
537
Bbitish Standabd Pipe Thbeads.*
g
« Tsi
in.
<M •
•
"S.
Nominal E
of Tube
5 S«
Is
5
n
in.
in.
in.
in.
per in.
i
a
•383
-0230
•337
28
i
H
•518
•0336
•451
19
1
«
•656
•0335
•589
19
i
SJ
•825
•0455
•734
§
«
•902
•0455
•fill
1
lA
1-041
•0456
•950
14
i
lA
1-189
•0456
1-098
1
144
1-309
•0580
1-193
u
144
1-650
•0580
1-534
li
141
1-882
•0580
1-766
li
2A
2 116
•0580
2-000
2
21
2-347
•0580
2-231
2i
21
2-587
•0580
2-471
2i
8
2-960
•0580
2-844
2J
34
3-210
•058t)
3-094
3
34
3-460
'0580
3-344
1 1
3i
3|
3-700
•0580
3-584
Si
4
3-950
•0580
3-834
H
H
4-200
•0580
4-084
4
4
4*450
. ^0580
4-834
4)
6
4-950
•0580
4-834
5
5i
6-450
•0580
5-384
5i
6
5-950
•0580
5^8S4
6
64
e-450
•0580
6-384
7
n
7-450
•0640
7-322
10
8 •
84
8-450
-0640
5-322
10
9
94
9 450
•0640
9-322
10
10
104
10- 450
-0640
10-322
10
11
114
11-450
•0800
11-290
8
12
124
12- 450
'0800
12-290
8
13
13i
13- 680
•0800
13-520
8
14
14}
14- 680
•0800
14- 520
8
16
15}
15- 680
•0800
15-520
8
16
16i
16-680
•0800
16-520
8
17
17i
17-680
•0800
17-620
8
18
18f
18- 680
•0800
18- 520
8
Length of screw on pipe end = ^cP - 4" (d = nominal bore).
* Ree Report No. 21 British Standard Pipe Threads for Iron or Steel
Pipes and Tubes, published by Messrs. Crosby Lockwood & Son, price 2«. 6d.
SS8 OASLBB, OHAIMB, AMD THIH&LBS.
iHiF nrmss.
Adicealty Lengibs of Chain Cables.
A ckbls ooiuiita o( eight lengthi ol I2i fathoma eaeh, with
A joining ihukle to Mch length.
Hawbb Pipes mo Dzok Pipes.
Eaww pipea should be 10 diameters, and deck pipes
8 dlAmeten at the ch^n Mble.
AsuKALTT Pbofosiiohs or Chains, Cabi.eb, etc.
Stud I^k Chain.
Eitieme length = 6 times dluueler of cable.
width =3-6 ,, ,,
Dituneter ol eta; pin at midille = -6 „ „
„ ,, ends = diuueker ot cable.
l^^jM of Cham. Open TAnk. I^ging. Cat.
Bxlremt Imfth —
Diameter of chain .6 IS 4{
Extreme tcidth —
Dlametei ol chain . 3 ' 0 0}
Mild Steel Thiublbs (Fig. SOa). |
Biie ot
Hope.
A.
B.
c.
D.
E.
F.
1
«
^
2
A
1»
A
1 „ 2
1*
t\
S
4
S
S'
2 „ 21
'a'
B
51
Si
Si"
3
7
f
4i
*
10
?
•1
IS*
6
4*
2^
II
1
n
2
6i
4S
a*
mil
1*
i
2A
6
•I
5
16i
il^
?l
BiaaiNG BLIPS.
^m
Slips fob Chain BiaaiNO, sto.
Fzo. 242.
kl-sS^lIl^l 3&I i
Fia.ao.
•«»-••>■«••-• 10* A
FlO. 944.
Sizes of Iron in Inches fob Stbaight and
M^ONKKY-TAILED
Slips. (Figs. 242, 243, 244.)
..1
•
a
'3
6
o
i
t
1
.a
QQ
I
•
Oi
Mi*
1
M
C
H
«
O
s
M
M
•
«
1
6
1
s
•
€
■s
s
8
i
18
•
1
11
•
M
1
S
1
•
M
a
M
bo
li
1
n
•
3
•
8i
1
h
i
i
1
8
li
f
1
li
i
8
A
I
1
18
a
li
li
8
li
11
\
li
11
I
2
8
II
li
2
Si
li
18
8
18
lai
?}
11
18
I
I
i
1
li
ai
98
18
JL
8
li
i&i
Note, — The pins of the shackles are i" larger than iron of shackle.
BLAKE'S STOPPEHa.
Blake's Stoppers.
Ro. vs.
SIZES OF BLAKE 8 STUFPER.
HISCELLAKEOUB FfTTtNQ9.
FiS. MT.— Wood BcLiTimi Oliut.
Shatch Bi^ok Bindings.
Fro. MB.
BiaaniQ Slips abd FiTTiuae fok lowbr past of
SORBWB. SOBBWB TO FBBTENT IHBX
(p^ 5j4 J woBKisa LOOSE.
Fia. 349. Fid. xO.
Tb«ee SttingB are aU in pioportjoa to the Eteel vore rope.
WOOD B&LAYINO flLBATS, ETC.
DiMBHBIONS IN
Inches op Wood Belaying Clbatb.
(&e Sieteh, fig. 247.)
1
2
3
*
6
6
7
8
9
8
10
l\
1
?
^
1
1
12
i
4t
1
14
n
4
4
Ifi
!|
B
^
a
6
2
20
7
Si
6
6
2
22
2
'l\
7
«
2
26
1
n
8
9
s
?
an
H
n
7
s
:i
32
3
m
H
loj
"! 1
J iNOBKa OP Snatch Block Bindikgb.
{See Shetck, fig. 213.)
\\ i
■it:)
6U
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
B
S
T
U
V
W
X
Y
Screw
RIGGING »LIPS AND SCBEWS.
(Fio. 249. p. 549.)
Wire
Ft. Ins.
1 6
1 sj
1
I
If dia.
For Si'' Steel
Wire
For 2" Steel
Wire
4 11
For li" Steel
Wire
4 11
jdia.
Stowage of Chain Cable
(Cubic Feet per 100 Fathoms),
Diameter in inches
i
14
li
80
2f
i
20
s
27
IS
105
25
1
35
15
IJ
44
2
160
3
li
1|
Cubic feet . ;.
55
66
Diameter in inches
li
93
2i
2J
2*
Cubic feet .
130
190
280
Diameter in inches
2S
3i
660
Si
Cubic feet .
270
315
355
395
480
650
RIGQING SLIPS AISTD SCREWS.
545
5cRKWs> Slips, &c., for setting up Shrouds and Backstays.
Fig. 251.
^OCCs:
I* U'-Cd-
-•t*"
•0 -H I
-»MB«K A->—- >4
t I
w^
A
B
0
D
B
F
G
H
I
J
E
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Screw I
For7&6i
Steel Wire
Proof
Strain
44ToDB
Ina.
For 6 & 5^
Steel Wire
Proof
Strain
36 Tons
Ins.
For1}&4i
Steel Wire
Proof
Strain
S4Tons
For 4 & H
Steel Wire
Proof
Strain
18 Tons
IHB.
For 8 AH
Steel Wire
Proof
Strain
13 tons
H
For2&li
Steel Wire
Proof
Strain
6 Tons
Ins.
i
n
1
1
2r
1
u
1
li
1
10^
lx2j
Nn
54G
CABLE AITACHMENT FOK ST0CKLE8S ANClRJRS.
Fio. 933.
I
I
•» f — .
fj^3 SHANK OFAriCHOR
>4....i^ -
SHACKLE OF ANCHOR
•^^""^ TT — 7g\ ^ /'\ \ — T r
)
I I
A = Anchor shackle; B — End link of cable (withoat stud) : G = Enlargred link;
X » Tinned Bteel pin secored with lead pellet. Unit >= size of cable.
SHANK /.'"'^^\ -H^r-+>-v^f N \ ^^^..^rr^^^
[ ANCHOR \^\^^y>>— f ■4--^.- ^^ V^ IV'^^"^^;
L -..::.-'/'' L.xQ ! '•/ ^! I ^ '^-^-
Table of Dimensions of Crosshbads and Screw |
Slips for Boats* Davits. (Fig. 253.) 1
m
89 ft. Steam Boat
86 ft. SaUJng Pbinaoe
83>80 ft. flaOin^ Pinnace
37-28 ft. Steam Cuttet
84-95 ft. Cutter
82 ft. Oftf and other
Sniai Boats
Inehea
Inches
Inches
A
8f a
10}
A,
8|
A,
0
4V
4
ii
It
1
1)
C
9)
%
D
6
5\
6
E
1*
11
1}
8
F
4 81
G
1} U
1
1
.3. '»
.»
3
K
L
11 diam.. 1 pitch 1 lidiam.7Vl«ths pitch
4* O
1 diam.. 1 pitch
19
M
IS 18
N
t}«8i Sx|
Ifxl
0
P
1
9 It 1
•8
8
lOxMyll
9«8^yli
8X8JKI
T
U
!}
li
V
«
6
4
w
X
Y
Z
If
iUnk
l{
7 I6ths link
luik
1
* See note on page 647.
It luling pinnule*, 1 ungls and 1 double
I eingle uid 1 daable
X^ofe.—Sleam boats ind 86 te
block, 8} inches di&i
30 fMt and SS feet Bailing piniMaes, I
bloch, 61 incbM diameter.
31 1Mb uid 35 fe«t cotters, 3 single blocks, 8} iaobes diameter.
!i3 teet gig and smaller boats, 3 single blocks, 6} ioches
diameter.
B Tbe number ol tinks to be nirsngwl BO that tlie boat n»j be tnrneil In
wLlJe auapeuiled b; the screw-BUp.
648
BOW AND STRAIGITT SHACKLES.
Bow Shackle wltji Fobelock.
Fig. 254.
Straight Bigging Shackle.
Fig. 265.
•-1 '^lA'\-">
( 8-6
Unit
Pnxrf
Load
in
tn
Inetaes.
Tods.
J
f
1
i
u
<l
2
2i
i
4
5
li
6S
75
If
%
1*
11:
If
13: r
16]
1}
17f
2
20
2i
m
SliftcUe
in
iMhes.
•55 x
FORELOCK
Proof
Load in
Tons.
U.MVBESAL JOINT.
Fio. 2£B.
Pork"X" FoPk'T"
SIDE rnONT SIDE FRONT
CLCVATION
Ddiensions (all in inohea). 1
Dlan. ol
rod.
B.
0.
D.
K
».
0,
=.]..
J.
E. L.
M.
N.
0.
'
M
II
li
li
1
!
18 111
19
li
J 1 7
1 \n
aft
2)
li
lA
IB
al
a]
A
1
)i
»
3
11
«
11
li
U
li
A
1
li u
li
'
.5
aA
u
1.
1
1
3 |lB
u
i 1 a
.1-
..
II
si
li
«
1
a
!!i|ll
li
II!
A\ 9
1 |ioi
11
u
m
ai
11
IH
a
'
2||li!
550 GENERAL FURM Of BIXK'KS FOB AKC'HOR OEAK.
enXBAt FOBM OP BLOCKS FOE ASCHOB OSAB.
Leading Block. Table op Diuensioho. 1
We
lU
130
90
SI
rhor
1
[11^. In!.
SO 6
Ine.j 1d>
SI 1 H
1
E, F
1 : s
Im.
i
1
'i
1 1
1«
IS
Q&N^ERAL FORU OF BLOCKS FOR ANCHOR G
Leading Block, Tablb of Dimbnsions (cmtmiitd).
Anchor
L,
M
M.
M.
::^i-iH:
Cirta.
Inn
li
3
It
s nsiii'i
}j>
4t
!f
is ", 19
1
1
i
^
A '
li
I'
1 1
1
Weight of
Anchor
X
.,!v
T,
.,
..!.
Tr,
J J
in
^ ,
im; »l
Jl
"f
S
!'•:
Ji
;l ;
■!
1 '
!• »
ii 't
2 S
! f
W " 4G
;l
s
:i
i
? f
i 1
h 1
a 1
n 1
11 ., 7
ji
'
__
*
. .1 .
^^^!,&'
.
'
Pnw
T«rt
F.a.W. rape
used
Qwllng-
,fr„
ilk
1
tna.
Idb
!■
Ton
Ins.
I,.
I..
Ji'
•A
;
li ," 'j
Yi
{
a
102 aSNEBAL FORM OF BUOCIU FOR ANCHOR ORAS.
Fia.K ,— Oatuud Bi/xnc
f^
Cathead Block. Table op Diuensions.
OEXERAL FORM OF BLOCKS FOR ANCHOR GEAR. 553
Cathead Block. Table of Dimensions (jocmtinued).
Weight of
Anclior
Owta.
136 to 131
120 „ 106
105 „ 91
90 „ 76
76 », 61
60 „ 45
44 „ 35
34 „ 26
86 „ 18
17 „ 12
11 „ 7
K.
L,
Ins.
Ins
2
6;'
IZ
6
5;
4
1
4i
*
4
3|
lif
sl
2$
1
n
f
2
M
Ins.
lA
If
t
M.
M., ; N
Ins.
lA
J*
lA
I*
I
H
H
*
A
Ins.
6{
6|
J^
4|
4i
4
84
8
2f
2i
N,
Ins
3
3
2
2|
2
2
2
2
1
1
1
0
Ins.
2
1|
ij
1
\\
n
1*
1
I
I
I
Weight of
Anchor
Cwts.
135 to 121
120 „ 106
105 .„ 91
90 „ 76
76 „ 61
60 „
44
45
» 35
34 „ 26
26
17
11
»»
18
12
7
Q
II
Ins.
2}
n
2
1«
n
11
1
Ids.
2J
27
2i
2
li
1;
IJ
ij
1,
1
U
Ins.
3i
3*
3
2*
2
1*
W
Ins
lOi
10
8j
7J
5
4i
3i
21
Z
a
Ins.
2i
2i
2
2
14
It
1
1;
ll
Ins.
II
1
1
I-
-*r
Ins.
■ ji
I
I
A
Weight of
Anchor
Cwts.
135 to 121
120 „ 106
105 « 91
90 „ 76
76 „ 61
45
35
26
18
12
7
Proof
Test
Size of Catting
Pendant used,
F.S.W. rope
60
44
34
25
17
11
Ins.
1
1
i
8
Ins.
Tons
1;
68
1;
60
1
55
1
45
tm
374
■
1 •
30
22
}
17
121
1
•
84
6|
Size of
Ground
Chain used
Ins.
it
1
1
I
I
i
Size of
Chafing piece
Ins.
X ^
X
X
X
X
X
X
X
X
X
X
24
24
24
II
14
14
1
554
WBOUOHT-iaON CLEAT.
PlO. aW.—WEOUOHT-IRON ClSAT.
Diameter of rope=2i incbes.
yoter—Th^ dinwnsions given are multiples of the diameter at the tip.
FlO. a60.--En-i»LATS fob GENSBAL FURPOfiRA.
iir«(tf.— The dimensions given are multiples of the eye.
The proof strain In tons of eye-bolts or eye-plates is 10 times the square
of the diameter in inches.
Fio.a(U.— Clbab Hawss Sup.
8T»L ^IN
LPIN
Hi!
1 Nt h
^^3^^x:
! !
42— --H U-.
I* 42 -*|
cnss
r
-8 25
-M
JIToic.— The dimensions given are multiples of the eye.
SWIVBL-PIECE, END LINK.
555
Fig. 262, .— Swivbl-pibcb.
4-
i-7
•TO i-Tfr-
-!-c«— ! »J k-
C' »'
D I
* T
ijo 6'7S— "■
T
0 c, eiUHrged Unks (vKth sUy-pInf),
s D, end links.
B, swivel.
J^o^f.^-Ttae dimenaious giren are multiples of the diameter of tli« ca^l^le.
Fia. 2tt. -End Link, Enlarged Link (with Stay-pijI), and Common
Link of Chain Cable.
-f-gTO<7S !
r^% r
6-0
TRANSVERSE
-WSECTION
B, end link (without stoy-pin).
c, enlarged link (with stay-pin).
D, common link.
Q, stay-pin.
Note.—Th^ dimensions given are muitiptes of the diameter of the cable.
556
DECK BOLTS. ETC.
At top
At bottom
Size of Baths.
Length
ft in.
5 4
4 ^
ft.
1
1
Breadth
In. ft.
10 I
4| 1
In.
Depth
ft. in.
1 lOi
Deck Bolts.
. Screw-bolts of ^" diameter, with hexagonal or square heads
and nnts, are to conform to Whitworth's standard gauges for
nuts and bolts of the respective sizes, and are to be round under
the heads. The diameter of square heads and nuts is to be
reckoned across the sides, the same as for the hexagonal form.
Bolts with round heads — deck bolts — are to have hexagonal
nuts, the nuts to conform to Whitworth's standard gauges for
the respective sizes ; the diameter of the heads to he |" more
than that of the bolt, and the thickness of the heads to be half
the diameter of ibe bolt. These bolts to be square under the
head for a distance equal to the diameter of the bolt.
Bolt heads to be let into deck ^ th^ thickness of deck screw-
bolts, wrought iron, |" to 1^'', for fastenins? the wood sheathing
of iron ships ; diameter of Dolt to be measured over screw part ;
plain part to be ^" larger, and round under the head. Heads
to be round, of a diameter -f" greater than the diameter of bolt ;
thickness of head to be half the diameter of bolt. The head is
to contain a slot equal in length to the diameter of the bolt, and
of a breadth and depth of A" for all diameters of bolts. Nuts to
conform as to diameter to Whitworth's standard gauges for the
respective sizes, but the thickness to be in all cases f of the
diameter of bolt.
Fig. 264.
•«>D'*'
Deck bolt
o
1
-!>•♦
-*B-
iMSa
a
Sheathing bolt
t-fe
5
The screwed part is to be truly concentric with the head and
plain part, for screwing into metal through wood without en-
larging the hole in the latter.
The screwing of all the above descriptions of bolts is to be
\\Tiitworth*s standard thread.
PROCESSES FOR SEASONING TIMBER. 657
fiEASOiriHG TIMBER.
Natural Seasoning,
This is performed by . exposing the timber freely to the air
in a dry place sheltered from the wind and sun, and so stacked
as to admit of the air passing freely over all the surfaces of the
pieces. Timber for carpenter's work will require afoout two
years to season it properly ; for joiner's work, about four years,
or even longer.
Seasoniiig hy a Vaonnm.
The timber is placed in a chamber from which the air is
exhausted, heat being at the same time employed so as to
vaporise the exuded juices, the vapour being conveyed away
by means of pipes surrounded by cold water.
Seasonitiff hy Hot Air (^Davidson).
The timber is placed in a chamber and exposed to a current
of hot air impelled by a fan at the rate of about 100 feet per
second, the air passages, fan, and chamber being so arranged
that one- third of the volume of air in the chamber is blown
through it per minute.
The temperature of the hot air varies for different kinds of
timber as follows : —
Oak of any dimensions . 105** F.
Bay mahogany 1" boards . 280^-300°
Leaf woods in logs . . 90M00°
Pine woods in thick pieces 120**
Water Seawnvng,
This is done by immersing the timber in water — if shallow
and salt it is better than fresh — and letting it remain there for
periods averaging from 10 to 20 years, but it is sometimes only
allowed to remain 14 days, when it is taken out and stood
upright in some sheltered place where the air can get at it
thoroughly, so as to render it quite dry. Sometimes it is
thoroughly boiled or steamed for a day or two instead of being
immersed in cold water for longer periods. All these processes
tend rather to injure the stren^h of the wood, making it
softer, although it tends to prevent cracking, warping, and
shrinking.
Nate* — Slowly seasoned timber is tougher and more elastic
than when it is rapidly dried.
Seasoning by heat alone is very injurious to timber, as it
produces a hard crust on the surface and prevents the moisture
from evi^rating.
For joiner's work and carpentry natural seasoning should
liave the preference.
558 i'KOCESSES FOR PBESERYIHG TIMBER.
F&xtsEviire TntBSB.
Crbosoting. (Bethell.)
The timber is first well dried, either by being freely exposed
to the thorough circulation of the air or dried in an oven at a
temperature varying from 90® to 100° Fahr., depending on the
kind of timber.
One process is then to place the timber in a strong iron
cylinder, and subject it to a vacuum of 6 to 12 lbs. per square
inch for 30 or 40 minutes. The creosote is then allowed to flow
in, and a pressure put upon it, varying from 100 to ] 60 lbs, per
square inch, for about 1 to 2i hours. The other process consists
in simply immersing the timoer in an open tank containing hot
creosote, the temperature being kept up to about 120° to 160° Fahr.,
and left for some time to the natural process of absorption.
Mte. — Ordinary fir timber absorbs from 8 to 10 lbs. of
creosote per cubic foot of timber ; red pine, from 16 to 16 lbs. ;
memel, from 10 to 12 lbs. ; oak, from 4 to 6 lbs. This method of
preserving timber is the most generally used ; it is a sure pre-
ventive against the attack of the teredo and other marine
worms.
iMPRBaNATION WITH METALLIC 8ALTS.
Kyan^s Process,
This consists in immersing the timber in a solution of bichlo-
ride of mercury diluted with about 100 to 150 parts of water,
or about 1 to § of a lb. of the salt to 10 gallons of water.
Twenty-four hours are usually allowed for each inch in thickness
for boards, &c.
Ma/rgary^s Process.
Margary employed sulphate of copper diluted with about
40 to 50 parts of water, applied with pressure varying from 16
to 80 lbs. per square inch for 6 or 8 hours.
Burnett's Process.
A solutioa of about 1 lb. of chloride of zinc to 4 or 5 gallons
of water is iiijected and applied with a pressure vatying from
100 to 120 lbs. per squars inch for about 15 minutes. The
timber is then taken out and allowed to dry for about 14 days.
The tilfiber should remain ittidiersed for ab6ut 2 days for every
inch in thickness.
Puj/ne^s Process.
Payne's process consists in impregnating th* tiiinb^r with a
strong solution of sulphate of iron, and afterwards forcing in a
solution of any of the carbonate alkalies.
MEASUREMENT OF TIMBER.
55D
TIXBEB XEASTTBE.
Ik estimating quantities of timber duodecimals are nsually
employed — that is, the foot, inch, seconds, &c., are each divided
into twelve parte instead of ten, as in common decimal fractions ;
so that by this means feet, inches, and seconds can. be directly
multiplied by feet, inches, and seconds. Thus : —
12 inches make 1 foot. I 12 thirds make 1 second.
12 seconds make 1 inch. 12 fourths make 1 third.
And—
Feet multiplied by feet give feet.
Feet multiplied by inches give inches.
Feet multiplied by seconds give seconds.
Inches multiplied by inches give seconds.
Inches multiplied by seconds give thirds.
Seconds multiplied by seconds give fourths, Sec.
To Multiply by Duodecimals.
Rule. — Place the several denominations of the multiplier
directly under the corresponding denominations of the multi-
plicand.
Then multiply each denomination in the multiplicand by
the number of feet in the multiplier, and place each product
under its corresponding denomination in the multiplicand,
always carrying one for every twelve.
In the same manner multiply by the number of inches, and
set each product one place farther to the right hand.
Then multiply by the number of seconds, and set each pro-
duct another place farther to the right hand.
Thus proceed with all the other denominations, and the sum
of all the producte will be the whole product required.
Example 1.
Multiply 3 ft. 6J ins. by
2 ft. 6i ins.
ft. in^. eeca.
3 6 6
2 5 3
- '■ ■■ <i "ill
7 10
15 8 6
10 7 6
Ant. 8 7 7 16
Example 2.
Multiply 2 ft, 7 ins. 4. sees.
8 thirds by 1 ft. 2 ins, 3 sees.
3 thirds.
ft. ins. fleet. tbrcU.
2
7
4 8
1
2
3 3
2
7
6
4 8
2 9 4
7 10 2
0
7 10
2
0
An8,
3
1
8 11 4
2
0
560
MEASUREMENT Or TIMBER.
To Find the Solid Contents of Bound ob Unsquabed
Timber.
Rule 1.— Multiply the square of the quarter-girt by the
length, and the product will be the solid contents.
Rule 2. — Find the area in the following table which cor-
responds to the quarter-girt in inches, and multiply it by the
length of the timber in feet ; the product will be the solid con-
tents in cubic feet and decimals of a cubic foot.
Examples.
What is the solid contents of a tree whose girt is 60 inches
and whose length is 18 feet ?
By Rule 1.
4)60 ft. ins.
ins. 15 = 1 3
1 3
1
3
3
9
ft. 1
6
9
Ins.
0
6
ft.
18
1
sees
0
9
18
9
1
0
0
1
0
0
6
A
ns.
28
1
6
By Rule 2.
4)60
15 ins.
Corresponding to 15 ins. in
the table is 1*562 feet, and
eq.ft
1-562
1£
12496
1562
Ant. 28-112
Table of Constan^is fob Measubino Timbeb.
Girt
4
Ids.
Area.
Sq. Ft.
•250
•271
•293
•316
-340
•365
•391
•417
-444
•473
-502
•532
•563
-594
-626
Girt
4
Ins.
APGft.
Sq. Ft.
•660
•694
•730
•766
•803
•840
•879
•918
•969
1-000
1042
1-085
1-129
1-174
1-219
Girt
4
Ins.
Area.
Sq. Ft.
1-266
1-313
1-361
1^410
1-460
1-511
1-562
1-615
•668
'723
'778
834
'891
•948
2007
Girt
4
Ins.
17i
17i
17|
18
18i
19
19i
20
20i
21
21J
22
22i
23
23J
Area.
Sq. Ft.
2066
2127
2-188
2-260
2-377
2-607
2^641
2^778
2-918
3063
3-210
3-361
3516
3674
3-835
Girt
4
Ins.
24
24^
26
26J
26
26^
27
27i
28
28^
29
2^
30
31
32
Area.
Sq. Ft
4-000
4168
4-340
4-516
4-694
4-877
6063
6-252
6-444
6-641
6-840
6043
6-2.50
6-674
7-111
MEASUEEMENT OP TIMBER, AND BRICKLAYING. 5)1
TlHBKB MSABUBES.
40 cubic feet of unhewn timber
60 „ „ squared „
600 superficial feet of 1-inch planks or deals
400
li
300 ,
4 %% ^
2
240 ,
« «• *
H
200
3
170
H
150 ,
4
100 ,
make
Is
iquar
e of b(
1 load.
120 deals = 1 hundred.
Battens are 7 ins. wide, deals 9 Ins., and planks 11 ins.
Waste on Converting Timber.
African oak • = 100 per cent.
American elm = 15
Dantzic fir plank = 25
oak = 50
„ plank = 40
English elm =200
»
))
»»
99
English oak «= 200 per cent,
„ „ plank = 50
Greenheart = 25
Mahogany = 30
Quebec oak » 10
Teak = 15
Dantzic fir, when cut from planks . • » 10 per cent.
Yellow pine, when cut for head and stem work = 200
decks . . . « 10
>»
>»
))
>»
»
Plastering.
1 In. Thick. i In. Thick. | In. Thick.
1 bushel of cement will cover 1^ sup. yd., IJ sup. yd., -2^ sup. yds.
1 do. and 1 of sand „ 2 J sup. yds., 8 sup. yds., 4*
1 » 2 ,, „ 05 ,f 4^ yf 6y
1
n
3
»»
>»
11-
>»
6
9>
»
1 cubic yd. of lime, 2 yds. of sand, and
3 bushels of hair will cover .
' 76 sup. yds. on brksk.
70 „ „ earth.
60 „ „ laths.
Bricklaying.
8i«e in Iqs.
London stock bricks 8 J x 4J x 2}
Red kiln
"Welsh fire
Paving .
Square tiles
>»
ditto.
9 x4ix2|
9 x4|xl|
9| X 9| X 1
6 x6 xl
height In Lbs.
6*81
700
7-84
500
5-70
2-16
00
^62
STOWAOG or YAKI0D3 SUB3TAMCE8,
Table showing the Nuhbbb of Cubic Feet bequibed
TO Stow Okb Ton Weight op Vabious SuBaffANOBS.
Sabetances
Cu.Ft.
to a Tod
Sahetances
Cn. Ft
to a Ton
Ashes, pot and pearl
40
Indigo, in cases
66
Ballast, Thames . . .
22
Linseed .
56
Birley ....
Bread, in bulk
47
Marl
28
124
Molasses .
60
Coal, Admiralt}'
48
Oats, in bulk
61
„ Newcastle
46
Rice, in bags
45
„ Welsh .
40
Rum, in casks
60
CofioA, in bags
61
Saltpetre .
36
Cotton, compressed .
50
Sand, pit .
22
Earth mould •
83
„ river
21
Firewood
288
Sandstone
14
Flax ....
88
Shingle, clean
24
Flour, in barrels
50
Slate
13
Freestones
16
Sugar, in bags
39
Ginger ....
80
Tares, in bulk .
48
Granite stone .
14
Tea, in boxes .
111
Gravel, coarse .
23
Timber, hard .
40
Hay, compresAed . «
105
„ soft .
50
„ uncompressed
140
Turmoric .
66
Hemp ....
64
Silk, in bales ,
128
Hides, well packed .
64
„ pieces, in cases
110
„ loosely packed
84
Wheat, in bulk
45
Table giving the
Vabious Substances which in India |
ABB beckoned
AT 60 Cubic Feet
TO THE Ton Mea-
8UBEMENT.
Apparel
Elephants* teeth
Roping, in coils
Arrowroot, in cases
Ginger, in bags
Sago, in cases
Bee's wax
Gums, in cases
Sal ammoniac
Blackwood
Guuny bags
Sarsaparilla
Books
Hemp, in bales
Senna, in bales or bags
Borax, in cases
Hides and skins, in
Shellac, in cases
Camphor, in cases
bales
Silk piece goods
Cassia, all kinds
Indigo, in cases
Skins
Cigars, in boxes
Mace, in cases
Soap, in bars
Cinnamon, in bales
Mother-of-pearl, in
Stick lac, in cases
Cloves, in chests
cases
Tallow
Cofiee, in cases or bags
Musk, in cases
Tea, in chests
Coir fibre, in bales
Nutmegs, in cases or
Timber, hewn
Colocynth, in cases
casks
Tobacco, in bales
Cotton, in bales
Nux vomica, in bags
Tortoise shells
Cowries, in bags
Raw silk, in bales
Wines, in casks
Cummin seed
Rhubarb, in cases
Wool, in bales
QUANTITY OF PROVISIONS.
56i
Quantity of Provisions allowed in the Royal Navy
PER 100 Men for 90 days.
Kind of Provision.
Quantity
or
Net Weight.
Gross
Weight
in lb.
including
tare.
Approxi-
mate
Measure-
ment in
cubic feet.
Biscuit (ships without bakeries)
1,500 lb.
2,200
108
„ (ships with bakeries)
220 lb.
320
16
Beans or peas ....
650 lb.
620
20
Celery seed ....
3 lb.
3
—
Chocolate
300 lb.
«
345
6
Flour (ships with bakeries)
8,500 lb.
11,200
360
„ (ships without bakeries)
4,000 lb.
5,260
170
Jams and pickles .
500 lb.
710
15
Lime-juice (home stations)
50 lb.
160
4
„ ' (foreign stations) .
90 lb.
275
8
Milk (condensed)
800 lb.
1,240
26
Mustard
15 lb.
27
1
Oatmeal
100 lb.
113
5
Peas (split) . . • .
500 lb.
570
18
Pepper
15 lb.
27
1
Preserved meats
1,500 lb.
2,090
45
Bice
300 lb.
330
9
Bum
110 gal.
1,260
34
Salt (ships with bakeries)
250 lb.
320
10
J, (ships without bakeries) .
200 lb.
255
8
Salt pork
1,050 lb.
1,900
48
Suet
80 lb.
116
3
Sugar
2,600 lb.
3,230
85
Tea
350 lb.
440
18
Vinegar .....
.15 gal.
190
5
Total, average (with bakeries),
ex rum
25,200
820
SCANTLINGS OF BOATS.
m
"O-
s.
8'S
^ o
15
4
n't ■
«»*«><aD H»
X X
X X
I**!
e3<4i 04<O RNM»
t"r-l rH
a .
.ja^c
HV'Mt
« ^« •*
"0 ^i<Bi^'~' ' "IMW ■ VM III »»■ r"! '
4* 43
00 fc" Vh
Is 1 -si
ftg •j's -s-s II
♦» — w-a-SwflXx -S-Sx XXX X
«
« «
« eB
oB «.
rHC>r«n rMi
I X !«• I
u
°-s
X X
•=|t -I
**d
X X
«•*•* 09"*
HE*
lis
_ _ "SI'S
§g;2 |S
og a « *
CT"*« «"*
1 I
X X
«0 r-<
I I
X X
vrM w 4) 4)
-« o a
HIS **
00 '^ to
»o
I I
•^■*"*»'^ . »Hi-ii-i e^ . ^^o
x**x|xxx7?x '*'«
C-MB rC« Hln H« Kji "^
o
c
«B
«
a
%
o
K
>2
g
•«H
J
as
si
UL(
Hin
Hjn
•8 <* X X **®
^XSXX,
X X'ff X
^g^
HRI
I
I t t
«o>o
r«4
CI 04
QQ
0)
11^
III
s
9Q
I
6tg
QQ **
LIFTING WEIGHT OF BOATS, ETC.
66
Lifting Weight
OF BOATS.
•
Boat.
No. of m«n
(life-saving
capacity) .
•
Breadth
iex rubbers).
Depth (top of
hog to top of
main gunwale).
Lifting w^ghl
iex slings).
50' Steel Pinnace
70
50
ft in.
9 9
ft. in.
4 8^
tons. cwt» qrs
Ifi 0 0
45' Steel Barge .
65
45
9 6
4 81
14 0 0
85' Motor Boat .
46
a*;
7 71
4 0
4^ 0 0
80 ,, ,, . . .
40
80
7 U
8 105
.400
Jo ti »i . . ■ •
83
25
6 9^
8 4
9 7 0
JU t, «, . . .
20
20
6 1|
2.9
1 10 0
42' Launch (auxiliary
motor).
ISO
42
11 2
4 8^
8 7 0
86' Pinnace.
85
86
9 9^
8 li
5 0 0
84' Cutter (drop keel)
66
84
8 10|
8 1
2 8 0
80' Gig
26
80
5 9
2.4J
19 0
27' Whaler
27
27
6 0
2 2
18 2
25' Whaler „ „
22
95
. 5 10J
2 2
16 3
16'IMlngliy .
10
16
5 6
2 ll
6 8
20^ Cutter Gig .
17
20
4 9
lllj
12 3
Colours FOR Working Drawings.,
Representative Colour
neutral tint.
gamboge or chrome yellow,
crimson lake or carmine,
neutral tint or Payne's grey,
burnt umber,
diluted Indian ink.
carmine or lake mixed with burnt sienna
green.
Indian ink tinged with PrusaiaQ blue,
pink (tinged).
pale blue tis^fed with lake or carmine,
green.
burnt sienna.
Prussian blue,
or mild steel
2foie, — ^The usual method is to colour ai least all thi
sectional parts ; when both parts are coloured the sectiona
are coloured much darker than the other parts.
Material.
Armour .
Brass . .
Brickwork
Cast iron .
Clay or earth
Coal .
Copper
Glass .
Lead
OU. .
Steel .
Water
Wood .
Wrought iron
566
TIDES.
1 General Tide Table. |
^^^ ^ ^F - 3 ff^^a - —
Rise. 1
To liOndon Times.
1 Spring.
Neap.
H. M.
Feet.
Feet.
Aberdeen . . <
, sub.
1. 7
12
10
Ardrossan
add
9 35
10
8
Ayr . . . ,
► 91
9 43
8i
n
Banff • • • «
. sub.
1 39
lOi
8
Belfast . . • .
add
8 36
94
8
Calf of Man .
• »
9 10
16i
13
CampbeLown .
• »
9 38
8i
6
Cantyre, Mull oi"
• f}
8 28
4
mm^
Cardiff ....
• i>
5 11
37J
29
Carlingford Bar
• »
8 53
14
11
Deal • . •
» w
9 8
16
12i
Donegal .
» »>
3 U
llj
8J
Downs
■ »
0 38
15
Dublin Harbour
• >J
9 2
13
10
Dumbarton
. sub.
1 47
9
_
Fleetwood
add
9 5
26i
191
Galloway, Mull of .
• »
9 8
15
12
Glasgow, Port
sub.
1 49
9
—
Gravesend
• ♦>
0 57
17i
14
Holyhead
add
8 4
16
121
Holy Island .
► »
0 23
15
Hi
Kinsale .
• j>
2 36
Hi
9
Lamlash . • <
► »
9 42
10
7
Land's End .
» »
2 23
—
«
Tjarne, Quay .
» »
9 13
30
27i
Leith, Albert Dock Sill
>}
0 32
28
24
Londonderry .
• »
5 54
7}
5i
Maryport . . ,
»
9 42
21i
15i
Movillo . .
» »
5 86
n
5i
Nore Light . • .
Plymouih (Breakwater)
» >*
10 24
15
12
» »
8 55
16
H
Portland Bill .
' i*
4 47
9
6
Portsmouth (Spithead)
»
9 46
13
16
Queenstown • • .
► »
3 34
11
Si
Euncorn . ' . .
» jf
9 42
16
8
Soul Hampton .
» It
9 2
13
9
Sunderland .
' »
1 30
14
11
Troon . .
• »
10 5
• 10
8
Tyne River Channel
• M
1 54
11
8
Whitehaven
• »
9 28
25
19
The tide is on the averasre 49 minates
later each tide. The times of high water
raately 1-45 for the nearest tide to new
nearest tide to first or last quarter. The
be obtained approximately by division.
iater each day, or 24i minateflj
at London Bridge are approxi-
or full moon, and 6-60 for thel
times of intermediate tides may]
WBSTONS DIPFBRBNTIAL PULLEY BLOCKS AND CHAIN. 56']
Sizes and Tests of WFi=tTON's Differential Pulley
Blocks and Chain.
jyiffererUial Pulley Blodki,
Description
Weight to
be lifted
in Tons
Tested to
Weight
in Tons
Upper pulley, with sprocket wheel *
4
6
Lower „.
• • • • .
4
6
Upper „
with sprocket wheel ♦
3
4i
Lower „
• • • • «
3
*i
Upper „
with sprocket wheel *
2
3
Lower „
• • • • •
2
3
Upper
■ « • • •
1
H
. Lower „
« • • • •
1
H
Upper „
• • • • •
i
i
Lower „
• • • • •
k
i
Upper
• • • • •
i
i
Lower „
• • • ■ ■
i
i
Cliain for JHtto.
Diameter of
Iron of Link
Length of
. Chain
Width of
Link
Chain
tested to
Block to
Lift
Inches
■ «
Inches
H
Inches
HI
Tons
4«
Tons
4
^
2M
1}
3f
3
• 7
Iff
2if
ift
2}
2
5
lo
m
1
1*
1
i
n
M f
i
32
Hn
23 23
32 40
1 1
i
* AU ajprooke/t wbeeto ue to woric wi^ ^tnoh chain.
68
SIZI8 AND TSSTS OF
BNCIKBBRS' TACKLK BLOCKS.
W
11^ M«.?
h
GO
*-4
64
Oi
c»
CO
^4*
»-4
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>
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•^
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5S
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to
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00
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la
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570
WROUGHT IRON LIFT BLOCKS, ETC.
SiNGLB Wrought Iron Lift Blocks with Padlock
Shackles and Gunmetal Sheaves
M
I
O
S
m
13
15
18
p)
in&Vbs.oz
6} 7 8
18 8
30 8
70 8
131 0
Flexible
Steel Wire
Rope
S
as
ins.
s
2J
{
f2i
3
(3J
4
'4i
II
tons
4
6J
7
9
a *
S a
"- a)
ins.
i)
17 IlJ
24
81
39
59
If
If
If)
2
Gnnmetal
Sheave
a
M
ins.
i
If
If
2i
i
ins.
5
Hi
14
1,
lbs. oz.
2 7
5 8
12 6
23 8
35 4
i
5
ms,
U
IS
Shackle
59
S8
ins,
I
1*
l/i
IS
Size
in
dear
ins.
IJxli
1^x3
2x2|
2^x3^
34x4i
Bolts
ins.
ixf
lx|
IJxi
2x1
6 o
S'S
ins.
U
H
Lug
ins.
fxl
ixiai
Jxli
|x2
I
torn
3
15
29
Wrought Iron Snatch Blocks with Gunmetal Sheaves
Weight
of Block
com-
plete
Weight
to be
Lifted
Size of Block
Gunmetal Sheaves
Proof
Strain
of
Blocks
Across
the
Sheave
Length of
Shell in
direction
of Strap
Thick-
ness
Dia-
meter
Finished
Weight
Dia-
meter
of Pin
lbs. oz.
55 8
37 8
25 0
tons
6
4
n
ins.
8J
7i
ins.
16
14i
13
ins.
H
U
ins.
n
n
6i
lbs. oz.
12 9
"7 12
6 1
ins.
If
1
tons
9
6
5
WBOUGHT IBON BLOOKS
571
^ •
pwiijsaji
^44
I— 1
;?
3"
eo
M
$
H
pt»xiJSupiioji\
00
HS
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au JO jaipuniia
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iM
572 >VEIQHT AND STRBNQTH OF OOVSRXMEyT CORDAGE.
Table of Weight and Strength of Government
Hawser Laid Cordage
Sixeof
Yam
8lM
of
Rope
in
Ins.
Threads
in
ropes
6
12
16
21
33
42
64
66
84
102
120
106
123
169
201
249
860
861
408
468
634
676
1,200
Approximate
Weight of
Tarndper
ooil of 113
Fathoms
Break-
ing
Strain
Approxlimttt
Weight of
White per
ooil of 113
Fathoms
Break-
ing
Strain
40 Thread
Yam
Hemp. J
Tarred-Riga^
White*
Italian
30 Thread
Yam
Hemp.
Tarrod-Riga*
White-
Italian I
26 Thread /
Yarn Hemp.
Tarred-
Petersburg
White-
Italian \
,i
u
1
1
2
J*
6
?
9
12
16
cwt. qrs. lbs.
124
— — 26
— 1 z\
— 1 U
— 2 11
— 3 1
— 3 26
1-21
12 1
13 9
2 — 17
2 2 1
2 3 21
8 3 6
4 3 6
6 3 21
8 2 9
10 — 4
11 2 18
16 1 1
19 1 6
tons cwt.
— 3
— 6
— 8
— 10
— 16
1 —
1 7
1 14
2 —
2 10
3 —
3 10
3 18
6 —
6 9
7 18
11 10
12 16
14 16
19 1
24 —
cwt. qrs. Ibfl.
loj
— — 21
26i
— 1 7
— 2 —
— 2 16
— 3 7
tons cwt.
= ?
— 12
— 16
1 1
1 8
1 17
111
2 18
18 6
2 1 22
8 — 18
4 — —
4 3 22
7 — 17
4 4
6 12
7 6
9 6
11 10
16 10
9 2 26
U — 17
12 2 26
16—9
28 2 9
20 —
23 —
26 —
83 —
68 10
110 —
Table of Weight and Strength op Government
Coir Rope (3 Strand)
Size
of
Rope
in
Ins.
24
3
34
4
6
Weiffht of coil
of 111 Fathoms
cwt. qrs. lbs.
— 2
— 3
1 —
1 2
2 1
II
12
19
2
14
Breaking
Strain
tons cwt.
2 —
Size
of
Rope
in
Ins.
6
7
8
9
Weight of coil
of 113 Fathoms
cwt. qrs. lbs.
3 1 21
4 2 19
6 0 11
7 2 24
Br«akiDg
Strain
tons cwt.
2 17
3 16
4 17
6 8
MANILLA HAWSER.
578
Table of Weight and Strength op Government Manilla
Hawser
Manilla Hawser^
3 Strands. In Coils of 12© Fathoms each.
Size
of
rarn
Sixe
of
Hope
in
Ins.
Threads
in Kopes
Approximate
Weight of
White
Breaking
Strain
Approximate
Weight of
Tarred
(
Breaking
Strain {
1
1
IS
Cwt. Qrs. Lbs.
— — 27
TonsOwt.
— 13
Cwt. Qrs. Lbs.
— 1 8
Tons Owt.
g
^
33
— 2 0
1 3
— 2 8
1 1
g ■
2
S4
- 8 7
S 0
— 8 20
1 16
9
n
94
111
8 0
1 1 21
2 18
3
120
13 6
4 12
2 0 6
4 4
n
123
2 1 22
6 3
2 8 6
5 12
"S
4
159
3 0 18
7 19
8 2 12
7 5
4i
201
4 0 0
9 13
4 2 8
9 5
H^
6
249
4 3 22
12 13
5 2 18
11 10
S
H
803
6 0 2
16 6
6 3 14
13 16
1
6
360
7 0 17
18 0
8 0 19
16 10
Manilla Hamser, 4 Strwnds, In CoUm ^113 Fathoms each.
4
i
2
"S5
m
I
s
H
4
44
6
H
6
a
i
CO
3
71
12
20
28
29
87
47
68
71
84
2
4
6
7
10
9
12
15
21
24
80
I
1
14
32
53
87
122
125
160
203
253
808
866
Approximate
Weight of
White
Gwt. Qrs. Lbs.
— — 24
— 1 25
— 3 4
1 1 4
1 8 6
2 1 25
3 0 17
4 0 0
4 8 25
6 0 7
7 0 21
Breaking
Strain
Approximate
Weight of
Tarred
Tons Gwt.
- 10
1
1
1
16
2 U
4 3
5
7
11
3
8 14
11 8
18 15
16 4
Gwt. Qtb. Lbs.
27J
- 2 4J
— 3
1 1
2 0
2 3
3 2
2
Tons Cwt.
— 9
4
5
6
8
164
244
7
8
114
8
2 20)
3 20
0 24
Breaking
Strain
1
2
18
13
9
3 15
6 0
6
7
9
17
9 18
12 8
14 12
574 WEIGHT AND STRENGTH OF QOVE&XME2(T CORDAGE.
Table op Weight and Strength
OF Government
Cordage (Various).
Siae
of
Bope
No. of
iThread!
1 in
Rope
1
• 1 DeacripUon of Rope 1 ^^^ '
Weight per
Oofl
Breaking
Strain
Ins.
1 1
Fathoms
Cwt. Qrs. Lbs.
Tons Cvi-t.
2
36
Lasso . . ' 102 1
— 2 24
1 10
U
27
Signal halyard .122
— 1 7
1 3
ll
9
^Vhite pocking— Bus- | 280
1 0 0
— —
sian honp
12
White packing— Bus- ; 200
10 0
— - — .
Bian hemp
1
1
2
16
White packing -Rus-
sian hemp
166
10 0
^^
White spun yard (flax),
6 thread
280
— 2 0
_
Hambro'Line(Bnssian ' 2U
hemp), 3 strand ! '
- - 3
— —
12 thread
1
White deep-sea line,
cable laid, 3 strand
9 thread
42
1
9J
—
—
Large, 3 strand 6 thread
196
— 1 0
— —
^^
Small, 3 thread .
280
— 1 0
TABTiK OF VVEI€»fT AND STRENGTH
of Government |
Bolt Rope Cordag
E.
Size
Approximate
Approximate
Size
of
Yarn
of r
Rope
in
Incheti
Fhreads
in
Bope
weight of
Tarred
per Ooil of
122 Fathoms
Breaking
Strain
weight of
White
per Ooil of
122 Fathoms
Breaking
Strain
; 0wt.Qr8.Lbs.
Tons Cwt.
Cwt. Qrs. Lbs.
Tons Cwt.
1
6
12i
— 3i
— — —
— —
12
— — 35
- 7*
— — —
— —
o.
1
16
— 1 3i
— 9|
— — —
_ —
n
21
- 1 14
I2I
— 1 7
— J6
£•4
U
33
— 2 11
19
— 2 0
I 2
k
H
42
- 3 1
1 3
— 2 16
1 10
2
64
— 3 26
1 10
— 8 7
1 17
53
H
66
1 0 21
1 18
3 27
2 7
»^
H
84
1 2 1
2 10
111
S 18
2|
102
1 3 9
3 0
._ — —
..— _-
\
3
120
2 0 17
3 10
13 6
4 8
p, '
3i
105
2 2 1
4 0
— — —
_— -i—
H
123
2 3 21
4 12
2 1 22
5 U
rS S"
4
159
3 3 6
6 0
3 0 18
7 5
44
201
4 3 6
7 12
— _- —
i
6
249
6 3 21
9 0
4 3 22
11 7
S «
7
489
11 2 17
18 10
_. _ —^
W ,
8
639
16 0 25 24 0
i
t
u
Table
0 Beeakiso Stsbnoths op Flexible Steel
Wke
BOF£S
FOR Cbahes, Slihos, Sheeblegs, LAUHcaraa
If exposed to the weather these should be galvanized.
E. S.
Newall Jt Son, Limited, Lrerpool and Glasgow.
Maae 6 Btranas. each conUiniDg
Clto.
Weight
Fathom.
19 wires.
97 wires.
37 wires.
ISniree.
ei wires.
91 wires
lb.
Ton..
1"
, -96
21
3
89
3 63
1-60
5
88
S
82
6 79
2-17
8
91
8
13
8-40
2-93
11
11
56
10-82
3-84
IS
61
14
62
14 41
13
62
4'8J
19
95
19
24
18-54
18
13
604
24
43
23-11
22
00
7-30
29
27
28
78
28-53
26
12
8'64
34
46
S3
76
32-20
32
23
30
40
29
24
10-10
40
22
39
04
38-31
37
21
36
86
33
38
1170
47
65
46
23
44-84
42
59
41
74
37
90
13-6
55
40
62
51-97
50
30
48
OB
45
00
1536
61
39
53
94
67-67
56
5S
54
50
15
19-50
78
66
74
86
74 00
66
67
00
64
22
2400
97-51
08
92-44
87
76
83
52
79
61
29'20
115-43
112-90
104
49
101
20
93
75
34 60
131 72
129 62
128
32
121
75
116
78
40-70
153 66
149
143
56
133
66
47'04
179 34
170
68
167
60
151
25
5400
208-12
201
57
192
27
180
00
6144
230 29
226
53
219
46
200
78
69'36
262 90
252
90
247
92
283
52
9
7776
—
—
296 '51
380-90
268-79
256-86
76
NEW all's F.S.W. rope FOR HAWSERS, ETC.
Galvanized Flexible Steel Wire Bopeb
(R. S. Ne\\'all & Son, Ltd.).
For Towlines, Hawsers, Warps.
To Lloyd's and Board of Trade Requirements.
6 Strands, 12 Wires per Strand.
Circ,
Diam.
Weight
per
Fathom.
Breakinc
Strain.
Diam.
of
1 Dmm.
in.
1
n
in.
•318
•858
•398
lb.
•67
•85
110
TCHIB.
2
2i
3
in.
3
3i
4
u
11
•437
•477
•517
•557
•5B6
•636
•676
•716
•795
1-25
1*50
1-77
31
4*
5
5
6
7
n
ij
2
205
2-36
270
309
342
415
5J
68
7
8
9i
12|
8
8|
9
10
12
21
2i
2f
3
3J
•875
t»4
1^03
600
6-00
700
15i
18
22
14
16
19
3|
4
1-11
119
1*27
80\1
950
1050
26
29
38
■
22
25 ;
30
4i
42
135
1-43
1-51
1218
1340
1500
35
39
47
83
36
40
NEWALL S F.S.W.R. FOR RUNNING RIGGING, ETC. 57'
Special Flexible Galvanized Improved Patent Steel
Wire Kope (B. S. Newall & Son, Ltd.).
For Hawsers, Cargo Falls, BunniD^ Bigging, etc.
Made 6 Strands of 24 Wires each.
To Lloyd's and Bo/jid of Trade Bequirements.
Size.
Weight
per
Fathom.
B.S.
Diam.
of
Drum.
Special
Flexible Steel
Wire KopB.
When an owner
Ciro.
1
Diam.
in.
1
in.
•318
•3S8
•397
lb.
•90
115
1-40
Tons.
30
37
4-5
in.
2
2J
3
prefers to substitute
Special Flexible
$teel Wire Ropes
lor wire rope of
ordinary make, the
sizes of the rope
1|
If
•437
•477
•517
1-70
195
235
5-4
63
78
. 4
5
5ir
may be reduced in
accordance with
following table :—
Circ.
24/6
If
2
•537
•596
•636
271
320
367
89
101
117
6
7
n
Circ
if made
32/6
2J
2J
21
•676
•716
•756
400
4-50
5-10
12'7
l4-8
16-4
8
9
10
in.
2
2i
in.
2
2i
2h
2* ,
21
2f
•795
•835
•815
575
620
675
l§-2
197
220
11
12
13
2i
2i
3
2| &3
3i
3i
3
3i
3J
•956
1*034
1-114
800
900
1090
26'2
80-7
S5'5
14
16
18
3i
3*
3^
31
4 &4^
.^^
1193
1-273
r352
■ 12-50
1400
1620
41-0
45-5
52-5
20
22
25
28
30
33
4
4i
4
4J
4i
4|
5 & 5i
4A
1432
1-512
1-591
1800
2000
2225
59^0
65-5
730
^2
4i
5
5i
51
6
51
6
1-671 •
1-750
1-909
2412
2670
320a
78-5
880
105-0
35
38
46
6
6i
7
7
pp
R ADMIRALTY REQUIAEMENTS FOR
ADMnULTY BBQUIBEMENTS FOE STEEL WiRE BOPE.
A.
B.
0.
D.
For Standing
Flexible for
Extra Special
Rigging and
Hawsers and
Extra Special
Flexible (tinned
Funnel Ouys.
Running Rigging.
Flexible.
for boat hoiste).
Wires in a
Btrand.
Weight per
Fathom.
Breaking
Load.
Wires in a
Strand.
Weight per
Fathom.
Breaking
Load.
Wires in a
Btrand.
Weight per
Fathom.
Breaking
Load.-
Wires in a
Strand.
Weight per
Fathom.
1
No.
lb.
Tons,
No.
lb.
Tons.
No.
lb.
Tons.
No.
lb.
Tod
•i—
—
30
63
170
—m
—
— .
—
—
_
—
_
30
36
110
—.
_
—
—
—
—
—
—
—
30
81
96|
^^^
—
—
91
61
36*6
32*6
124
118
—
—
—
24
26
81i
—
—
—
61
61
29*7
269
108
99
—
—
—
24
211
68
37
23*6
90
61
61
24*3
21*8
90
81
19
19J
67
12
14
43J
37
19
73
61
61
20*2
17*8
73
65
19
16
45
12
11
83
37
15*3
58
61
157
6S
7
11
36
12
26
37
ll'S
46
61
11*9
45
7
8
26
12
6
18
37
8*6
33
_
.«
..
..
_
—
— _
—
37
7*2
27
.^
_
__
7
6f
17i
12
4J
13
37
5*8
22
—
—
—
— .
~-
— .
*-
37
4*6
18
—
..
—
7
s
U
8
12
2*
8
37
3-8
147
—
_»
_
7
12
2
&
37
29
11:
8
.-
^
—
7
2
6:
12
1*
4
37
21
—
_
—
— .
— .
— .
_
37
1-7
7
—
«_
_
—
~>
—
12
1
^
37
1'4
6|
—
..
..
— .
-~
>•
—
..
24
1*06
4-3
.^
.iw
mm
—
—
— .
12
i
2i
24
•876
3*6
_
._
m~
— .
—
—
— .
24
•676
2*06
-.
_
«.
— .
—
—
...
.»
..
19
•6
_
"—
«.
—
_
—
—
_
._
19
•35
1*4
~.
,
m.
— >
_
.—
_
.^
_
19
•22
•9
— .
_
m.
*""
"
^■"
^^
•^
"~
19
•12
•6
~ 1 ~
"■
Qualities A, B, C, to be g^alvanized by the hot process.
l11 to be made of the best quality steel. The wire to stand
be following ductilitj tests :-»
Test A, — To have the number of turns taken in itseU indicated ,
)r each gauge of wire in the table below ; distance between grips
3 be 8" for sizes above 036", 4" for sizes below -026", fcnd 6" for '
itermediate sizes. i
ADMIRALTY REQUIREMENTS FOR STEEL WIRE ROPE. 579
Diank. In In.
•128
•193
•116
•110
•104
•098
•092
•066
•080
•076
•072
No. of tarns
16
17
18
19
20
21
23
%i
26
27
29
Diam. in in.
•068
•C64
•C60
•066
•052
•048
•044
•040
•088
•086
•084
No. of tarns
80
83
84
87
40
48
47
62
66
48
46
Diam. in in.
No. of tarns
•oaa
48
•080
6S
•028
66
•026
60
•024
44
•022
48
•020
52
•a8
68
•0148
71
•0U6
90
•0088
and
less
100
Te$t B. — ^To have eight tarnA taken round its own part
and back again. The tensile tests on the ropes on a length
of not less than six circumferences. The wires and strands
to be laid in opposite directions.
The number of wires in a strand to be as follows : —
Quality A : 19 for 4'' and 4^", 7 for other sizes ; quality B :
80 for 6" and over, 24 for 5" and 5^", 12 for other sizes ; quality C :
87 for li" and over, 24 for J" to 1 J" 19 for other sizes ; quality D :
91 for 6", 61 for other sizes.
Useful Dimensions.
SiieM Canal, — Minimum depth 30'; breadth 120' on floor,
148' at water level.
Manehestar Ship Canal. — Minimum clear height of bridges
above mean water level 74' 6" ; allow 2' or 8' for flood ; size of
locks 600' X 65'.
NeweastU High-level Bridge, — Clear height above water at
B.W.O.S.T. 88' 1' . ^dd about 14' at low water springs.
BO
BUUilVAinn 8TBEL WIEB HOW.
BuLUVAMTtt' St£el Wirx Bopes (Galvanized),
Flejuble Steel Wire Rope,
6 Strande, each 19 Wires
Extra Flexible
Steel Wire
Rope,
6 StraDds,
each U Wires
a
Lbs.
•88
1-
2
S
4
&
6
8
88
68
78
75
SI
12
0
9-37
10-76
1219
13 62
lS-69
17-76
19*88
22-6
23-26
24-6
Tons
S'M
4-40
«-7
8-78
u-m
146
18*65
21*96
86-7
30-8
86-3
411
46-3
62*9
68-6
66*4
74-2
82-88
91-55
Special ExtM Flexible
Steel Wire Rope
6 Strands, each' BuUirants'
87 Wires Special Maki
Lbs.
1*0
1-M
2*00
8*88
4-0
6-2
6-3
6*81
8*81
10-88
11-9
18-5
15-8
17-»
19-0
3189
84-38
27-69
810
38*75
l&i
42*6
48-6
66-0
63-0
79-0
98*0
120-0
1420
Tods
7*26
10*0
13-0
15-76
19*75
34-0
39*0
38*5
88-6
44*6
51-0
58^
68*6
71*35
7935
87*76
96*76
108-75
113*76
132*0
1540
178-5
198H)
2600
3060
Tons
Note. — ^In these flexible rope tables the wire is calculated as taking a
reaking strain of 90 tons to the square inch ; ropes made of wire which is
alculated above that will take a proportionately higher strain.
Tn crane ropes (black) the weights are the same, but the breaking loads
r extra ilexible and special ropes are about 11 per cent higher.
BULLIVANT'S steel wire cord for AIRCRAFT. 581
BuLLiVANT's Plough Stekl Wire Stranded Cords, Tinnej.
FOB BRAOINO
, OasTROLR, ETO.> FOR AmCRAFT.
Approx.
Circum.
in
Inohes.
Flexible.
Approx.
Breaking
Load in
Lb.
Extra FLEXiBiiE.
Approx.
Weight
in Lb. per
iJoeoiL
Approx.
Circum.
in
InolMS.
Approx.
Breaking
Load in
Lb.
Approx.
Weight
in Lb. per
1000 &.
A
500
8
il
850
IS
i
6.50
10
A
1200
18
A
1120
14
ii
1500
20
A
1500
17
il
dooo
27
a
1680
20
A
2450
32
a
1750
23
i
2900
88
a
2300
30
ft
3400
50
a
. 2450
33
a
4200
56
A
2600
36
t
5000
64
«
3600
47
a
5500
80
a
4500
55
u
6250
90
a
6300
74
f
7250
100
«
8500
117
a
9600
124
1
1
11000
151
Note. — All these cords have 7 strands ; the flexible cord*;
have 7 wires per strand, and the extra flexible 19, except the
W' and A" which have 12, and the |}" which has 14.
Tlie extra flexible is preferred in England, but the flexible
is usually used on the Continent on account of its superior test,
and its smaller liability to fray, kink, or corrode ; the cost
of flexible is also less by about 25 per cent.
The elongation at the breaking load lies between 1 per
cent and 3 per cent j at any ordinary working load it is
negligible.
082 GENERAL N0TB8 ON WIRB BOPB,
General Notes on Wibe Bope (Bullivant & Co.}-
The diameter of palloTa and sheaves given on p. 580 i^
the minimum for slow speeds ; with extra flexible and
special extra flexible ropes the diameter can be somewhat
reduced, but the best working resnlts are always obtained with
tho diameter as large as possible.
Pulley groves should be so constructed that one-third of
the rope s diameter is fully supported. The depth of groove
should be 1| times the ' diameter of the rope.
When reeving sufficient turn sbould be put in the rope
io make the strands Imd wires " snug and tignt ". It is very
detrimental to a wire rope to allow it to chafe or ride on
its own part. All jdnxsks shorten the life of a rope ; the
strain should be pnt 0n as gradually as possible.
Bunning ropes shoold preferably be ungalvanised. ^ All
ropes should be well and frequently lubricated ; the lubricant
should not contain aoid or alkali, and the grease should be
well worked into the interetioes of the strands. Stook ropes
^ouid be kept in bl dry place.
For ordinary purposes a factor of safety of eix is sufficient ;
|>ut in shafts and other high speed workings 30 is frequently
sdopted.
Crane ropes made of wires whose strength is up to
135 tODS per square Inch (instead of 100 tons as uflual) can be
obtained ; but larger eheaves and barrels should be used.
A moderately tempered wire gives a better working result
than a highly tempered one.
Description of Hemp Cables.
S9tnp is laid up rigJit-kand^d into yarns.
Yarns are laid up left-handed into strands.
Three strands laid up right-handed make a hawser.
Three hawsers laid up left-handed make a cable.
Shroud'laid rope has a core surrounded by four strands.
bullivant's standard ckab-winch.
588
BuLLivANT*s Standard Crab-winch.
Lift from barrel in
tons
Circumference of
rope in inches
Extreme axial
width between
tips of handles
Extreme height .
Length between
centres of bolt-
holes in base
Width . between
. centre of bolt-
holes
Extreme width of
base
Extreme length of
base (in direction
of rope)
Height of axis of
handles
Diameter of barrel
Approximate
weight in cwt.
i
u
ft. in.
4llj
2 10}
1 6
2 li
2 3}
2 0
2 4J
0 6
2-9
If
ft. in.
5 8j
2 10}
llOi
2 5|
2 8}
2 4}
2 H
0 8
4-8
2
2i
ft. in.
6 10}
3 5
2 1|
3 li
3 4}
2 8
2 9}
010
7-8
2}
2}
ft. in.
7" 21
4 0
2 41
3 if
3 5
3 0
3 3}
0 10
10-0
3
2f
ft. in.
7 6i
4 3
2 6i
3 5}
3 9
3 3
3 3}
1 0
125
31
ft. in.
8 3^
4 6
2 11}
3 6}
3 11
3 8
3 3}
1 1
17-0
3}
ft. in.
8 6}
4 10
3 1}
31 }
4 3
3 11
3 6
1 3
190
584
LLUTDS &ULS8 FOB YARDfit, BXC.
Lloyd's Sizes and Scantlings for Yards and Topmasts
YARDS
SndQBftTter
SrdQnarter
M
1 TUiek&eflB
I
Iron Steel
Ins.
3
IS
4
7
T«
Te
•4
4
Ins. '
Iff
3
?
36
5
6
1
6
50
Topmasts. — The plating should be of the thickness given in
the table. The seatns of topmasts may be single riveted ; the
butts should be treble riveted, and their straps ^ of an inch
thicker in iron topmasts, and ^ thicker in- steel than the plates
they connect. There should be doubling plates in the way of the
lower mast cap. Topmasts should be efficiently strengthened in
the way of the fid holes and in the way of sheave holes where
such are cut, by the doubling plates, iron hoops, or by other
approved methods.
Lower Yards. — ^The plating should be of the thickness given
in the table. The seams of yards may be single riveted ; their
ISiOYD*B RULES POR YARDS, ETC.
585
OF Sailing Vessels and Full-bigged Steam Vessels.
YARDS
Ends
At Olkats
Ins.
4
5
H
6
6J
7
n
8
H
9
•'2
10
lOi
u
Hi
12
Thickness
Iron Steel
3
IS
a
S
16
I
2
¥
5o
Feet
32
34
36
38
40
42
44
46
48
60
52
54
56
58
60
62
64
TOPMASTS
Hebl
I
Ins.
2
2J
3
4
5
6
6}
7
8
BJ
9
20
20J
21
22
23
Thickness
ffon
Ins.
4
la
5
15
e
e
2
16
Steel
LowOT Part
of Head
ns.
I
1-1
2i
H
41
5
6
H
7
8
9
20
21
Thickness
Iron
Steel
Int.
9
0
1
H
2
3
H
41
*2
6i
6
6i
Hbad
Thickness
Iron Steel
Ins.
8
13
16
4
Te
4
16
ft
5
1
^«
2
16
Ins.
butts shonld be treble riveted, and coiinected by being over-
lapped, or by efficient butt straps. The plated should be doubled
at the centre, and the doubling plates should extend beyond the
truss hoops.
Where iron ot steel masts and yards are to be cooBtruoted
9therwi«» than in accordance with the tables* plans and parti-
culars of the same must be submitted for the approval of the
Committee.
Where steamers are intended to be fitted with topmasts for
auxiliary purposes, they might be one eighth less in diameter
than prescribed by table.
86
UiOTD S BULE8 FOR RIGOIKO, SVO.
Lloyd's Table of Sizes fob the Steel
Begteter Tonnage
under Deck
Plating Number
Fore & Main Shrouds
ft
n
»
n
„ Ohain plates
n Dead-^w •
„ lianyards O^emp;
„ BigglngSorewB^
(Diameter at
bottom of thread)
„ RigglngScrews )
(Diameter of Pins) [
Topmst. bckstys.
Top-gllkbckstys.
Lower stays
M n Topmast stays
„ „ Top-gUt stays
Mizen Shrouds . •
„ Topmast backstays
^ Top-gallant backstays
„ Lower etayH .
„ Topmast stays •
M Top-gallant stays
Bobstay Bar , , •
n Pin • • •
„ Chain •
Bowsprit Shrouds (Ohain)
Tons
8,000
andnoder
IJSOO
M.900
and nfiiliHP
97.900
No.
Size
Ins.
6 H
and Soap
3
2
3
2
H
H
H
H
H
H
5 4}
andoap
8 4\
3 Si
2 ^
8 4i
H
H
81
2 1*
Tons
2,600
and under
8,000
31.900
and under
No.
EHze
Ins.
6 H
and a cap
n
8
2
2
2
H
H
5 4|
andoap
3 41
2 3|
2 4|
2 4|
Si
4|
8 lA
Tons
2,800
andpnder
2.600
so/no
■ad under
ai.900
No.
Size
Ins.
6 6
aadeap
6
5
6
• 3|
5 4^
aadeap
3 4i
8
2
2
2
2
2
2
3
4i
41
8
4
8
8
1*
Tons
S,000
and under
S,800
18.400
and under
90.000
No.
6
Ins.
4|
ap
2i
3
2
2
2
andoap
1ft
u
4|
H
4i
8
2
2
8
4J
2ft
4J
4i
2|
8|
1. The abore requirements are intended to apply to veisels in which tlte
Imensions of the masts and yards are such as would not be deemed unusual for
essels of the respective tonnages ; where these dimensions are extreme, or in
:her exceptional cases where deviations from the above sizes are required,
gglng plans showing the sizes and arrangements of the several parts drauld be
ibmitted for the approval of the Oommittee.
2. Where four masts are adopted instead of three, the tonnage of the ves^sel
lay be reduced one-fifth, and where five masts are adopted, one-fourth, in Obtaimng
le sizes of rigging, &o., from the ubove table.
-I
LLOTBS RUIiES FOR RIQOINO, BTC,
§87
Wire Standing Bigging, etc., of Sailing Snipe
1.
Tons
1,800
andnnder
2,000
Tons
1,600
and under
1,800
Tons
1,400
andund^:
1,600
Tons
1,200
andnnder
1,400
Tons
1,000
andnnder
1,200
17.000
and un^er
18.400
16.600
and under
17.000
14.900
andnnder
15.600
12.800
andnnder
14,900
11.400
and under
12.800
*r^ Size
N**- Ins.
6 42
cAdcap
Nr« Size
^^- Ins.
5 ^*
andeap
andoap
■w^ Size
N^- Ins.
6 4J
andoap
v^ Size
^®- Ins.
6 4
andoap
n
n
2
n
1*
12x7
lliX6i
11x6
104x6
10X6
6
52
H
H
5
n
U
n
i|
1|
H
li
U
11
If
3 42
8 44
S 41
3 44
3 4
2 3i
2 82
2 8
2 22
2 2i
2 42
2 4i
2 42
2 44
2 4
2 42
2 4J
2 42
2 44
2 4
34
H
3
22
2|
6 4
ando&p
5 32
and cap
5 Si
and oav
5 32
andoap
5 3
8 4
3 32
3 3|
3 82
3 3
2 22
2 2i
2 22
2 24
2 3
2 4
2 32
2 8i
.2. 82
2 3
2 4
2 82
2 34
2 . 82
3
22
H
H
H
2
H
H
H
32.
8
22
^i
H
. 22
24
m
MS
IH
IH
lA
2 1
2 1
2 i
S 4
2 it
8. Where pole masts are adopted in vessels requiring one oap shroud only, an
idditiooal cap shroud is to be fitted, when the number of lower shrouds may be
jorrespondlngly reduced.
4. Where double top-gallant yards are to be adopted, a topmast cap backstay
should be fitted in addition.
)8
Lloyd's rules for riooino, etc.
lAOYD'S Ta6LB op S155E8 FOR THE STBEL
Register Tonnage
under Deck
Tons
800
and under
1,000
Tons
700
And under
800
Tons
600
and under
700
Tons
500
and under |
600 !
natlng Kdinber
10.000
and under
11,400
9.000
and under
10.000
8.000
and under
9.000
7,100
and under
8.000
w
v*» Siae
^•' Ins,
^°- Ins..
^'»- Z
F'ore & Mofai Shrouds .
andeap
1 ^*
andeap
and cap
» S
„ „ Chain plates
n
H
IS
if
„ „ Dead-eyes .
9|xH
9x5i
8|x5
8x5
,. „ Lanyards (hemp)
4f
4i
n
4
» » Rigging Screws
(Diameter at
bottom of thread )
H
li
If
n
„ Rigging Screws)
(Diameter of Pins) |
18
n
U
u
n ,1 Topmst. bckstys.
2 8J
2 8i
2 8i
2 3
» » Top-gllt.bcksty8.
2 2i
d|
2i
n
„ „ Lower stays
2 3}
2 fti
2 3i
2 3
„ „ Topmast stays .
2 3i
2 8i
H
3
„ Top-gUt. stays .
n
2|
n
n
Mizen Shrouds . .
5 n
i 21
4 21
4 2*
„ Topmast backstays
2 2|
2 22
2 2S
2i
„ Top-gallant backstays .
15
U
1|
U
„ Lower stays . •
2i
2*
H
n
„ Topmast stays .
2}
n
H
n
„ Top-gallant stays .
1*
ij
i|
U
Bubstay Bar ....
2i
2i
2
2
„ Pin ... .
1|
i«
li
U
„ Cliaiu
lA
lA
lA
1.1
Bowsprit Shrouds (Chain) .
2 H
\l
ii
' "1
5. The steel wii-e ropes are to be guaranteed to withstand the btvaking sties
iven in the table, and no henp is t« be used la the strands, a hemp ee^ onlr
) be fitted. '
«. A short leMgtfa of each of tbe wires composing the rigging wffl be f«Quini
ter being galvanised, to withstand a tensile stress equivalent to that act forti
I the table, and the aggregate strength of the wires must not be len thu
) per cent, in excess of that stress.
LLOTD*8 RULES FOB RIGGING, ETC.
M9
WiKE Standing Rigging, etc., gf Sailing Ships.
Tons
400
and voder
MO
6.200
»nd oniiex
7.100
No.
«se
in.
2
a
7ixi|
81
U
1
a|
3
SS
U
at
21
u
9
Tona
800
And under
400
4900
«Bd under
6,900
Ko.
2
2
8
8i£e
In.
2i
7x4*
8§
1^
2&
II
14
s4
If
91
9i-
u
21
2i
a
2
1»
lA
U
I
Sim
in.
9}
61
6
42
41
48
ik
43
4i
4i
4
8i
81
81
8i
8g
8J
Steel Wire Standing Rigging.
U).
29-2
96i-6
24-0
28*a
92*0
20-75
19*5
18 C
17-8
lA-4
16*4
14-5
18-6
12*7
Il«7
10-9'
loa
Is
Siee
Tons
in.
58
8i
58
8
48
22
U
92
49
28
40
21
88
98
86
9i
S4
2i
89
9
80
12
28
12
26
18
24
n
22
li
20i
u
19
••/:
.lb.
9-80
8-64
7-90
7-80
6-69
6-04
5*6a
4-84
4-43
8-84
8-80
2-92
2*M
2-17
1*80
X-50
^1
a 9
n
Tons
17i
16
14»
19
19
11
10
9
8
7
6
5i
5
4
82
8
Note.— The weights per fftthom *re not epecifled
by Lloyd's, bvt are in aoeordanoe with information
supplied by Messrs. R. 8. Nerwall A Son, Glasgow.
7. ESaoh wile wUl be required to be capable of being twisted around itself
lot loss than eight times, and of being untwisted and straightened before
breaking.
8. Where it is proposed to adopt iron wire rigging, the sizes proposed
.nd the guarantead tests should be submitted for the consid«ratioii of the
Committee.
LUiVlXt KVUSB FOB MASTS AND TJl
tu (Whu, hud la BtaDd M bsm
iSirr^s
Tblckw
ffifiUa'
GnlD
2
•
tt°
8=
u
<>
70=
«
?«•?
L'Slss
uS^^tiiriinnr dHk l^ibwrTAi uj
■J^ BUflKud br Uma aull* u Jpr
■lUnlUu tka wtaoh loulh of Uw
[t bha piktaa ba hmncad ■* Aa
lA^ AllbfnvpEj
tic v«dslng to Uie ^mn
SStes
U. Tkl* Miawimnta te Iwodh mar ba it
laaU ma; ba alnfle^atad.
naJly BlienffUiBnad, by an^w (
LLOTB S RULES FOR BOWSPRITS, ETC.
6dl
Lloyd's Sizes fob Bowspbits op Sailing Vessels and
FuLL-BiGGED Steam Vessels.
iron AND steel BOWSPRITS.
o
s
«
Thick-
ness
Peet
14
Ins.
164
15
174
16
19
17
20
18
214
19
23
20
244
21
254
22
264
23
28
24
29
25
30
26
314
27
33
I
QQ
Ins.
5
6
13
s
Iff
e
I?
A
Iff
17
T
T5
7
1?
7
18
8
15
a
T8
IS
8
IS
A
Ins.
6
2S
e
2S
7
S7
20
JL
20
8
20
8
So
8
29
9.
20
&
0
2(5
9
15
Hbrl
Cap
Thick-
Thick-
1
ness
n
ness
1
5
Ins.
1
Iiu.
CO
Ins.
1
QQ
Ins.
Ins.
Ins.
14
6
6
9o
12
4
15
5
20
15
^
igi
124
5
15
&
16
A
!>
13
6
15
fo
17
&
i^
14
A
&
18
h
ilb
15
5
15
e
25
19
6
15
&
16
&
e
20
20
A
^
164
6
15
J;
21
6
le
7
30
174
6
i5
7
S3
22
«
le
7
3o
184
e
15
^
23
^
8
25
19
e
15
&
24
7
la
jk
20
e
15
^
25
A
lib
21
6
15
7
55
26
A
8
35
214
A
7
20
27
7
15
&
22
6
15
^
Sizes of Angle Bars
Iron
Inches
24x2 x^
ijQ 2^ X 2 X ^
3 x2 x^
3 x2 x^
3 x24x^
3 x3 x^
84x3 x^
34x3 x^
4 x3 x^
4 x3Jx^
4 x34x^
^2 ^ ^a ^le
44x34x,%
Steel
44x34
^15
Inches
24x2 xA
24x2 x^
3 x2 x^
3 x2 x^
3 x24x^
3 5c3 x/5
34x3 x^
34x3 x^
4 x3 x^
4 x34x^
4 x3jx&
44x34x^
44x3jx^
44x34x5*5
592
LLOYD'S RULES FOR MA8IS, ITC.
Lloyd's Sizes and Scantlings fob Masts of Sail-
IBON AND
Partnbrs
Hbrl
HOUNDB
Hbao
-1
i
1
1
EXTBXMB
Lexqth*
«
s
Ins.
16
Tna
13
Thiokneas
1
In&
13J
ThiokneaB
Ins.
11
Thickness
Iron
Ins.
Steel
Ina.
Iron
Ins.
Stcd
Iiu.
Itmi
Ins.
Steel
Ins.
Iron
lat.
Steel'
1
Feet
48
Ins.
s
le
'^
51
17
6
le
h
13i
^
^
14
^
1^
Ui
^
^
«
54
18
A
6
So
14
A
5
25
15
A
^
12
A
6
20
57
19
16
A
15
A
fl
30
15i
A
is
12^
^
5
55
tin
60
20
6
T
15
16
A
&
16i
A
^
13J
A
ft
^
63
66
21
22
6
T
SO
17
A
A
e
55
6
SO
18^
*
&
14
A
fl
55
6
2S
69
2S
9
6
Iff
h
18
ft
^6
19
A
&
16J
6
l6
6
20
72
24
fl
16
7
Ito
19
ft
fl
25
20
A
&
16
B
16
6
20
76
25
T
16
8
20
19^
ft
^
21
A
T
13
16i
e
Te
7
20
1
78
26
T
le
8
20
20
ft
7
35
21i
A
&
17J
fl
le
7
20
1
J
81
27
8
Te
&
21
ft
T
25
22i
A
^
18
e
16
7
20 .
84
28
16
9
20
22
6
16
T
20
23
T
20
18J
e
16
7
20
5§1
87
90
29
30
IB
8
16
9
26
22i
23
^
A
r
lo
ft
20
24
25
1^
19J
20
n
16
ft
7
20
^
93
31
9
15
10
20
24
t'8
^
26
A
&
20J
6
Te
7
20
I
96
32
9
Te
¥o
25
A
8
20
26|
A
&
21
Tff
T
20
^
* The length for regulating the scantlings of the mast to be taken,
LLOYDS RULES FOR MASTS, ETC.
598
iNG Vessels and Full-rigged Steam Vessels.
STEEL MASTS
Chesks
Sizes of Ang^e Bars in Masts
Thiol
ofP
mess
late
Sizes of Angle Bar
Iron
Steel
•
Iron
Ins.
Steel
Ins.
8
20
Iron
Steel
Inches
Inches
Inches
3jx2ixjL
Inches
3ix2ix^
—
7
8
20
3ix3 x^
3jx3 x^
—
—
7
18
1&
3ix3 xA
3ix3 x^
—
—
8
IS
9
30
4 x3 x^
4 x3 x^
—
—
8
Te
/«
4 x3 x^
4 x3 x^
—
—
&
ft
4 x3 x^
4 x3 x^
—
—
&
B
55
4jx3 x/g
4Jx3 x^
—
«
55
4^x3 x^
4^x3 x^
—
—
ft
It
ft
4ix3 x^5
4jx3 x^
—
—
ft
i§
5 x3 x^
6 x3 x^
—
—
ft
}&
5 x3 x^
5 x3 xj§
—
—
ft
ht
5 x3ix^
6 x3JxJJ
S^xSx^
3ix3x^
^
a
6 x3jx^
5 x3jxl§-
4 x3x^
4 x3x^
tl
M
5^x4 xJS
6jx4 xJJ
4 x3x^«5
4 x3x5«5
ii
H
6 x4 xiS
6 x4 xi^
4ix3xA
4^x8x555
«
in
6 x4 x^J
6 x4 yii
6 x3x^
5 x3x^
M
)2
So
6 x4 xig
6 x4 xij
In aU cases, from the cap to the top of the keelson.
Qq
OF KOKEIGK PORtU FBOV I
TnrKlud (Uidelt
GibTRltsr .
no
Ne-Z«iuid
'am
ougo : :
"aaj
Pekln (Onlf)
e,*"!!
l.SM
,!S
PortJKkBn
s]rMG
PoloPmaBg
7^111
Quebec . .
I.MI
E^Bo™ .
Ei~ Jnnf iro .
iottem&m .
»',c«s
e«iFr.DolK(.
Ml
SlLUIgbBl .
Sh«n.« .
asm Leone'
1.8»7
BI>K.pon .
i.aao
at. Helena .
J,«W
St Iflgo (Cspe
et, John (New
4,Me
8».PeUnH««
0,»1
Stockholm .
'i;^1
ewMBl™.
1S,9I0
Sjdiiey. ' ,
B^g
Venice. .
WiabiagUm.
1,083
Tokobum. .
M],«tl
(U),8»
rAINTS, RTO. 595
PAINT8, VABNI8HS0, ETC.
Paint Allowajtob.
Allowance per coat per 1,000 square yards.
1. Outtide surface.
Cr^y.— White lead 2001b., black painfc 221b., dryers
15 U>., Bpiritg of turpentme 15 pintp, lin^ed-oll 45 pinte,
irA«<?.--White lead 671b., ainc white 13.41b., dryers
151b., spirits of tarpentine 15 pints, linseed-oil 50 pints.
Yellow, — ^Yellow ochre 1501b., dryers 10 lb., spirits of
turpentine 10 pints, linseed-oil 40 pints.
2. Varnish and Inside Paint.
Black.— BlsLck paint 1251b., dryers 101b., boiled linseed-
oil 40 pints, linseed-oil 10 pint*.
JTAi^^.—White lead 671b., zine white 1331b., dryers
15 lb., spirits of tarpentine 15 pints, linseed-oil 100 pint^.
Bed. — ^Venetian red paint 1501b., dryers 101b., spirits
of turpentine 10 pints, Imseed-oil 40 pints.
rW/iHi'.-r-Eiiglish yellow paint 1501b., remainder as red
(above).
Green, — Green paint 1701b., dryers 151b., boiled linseed-
oil 40 pints, linseedroil 40 pinto, spirits of turpentine 5 pints.
Confined Spaces. — Iron oxide paint 2801b., dryers 151b.,
boiled linseed-oil 60 pints, linseed-oil 40 pints.
Other internal portions. — ^Red lead paint 280 lb., white
lead paint 701b., dryers 801b., boiled linseed-oil 60 pintail
linseed-oil 40 pints.
Copal varnish for inside and outside. — ^200 pints.
White pr grey enamel. — 100 pints.
Flatting for grey enamel. — White lead 1301b., ordinary
block 151b., dryers 12} lb., spirits of turpentine 32J pints,
linseed-oil 7} pints.
Flatting for white enamel. — Zinc white 1001b., white }ead
501b., dryers 12} lb., spirits of turpentine 30 lb., linseed-
oil 7} lb.
Bituminous Paind for I!Nai^E«-fioosi a^ BoiLEBrSooM
Bilges.
(JTaral Constructor, H. WjIHams, U.S.Nv, A.S.N.J).)
Solution.^ldQlh. eoal-tar pitch and 501b. of Trinidad
asphalt melted together. When cool add 65 lb. coal tar naphtha
and 15 lb. of miner^ oil.
596 FAULTS, ETC.
Enamel. — 3501b. coal-tar pitch and 3501b. Trinidad
asphalt, melted together and boiled aboat 8 hours.
Cement. — 1501b. coal-tar pitch and 1201b. Trinidad
asphalt melted together and boued about 3 hours ; then add
and stir in 150 lb. Portland cement.
First apply solution cold, after ihoroughly cleaning the
metal ; 48 hours afterwards apply cement on horizontal
and enamel on vertical surfaces. On overhead surface substi-
tute I in. Portland cement for the enamel or cement. Both
these last to be applied hot ; the cement about | in. thick,
the enamel (with a brash) ^" to }".
Good Dbyebs for Coloubbd Paints.
3 galls, of linseed oil, 1 lb. of manganese, 1 lb. of red lead,
1 lb of litharge. To be left for three hours.
Distemper.
112 lbs of whiting, 28 lbs of dry white lead, 7 lbs. of glue.
To be mixed with boiling water.
Hammock Clotub. .
46 lbs. black, 3 J galls, of boiled linseed oil.
2 lbs. litharge will paint about 100 yards running measure.
1 lb. white paint will cover about 3 square yards.
1 lb. black „ „ „ » 6
t» y»
Habmont of Coloubs.
Red looks well with white, black, or yellow. .
Blue „ „ „ white or yellow.
Green „ „ „ black, white, or yellow.
Gold „ „ „ white, black, brown, blue, purple, and
pink.
Mixing Paints.
White lead and lamp black mixed together make an ash
colour.
White lead and ochre make the colour of new timber.
Yellow ochre and white lead make a buff colour.
White lead, vermilion, and lake make a flesh colour.
Lake and white make a carnation.
Yellow ochre and red lead make an orange.
DATA FOH CAULKING
597
Red lead, yellow ochre, and a little white make a brick-
colonr.
Burnt umber and white make a walnut-tree colour.
Yellow spruce, white lead, and a little black or burnt umber
make a stone-colour.
This and experience will show the result of many other
oolours.
1 lb. of verdigris to 3 lbs. of white lead.
1 „ mineral „ 2
1 .. Antwerp „ 1|
w
)
28 lbs of white lead
1 lb. of litharge
6 pints of linseed oil
2 „ turpentine
28 lbs. of black paint
1 lb. of litharge
10 pints of linseed oil f
2 „ turpentine )
46 lbs. of black
3^ galls, of linseed oil
2 lbs. of litharge
[ will cover about 100 superficial
yards.
will cover about 160 superficial
vards.
will paint about 100 yards (running
measure) of hammock cloths.
1 lb. of white paint will cover about 5 square yards.
1 lb. of black paint (thin) will cpver about 7 square yards.
Weight op Oakum and Pitch
y IN LBS.
, BEQUIBED FOB 1
EVERY 100 FKBT OF SeAM IN LENGTH.
•
Wales,
Materials
DMka
Top
' Sides
Channel,
and
Main
Wales
bottom
Middle
_
Oakum —
Very slack
seams .
8
8
11
15
8
Ordinary slack
seams .
5
6
7
10
5
Pitch—
Middling-sized
seams .
24^
14^
18^
I8i
18^
Over spun-yam
when used .
■^**
m^a^
n
598
DATA FOR DECK CAULKIHG.
Oakum,
Pitch, &c
., FOB Woodwork.
Table showing
THE Quantity ake
1 Description of 1
Oakum, &c., used in Caulking * New Wobk ' ik H.M. I
Dockyards.
1
Thlokneas
of Plank
Donble
Threads of
Oakum
Single
Threads of
Spun Yarn
Thickness
of Flank
DouUe
Threads of
Black
Oakum
Doable
THrRftdsof
White
Oakam
M
I
^
IS In No.
Sin No
SB
Ins.
9
U in No.
9
la „
2 „
1
8
W n
w—
g
8
11 „
2 „
7
» «
__
1
7
10 „
2 „
S
6
7 ,,
_
6
5
I :;
2 „
2 „
5
4
5 «
4 »
1 in No.
1 „
1
4
3
4 n
2 „
1 .,
1
3
2%
3 „
2 »
I I
S
ik
8 „
■g
3
2
2 „
—
^
m
^
I
1 »
—
S
Gnn
decks
4
3
3 «
2 „
—
1 „
1 .,
Single
Single
Threads of
Threads of
Bkok
White
Oalnim
Oakum
^a
8
2 lu No.
1 in No.
Weat
dec]
2i,
2
2 „
1 »
1 „
1 „
Weight op Spun-yarn of different Sizes, in lbs.,
REQUIRED TO FILL EVERY 100 FT. OF SEAM IN LENGTH.
Materials
Number of Yarmi
.J . ,
13
9
6
4
3
2
Spiih-yarti
Lbs.
Lbs.
Lbs.
Lbs.
If
Lb&
1*
Lb.
i
YABNISHBS. 599
Vabnishss.
Black Japan for Metals. -^Bnmt nmber 4 ozs. , asphaltom
1^ oz., boiled oil 2 quarts. Mix by heat and thin with turpentine.
Anothef M^ei^. — Amber 13 oi8.» atiphaltam 2 ozs. Fuse by
heat ; add boiled oil half a pint, resin 2 oss. ; when cooling add
16 OES. of oil of turpentine.
Black Japan Varnish. — Bitumen 2 oss., lamp-black 1 oz.,
Tarkey umber ^ oz., acetate of lead i oz.« Venice turpentine i oz.,
boiled oil 12 bts. Melt the turpentine and oil together, carefully
stirring in the test of the ingredients, previously powdered.
Simmer all together for ten minutes.
Cabinet-maker' B Varfnsh, — Pal« ahellac 760 parts, mastic
65 parts, strongest alcohol 1,000 parts by measure. Dissolve and
dilute with alcohol.
Cahinet Varnish, — ^t^used copal 14 lbs. , hot linseed oil 1 gallon,
hot turpentine Sgallons. Properly boiled, dries very quicMy.
Cheap Oak Varnish. — Dissolve 31 lbs. of pale resin in 1 gallon
of Oil df lurpentine.
C&mmc^ Va/miah, — ^Dissolve 1 part of shellac in 7 or 8 of
alcohol.
Copal VamisTt.— Copal 300 parts, drying linseed-oil 126 to 260
parts, spirit of turpentine 600 parts. Fuse the copal as quickly
as possible ; then add the oil, previously heated to nearly boiling-
point ; mix well ; then cool a little and add the spirits of turpentine ;
again mix well, and cover up till it has cooled down to 130®
Fahrenheit ; then strain.
Cdpal Varnish for Metals^ CJiainSf etc. — Copal melted and
dropped into water 3 ozs., gum sandarach 6 ozd., mastic 2) Ozs.,
powdered glass 4 ozs., Ohio tnrpentine 2^ ozs., alcohol of 85 per
cent, 1 quart. Distolve by g^itle heat.
Gold Varnish* — Turmeric 1 drachm, gamboge 1 drachm, oil of
turpentine 2 pints, shellac 6 ozs. , sandarach 6 ozs., dragon's blood
7 drachms, thin mastic varnish 8 ozs. Digest with occasional
shaking for 14 days in a warm place ; then set it aside to fine and
pout off the clear.
Mastic Varnish. — Gum mastlo 5 lbs., spirits of turpentine
2 gallons. Mix with gentle heat in a Olose vessel : then add pale
turpentine varnish 3 i^nts.
Table Varnish. — Dammar teein 1 lb., spirits of tnrpentine
2 lbs., camphor 200 grains. Digest the mixture for 34 hours.
The decanted portion is fit for inunediate use.
Another Becipe. — Oil of turpentine 1 lb., bee*s wax 2 ozs.,
colophony 1 drachm.
Tnrpentine Varnish. — ^Kesin 1 part, boiled oil 1 part. Melt
and then add turpentine 2 parts.
Varnish for ironwork. — Dissolve 10 parts of clear grains of
mastte, 5 plLrts of camphor, 15 parts of sandarach, and 5 parts of
elemi in k sufficient quantity of alcohol, and apply cold.
600 LACQUEfiS.
Afufthsr Reeipe, — Dissolve in about 2 lbs. of tar oil \ lb.
of aspbaltum, \ lb. of powdered resin. Mix hot in an iron
kettle and apply cold.
Varnish for Metalt^^Disaolye 1 part of braised copal in
2 parts of strongest aloohoL It dries very quickly.
Another Recipe.— Oo^ 1 part, oil of rosemary I part,
strongest alcohol 2 or 3 parts. This should be applied hot.
White Copal Varnish. — Copal 16 parts ; melt, and add hot
linseed oil 8 parts, spirits of turpentine 15 parts. Colour with
the finest white lead.
White Priming for Japanning, — Parchment size |, isin-
glass J.
White Varnish, — Tender copal 7^ ozs., camphor 1 oz., alcohol
of 95 per cent. 1 quart ; dissolve, then add 2 ozs. of mastic, 1 oz.
of Venice turpentine ; again dissolve, and strain.
mute Spirit Varjiish. — Sandarach 25 parts, mastic in tears 6
parts, strongest alcohol 100 parts, elemi 3 parts, Venice turpen-
tine 6 parts. Dissolve in closely corked vessel.
Laoquebs.
To mahe Lacquer, — Mix the ingredients and let them stand in
a warm place for 2 or 3 days, shaking them freely till the gum
is dissolved, after which let them settle for 48 hours, when the
clear liquor may be poured off ready for use. Pulverised glass
is sometimes used to carry off impurities.
Oold Lacquer. — Ground turmeric 1 lb., gamboge \\ oz.,
powdered gum sandarach 3^ lbs., shellac f lb., spirits of wine 2
gallons. Shake till dissolved, then strain and add 1 pint of
turpentine varnish.
Gold Lacquer for Brass not Dipped. — Alcohol 4 gallons,
turmeric 3 lbs., gamboge 3 ozs., gum sandarach 7 lbs., shellac
\\ lb., turpentine varnish 1 pint.
Gold Lacquer for Dipped Brass, — Alcohol 36 ozs., seed-lac
6 ozs., amber 2 ozs., gum gutta 2 ozs., red sandal- wood 24 grains,
dragon's blood 60 grains, Oriental safiEron 36 grains, pulverised
gla^ 4 ozs.
Good Lacquer.— K\x^^fA 8 ozs., gamboge 1 oz., shellac 3 ozs.,
annotto 1 oz., solution of 3 ozs. of seed-lac in 1 pint of alcohol;
when dissolved, add Venice turpentine \ oz., dragon's blood }
oz. Keep in a warm place 4 or 5 days.
Good Lacquer for ^r/M*.— Seed-lac 6 ozs., amber or copal 2 ozs.,
best alcohol 4 gallons, pulverised glass 4 ozs., dragon's blood 40
grains, extract of red sandal- wood obtained by water 30 grains.
Lacquer for Dipped Brass. — Alcohol of 95 per cent. 2 gal*
DIPPING ACIDS. 601
ions, seed-lac 1 lb., gum copal I oz., English safih-on 1 oz., an-
notto 1 oz.
Anotlier Recipe, — Alcohol 12 gallons, seed-lac 9 lbs., tur-
meric 1 lb. to a gallon of the above mixture, Spanish saffron
4 0Z8. The saflfron is only to be added for bronze work.
Zacq'iier Va7*nish. — Add so much turmeric and annotto to lac
varnish as will give the proper colour, and squeeze through a
cloth.
Pale Zaeqiter for Brass, — Alcohol 8 gallons, dragon's blood
4 lbs., Spanish annotto 12 lbs., gum sandarach 13 lbs., turpen-
tine 1 gallon.
Dipping Acids.
Aquafortis JSnmze Dip, — Nitric acid 8 ozs., muriatic acid 1
quart, sal ammoniac 2 ozs., alum 1 oz., salt 2 ozs., water 2 gallons.
Add ttie salt after boiling the other ingredients, and use it hot.
Brown Bronze Dip, — lion scales 1 lb., arsenic 1 oz., muriatic
acid 1 lb. ; a piece of solid zinc, 1 oz. in weight, to be kept in
while using.
Brown Bronze Paint for Copper Vessels. — Tincture of steel 4
oz8,» spirits of nitre 4 ozs., essence of dendi 4 ozs., blue vitriol
1 oz.jwater J pint. Mix in a bottle. Apply it with a fine brush,
the vessel being full of boiling water. Varnish after the appli-
cation of the bronze.
Bronze for all kinds of Metals. — Muriate of ammoniac (sal
ammoniac) 4 drachms, oxalic acid 1 drachm, vinegar 1 pint.
Dissolve the oxalic acid first.
Dipping Aeid. — Sulphuric acid 12 lbs., nitric acid 1 pint,
nitre ' 4 lbs., soot 2 handfuls, brimstone 2 ozs. Pulverise- the
brimstone and soak it in water 1 hour ; add the nitric acid
last.
Another Beeipe* — Sulphuric acid 4 gallons, nitric acid 2 gal-
lons, saturated solution of sulphate of iron (copperas) 1 pint,
solution of sulphate of copper 1 quart.
Good Dipping Aeid for Cast Brass.-^^xial quantities of sul-
phuric acid, nitre, and water. A little muriatic acid may be
added,
6rreen Bronze Dip. — Wine vinegar 2 quarts, verditer green
2 ozs.) sal ammoniac 1 oz., salt 2 ozs., alum | oz., French berries
8 ozs. Boil the ingredients together.
Ormolu Dipping Acid for Sheet Brass. — Sulphuric acid 2
gallons, nitric acid 1 pint, muriatic acid 1 pint, water 1 pint,
nitre 12 lbs. Put in the muriatic acid last, adding a little at a
time, and stir with a stick.
Another Bedpe. — Sulphuric acid 1 gallon, sal ammoniac 1 oz.,
602 CEHEHT8 AND GLUSS.
fioweni of sulphur 1 os., blue vitriol 1 oa.^ saturated solution of
zinc in nitric acid mixed with equal quantity of sulphuric
acid 1 gallon.
ViJiegwr Bronze for Brasi. — Vinegar 10 gallons, blue ritriol
3 lbs., muriatic acid 3 lbs., corrosive sublimate 4 grains, sal
ammoniac 2 lbs., alum 8 ozs.
Cements and Glues.
Cement for Earthen and OUus Ware, — ^Isinglass dissolviBd in
proof spirit and soaked in water 2 ozs. (thick) ; dissolve in this
10 grains of very pale gum ammoniac (in tears) by rubbing
them together, then add 6 large tears of gum mastic dissolved
in the least possible quantity of rectified spirit.
C&ment for Iron Tuhee, ^f?.— Finely powdered iron 60 parts,
sal ammoniac 1 pint, sufficient water to form into a paste.
Cement for Plumbers. — Black resin 1 part, brick dust 2 parts.
Melt together.
Cement for Leaky Boilers, — Powdered litharge 2 parts, fine
sand 2 parts, slaked lime 1 part.
Cement for Joinbig Metals and Wood, — Stir calcined plaiiier
into melted resin until reduced to a paste ; add boilfed oil till
brought to the consistency of honey. Apply warm.
Cast-iron Cenient. — Clean iron borings or turnings pounded
and sifted 60 to 100 parts, sal ammoniac 1 part. When it is to
be applied moisten it with water.
Turner's Cement. — Bee's wax 1 oz., resin J oz., pitch J oz.
Melt and stir in fine brick dust.
Coppersmith's Cement. — Powdered quick lime mixed with
bullock's blood and applied immediately.
Migineer's Cem&nt, — Equal weights of red and white lead
mixed with dryin|f oil. Bpread on tow or canvas.
Cement for Joining Metal and Olass. — Copal varnish lb parts,
drying oil 6 parts, turpentine 3 parts, oil of turpentine 2 parts,
liquid glue 5 parts. Melt in a bath and add 10 parts of slAked
lime.
Gasfittei''s Cement. — Resin 4| parts, wax 1 part, Venetian red
1 part.
Cement for Ihstening Blades into Ifandles^^^Shti^ac 2 parts,
prepared chalk 1 part, powdered and mixed.
Cement for Pots and Pans. — Partially ttielt % parts of sulphur
and add 1 part of fine blacklead« Mix well. Pour on stone to
oool, and then break it in pieces. Use like solder with an iron.
Cement for Crachs in Stoves. — Finely pulverised iron made
into a thick paste with water glass.
WOOD-STAINIKQ AMD E^'AMSLS. 6^8
Vtfr^ Strong Ohie, — Mix a small quantity of powdered chalk
with melted common glue.
Glue to JResi^ Moistttre. — Boil 1 lb. of common glue in 2
quarts of skimmed milk.
Marine Ghis. — Cut caoutchouc 4 parts into small pieces and
dissolve it by heat and agitation in 84 parts of coal naphtha, add
to this solution 64 parts of powdered shellaci and heat the whole
with constant stirring until combination takes place, then pour
while hot on to metal plates to form sheets. When \ised must
be heated to 280' Fahr.
Zifiiid (?Zi^*^— Dissolve 1 part of powdered dlum in 120
parts of water ; add 120 parU of glue, 10 parts of acetic aoid,
and 40 parts of alcohol. Digest.
Another Recipe, — Dissolve 2 lbs. of good glue in 2J pints of
hot water, add gradually 7 ozd. nitric acid, and mix well.
Poffekment ^?ti^.— Parchment shaving:s 1 lb., water 6 quarts ;
b<>ll until dissolved, then strain and eTaporate slowly until of
proper consistency.
Draughtsman's or Mouth Gltte. — Glue 6 parts, sugar 2 parts,
water 8 parts. Melt in water bath and cast in moulds. For
QM dissolve in warm water or moisten in the mouth.
WaOD-STAINDTG.
Mahogany Colour (^Dark). — Boil together in a gallon of water
J lb. of madder and 2 ozs. of logwood. When the wood is dry,
after having been washed over with the hot liquid, go over
again with a solution of 2 drachms of pearl ash in a quart of
water.
Mahogany Colour (Light). — Wash the surface with diluted
nitrous acid, and when dry use the following : — dragon's blood
4 ozs., common soda 1 oz., spirits of wine 3 pints. When well
dissolved, strain.
Hose Wood. — Boil 8 ozs. of logwood in 3 pints of water until
it is reduced to half. Apply boilii^ hot two or three tiipes.
The stain for the streaks is made rrom a solution of copperas
and verdigris in a decoction of logwood.
Ebony, — Wash the wood with a solution of sulphate of iron ;
when dry, apply a mixturfe of logwood and nut gstlls ; when drj%
wipe with a sponge and polish with linseed oil.
Enamels*
White Ihameh*-^o\9&\i 2£) parts, arseniq 14 pajrts, glass 13
parts, saltpetre 12 parts, flint 5 parts, and litharge 3 parts.
614 RECIPES, CASK-GAUQING, ETC.
Blaek Ekomeh — Clay 2 parta, protoxide of iron 1 part.
Blue Enamel, — Fine paste 10 parts, nitre 3 parts ; coloar with
oobalt.
Orcen Enamel. — Frit 1 lb., oxide of copper \ oz., red oxide of
iron 12 grs.
YelUne Enamel. — White lead 2 parts ; alum, white oxide of
antimony, and sal ammoniac, each 1 part.
Tracing Papsb. .
Nat oil 4 parts, turpentine 6 parts ; mix and apply to the
paper, then rub dry with flour and brush it over with ox gall.
Indian Ink.
Finest lamp black made into a thick paste with thin isin-
glass or gum water, and moulded into shape. It may be scented
with essence of musk.
Copying Ink.
Add 1 oz. of moist sugar or gum to every pint of common
ink.
Staiucassb OB Companion Ladders.
The ordinary tread of a stair or step is 8 ins., and rise 74 ins. ;
above or below that j in. rise must be subtracted or added for
every inch added to or taken from the width of tread, as the
case may be.
Cabk-gaugikg.
C = contents of cask in gallons.
I) = middle or bung diameter in ins
L = length in ins. *
d ^ end or head diameter in ins.
c = •0009442l(2d2 + iP) considerably curved.
c = 000944 2l(2d' + <?*) -Kd - dTf moderately curved.
c « -00141 j2L(D2 + rf*) very little curva.
c = •0000816Lv;39d» + 25<f» + 26Drf) any form.
Variations ot* Tides.
The difference in time between high wnter and higrh wa*er
averages about 49 minules.
galvanising. 605
Galvanising Ibon A&ticles.
1. Cleaninff the Work. — Any paint or old work should be
burnt off before being placed in the acid bath.
2. The acid bath for cleaning the work should consist of
water 40 parts, muriatic acid 1 part.
3. Two or three hours in the acid bath will remove dirt and
oxide. Cast articles generally require longer time than wrought.
Work put in at night may remain without injury till the next
morning.
4. When the bath ceases to act fresh acid should be added till
it acts as at first. If the acid solution has become thick from
the work, it should be allowed to settle, and the clear liquid
syphoned off, the bath cleansed, and the liquid again returned.
6. When the work is removed from the acid bath it should
be washed in water, and all dirt or oxide removed by brushing
or scouring. It should then be placed for two or three minutes
in a bath composed of water 6 parts, muriatic acid 1 part. On
removal from this bath it should be placed in a clean, warm place
to dry.
6. As far as practicable the work should be taken warm from
the drying-furnace to the zinc bath. It should be lowered end-
ways slowly into the molten zinc, so as to remove anj' loose oxide
or air from the surface, and allowed to remain long enough to
become as hot as the zinc. When it is considered this has been
effected it should be raised slowly ; if the zinc does not com-
pletely cover and flow freely over and from the surface, it must
be again lowered and allowed to remain longer. On its final
removal some powdered sal-ammoniac should be thrown from
the hand on the surface, which greatly helps to make the surface
smooth and remove the surplus zinc. When this is done the
work should be put aside to cool gradually. It must not be
dipped in water when hot, nor chilled in cold air. Articles with
joints should be worked whilst cooling, to prevent being set fast.
ZiNO Bath.
1. The metal must never be allowed to cool and set fast in
the bath.
2. At night the bath should be covered by a sheet of iroi^ to
prevent loss of heat, the fire made up, and a^in seen to once in
the night..
3. When from Want of work the operation is to be suspended
for a time, the zinc should be ladled out into ingots or cakes.
In the OBa%.oi small bs^bs this is done- every- night to save atten-
tion and fuel in the night:
4. The sine bottoms which form in the bath should be raked
out as soon as they form to the depth of two inches in a large
bath or one inch in a small one ; and in dipping articles care
should be taken not to lower them down far enough to touch
the zinc bottoms.
uv/u
f < t^ VI Lj 1 17 <J.
1 T J a 1 \ J <J. X O
^AXJXkO U MtM^t:',
EK0LI8H WEIGHTS AND XEABUBES.
AvoiRDirpois Weight.
bnans \ Ozs.
Lta. Qn.
OwtB.
Ton arannies |
1 *0625
•00S9063 -0001396 0000349 00000174 1-771846 1
16
= 1
•0626 ;002232l!000668 •
00002790; 28-34954
266
16
= 1-0367 14a*0089286-
00044643 453-6927
7168 ! 448
28 =1
•26
0126
12700-59
28672 1 1792
112 4
«1'
06
50803*38
673440 , 36840
2240; 80
20
= 1
1016048
A Btone of iron, coal, &c. «= 14 lbs.
Trot Wwght.
Avoir. Dn.
Grains
Dwt8. Ozs.
Lbs.
Onumnes
32 + 876
= 1
•0416667 0020833
•0001736
•0648
768 -r 876
24
»1
•06
•0041667
1'5662
17 + (97 -r 176)
480
20
«1
•0833333
311035
2104(114 + 176)
6760
240
12
= 1 3732420
176 lbs. Troy =144 Ibg. Avoir. 176 oz. Troy » 192 oz. Ayoix.
Avoir, lbs. x 121627 = lbs. Troy. Troy lbs. x 823 *» Avoir. lb».
In Uio " ApotbeoaiiM' ** fystfia (h« po«vd, lh« omwe (S) •n^ *b« vrtdn
are the same as in the " Troy " system ; bat otbitr Eubdi visions are diUMsent :
e.g. 20 grains make 1 icraple Oj, 9 scmplM mal^e 1 dram (5)1 an4 8 flrams
make 1 ounce.
LiNBAL MeASUHS.
Indies
Feet
Yards { Paths.
Poles
Purls.
Mile
Metres
1
08333
•02778 1*013889
•006061
•000126
•000016
•0254
13
«1
•33333 •166667
•060606 I0OI6I5
•000189
•304797
36
3
«1 -5
•181818 ^•004646 000668
•914392
72
6
2 :«1
•363636 -009091 001136
1-82878
198
16*
H
n
« 1 026
•003126 ;602915
7920
660
220
110
40' «1
•126 201-166
63360
6280
1760
880
330 i 8
^l ^609-33
T}ie palm = 3 in.
The span - 9 in.
The common military pace * 80
A cable's length ^iv 120 fathoms.
in.
Thehandw4u»-
The cubital 18 in.
An itinerary pace m 6 fe&t,
A le^^ue s 3 mUs.
Land Mxasvbs (LnrsAi;.).
Inches
Links
Feet
Yards
OhaiBS
}m .
IMii
I
1261861
•OMSBSB
•0277778
•0012686
•0000158
0264
m
**1
•6666667 -2222222
•01
•000185
•301166
12
IM
ar 1 |'83S3S33
•0161616
•0001 8»4
•304797
36
^
3
«1
•0454646
•0005682
•914392
792
100
66
22
«1
•0125
20-1166
63860
8000
6280 1760
80
t»l
1609-33
knglish weights and mluasurss.
Square Measfre»
607
bebcB
Feeb
Yards Perches
Roods 1 Acre Sq. Metres 1
1
144
1296
39204
1668160
6272640
•0069444
= 1
9
2721
10890
43560
•0007716 -0000255 •00000064'-00000016 0006452
•1111111 -0036731 -0000918 000023 -0929013
«1 -0330679-0008264 0002066 -836112
80i =1-025 -00625 25292
1210 40 =1-25 1011-696
4840 160 4 «= 14046-782
Acres X -0016626 = sq. miles. Sq. yards x -000000323 «8q. miles.
Land Measttbe (SftUAiEtE).
links
Fercties
Chains
Hoods
Aero
Sq. Ketros
1 -0016
•0001
•00004 -00001
•04046
625
= 1
'0625
-025 ,-00625
25-292
10000
16
-1
•4 1
404-6782
25000
40
n
«= 1 -25
1011^696
100000
160
10
4i =1
4046^782
A bideof land « 100 aeres. A jard of land - SO acres.
A chain wide » 8 acres per mile.
Ottbic Meabube.
Imperial Gallons
-003606640822
6^232102641168
168-266768641654
Cub. Ins.
»1
1728
46666
Cub. Feet
•0006788
=.1
27
Cub. Yds.
•00000214
•0370370
«1
Cab.|Cetre
•000016387
•0283161
•764634
A cubic yard of earth » 1 load. A barrel bulk » 6 cab. ft.
Ton of displacement of a ship ^36 cub, ft. s '9910624 cub. metre.
WjJTB MliABUSE.
Cub. Ins.
8-664JI
34-659i
69^dl8}
277-274
2772-740
4990-932
8734-131
116^5-608
17468-262
3291-016
4936-524
9873048
One gallon of water weighs 10 lbs. 20 fluid ozs. make 1 pint.
608
ENGLISH WEIGHTS AND MEASURES.
Ale AiTB Beer Measure.
Cub. Iiw.
1
1
1
P4
so
R
Barrels
1
1
H
J
34-669i
= 1
69-318i
2
= 1
1
1
277-274
8
4
= ll
1
2495*466
72
36 9{»i
4990-932
144
72 18 1 2
»1
9981-864
288
144
36
4
2
»1
14972-796
432
216
64
6
3
H
«1
19963-728
576
288
72.
8
4
2
H
«1
29945-592
864
432
108
12
6
3
2
H
«1
59891-184
1728
864
216.
24
.12
6
4
3
2
«1
119782-368
3456
1728
432 1 48
24
12
8
6
4
>.,|
Corn asi
> Dry Measure.
Cub. Ins.
1
1
Ph
1
1
1
1
1
J3
5
08
1
1^
34-659i
69-318i
138-637
= 1
2
4
-=1
2
«1
277-274
8
4
2 =1
654*648
16
81 4' 2
«1
2218192
64
32
16
8
4
= 1
4436-384
128
64
32
16
8
2
= 1
•
8872-768
256
128
64
82
16
4
2
«1
1
t
17745-536
512
256
128
64
32
8
4
2
«1 ;
88727-680
2560
1280
640
320
160
40
20
10
5
= 1
177455-360
5120 2560 1280 1 640
320
80
40
20
10
2
= 1
Goal Measure.
Cub. In=.
Heaped
d
1
1
^1
1
1
^1
Measure
s<
P^
m
«
l>£
iSo
M
s
S^
703-872
ISi
»1
2816-487
iq
4 -1
8446-461
224
12 3
»1
25339-383
672
36]
9
3
ml
101357-532
2688
144
36
12
4
«1
'
196380-2181
5208
279
69f
23^
H
HI
»1
1571041-746
41664
2232
668
186
62
15.}
8
= 1
2128508-172
56448
3024
756
262
84
21
lOM iH
= 1
BI420834-92
833280
44640 11160
3720
1240
310
160 ;20
14|f
«1
ENGLISH WEIGHTS AND MEASURES.
Wool Weight.
609
Fonndfl
Cloves
Stones
Tods
Weys
Packs
Sacks
Last
7
= 1
14
2
= 1
• 28
4
2
= 1
182
26
13
H
*1
240
34f
17^
H
1|?
«1
364
52
26
13
2
if
«1
4368
624
312
156
24
12
= 1
Measure of Time.
Seconds
Minates
Hours
Days
Weeks
Montbs
Galend.
Year
Julian
Year
Leap
Year
60
= 1
<
3600
60
= 1
86400
1440
24
= 1
604800
10080
168
7
= 1
2419200
40320
672
28
4
= 1
31536000
526600
8760
365
62i^
135V
= 1
31557600
31622400
525960
527040
8766
8784
365i
366
62|
52f
is-A
13i
1 1
^1460
Isfe
= 1
1 1
= 1
Angulab Measure.
The Geographical Division of any Line round the
Circumference of the Earth
60 seconds = 1 minute
60 minutes = 1 degree
16 degrees s J sign of the zodiac
30 degrees =% 1 sign of the zodiac
90 degrees = 1 quadrant
1 revolution or 4 quadrants or 360 degrees = the )
earth's circumf., or 12 signs =sl great circle . J
Diurnal Motion
of the Earth
reduced to Time
=:= 4 seconds
= 4 minutes
= 1 hour
! = 2 hours
' = 6 hours
= 24 hours
OOKB.
4 bushels := 1 sack. 12 sacks = 1 chaldron. 2 1 chaldrons - 1 score.
Miscellaneous Weights and Measures.
Aume of hock
Bag of cocoa
coifee
>»
It
hops ....
pepper (black), company's
„ free-trade bags
„ (white) .
rice ....
sago ....
28,
. 31 gals
. 112 lbs
140 to 168
. 280
. 316
56, and 112
. 168
. 168
. 112
it
tt
It
tt
tt
tt
Ur
610
ENGLISH WEIGHTS AND MElfintSS.
Miscellaneous Weights and Measures (continued).
Bag of saltpetre (East India) 168 lbs.
sugar or malt (Mauritius)
„ (East India) .
biscuits (Admiralty) .
Bale of coffee (Mocha) .
1>
»»
»
cotton wool (Virginia,Carolina,& W.Indies) 800 to 310
If
7t
(Brazil)
(Egyptian)
rags (Mediterranean)
Bar of bullion
Barrel of raisins .
soap
anchovies
coffee
tar .
turpentine
flour
pork
Boll of flour .
Box of camphor .
„ raisins (Valencia)
Bushel of wheat .
flour
rye
barley .
oats
oatmeal
peas
beans .
rape seed
malt ,
salt
clorer (red)
„ (white]
linseed
chicory (raw)
(kiln-dried)
(powdered)
coffee (raw) .
„ (roasted)
„ (ground)
buck wheat .
canary seed .
hemp
lentil
linseed (Bombay)
(New Orleans and Alabama)
(East India)
»
»»
»>
ft
»»
•>
»»
»♦
>»
»
»»
j»
»»
5»
>»
>»
»>
♦ >
»»
»>
»>
»
»♦
112 to 168
112 to 196
. 102
224 to 280
If
f?
)}
It
tt
It
11
400 to 600
32010 360
160 to 200
180 to 280
448 to 476
15 to 30
. 112
. 256
. 30
112 to 168
26*5 gals.
224 to 280 lbs.
220
224
140
112
30 to 40
60
56
58
47
40
51
64
63
50
38
56
• 64
62
52
50
28
38
61-25
32*25
36
60 to 56
53 to 61
42 to 44
60 to 62
50 to 52
ti
ti
it
>j
It
tt
tt
tt
tt
n
tt
tt
tt
tt
n
n
11
n
*.
11
It
tt
11
11
11
>»
ENGLISH WEIGHTS AND MEASUKES.
611
MiscBLLANBOUB Wbights AND MEASURES (contdnued).
Bushel of onion seed
millet
j»
it
*>
»
»
»
>»
>»
poppy
rape
tare
turnip
cabbage,,
Bntt of currants .
,, Cadiz .
„ sherry
Cask of cocoa
mustard .
nutmegs .
rice (American)
tallow
Catty c^ tea .
Chaldron of coals
Chest of tea (Congou) about
(Souchong)
(Pekoe)
(Hyson and Hyson skin) about
(Gimpowder) abouit
(Imperial) about
(Young Hyson)
pran of herrings .
|Firkin of butter
„ soap
Hogshead of brandy
rum .
tobacco
sugar
whisky
burgtmdy
claret
lisbon
port .
sherry
Jar of olive oil
Last of salt .
potash, cod fish,
flax or feathers
ale or beer
„ gunpowder
Load of hay or straw
„ bricks
„ tiles
Pig of ballast
Pipe of Cape wine
Lisbon or Bucellas
99
»
99
99
»»
>»
»
99
99
»
1)
9)
herrings,
II
meal
soap;
36 to
66 to
88 lbs.
64
48
53
66
66
66
»
99
»
48 to
62 to
60 to
60 to
1,680 to 2,240
. 108 gals.
. 108 „
. 140 lbs.
9 to 18
. 300
. 672
.1,008
. 1-33 „
. 2*63 tons
. 82-6 lbs.
. 810
. 66-6
. 65
. 109
. 95-7
. 94
. 37*6 gals.
. 66 lbs.
. 64 „
45 to 60 gals*
45 to 60 „
1,344 to 2,016 lbs.
1,456 to 1,792 „
65 to 60 gals.
. 44 „
. 46
. 58
. 67
. 64
. 26
18 barrels
12 „
. 1,904 lbs.
12 bairels-.
24 „
36 trusses:
600 number
1,000 „
56 lbs..
. 92 gals.
- 117 ,.
99
»
tar
»>
»»
»>
I)
>»
612
ENGLISH WEIGHTS AND MEASURES.
HlBOBLLANBOUB WBIGHTB AND llSABUBBS (concludedj.
»
»
»
«»
if
if
Pipe of madeira
malaga
marsala
port .
sherry or tent .
teneriffe or vidonia
Pocket of hops
Puncheon of brandy
rum .
whisky (Scottish)
prunes
„ molasses .
Quintal of fish
Roll of parchment
Sack of coals
„ flour of 2 bolls .
Tierce of beef (Irish) of 38 pieces
„ coffee
„ pork (Irish) of 80 pieces
Truss of straw
„ old hay .
„ new hay .
Tub of butter
Tun of oil (wine gals.)
Miscellaneous Numbers.
»j
>?
1,
110 gala
. 105 ,.
. 108 ,.
113 to 115
92 to 108
. 100
168 to 224 lbs.
110 to 120 gals.
90 to 100 ,.
112 to 130 „
1,120 \U,
120 to 1,344 ,.
. 112 „
* . 60 skins
. 224 lbs.
. 280
. 304
560 to 784
. 320
36
66
60
84
. 252 gals.
J?
n
»
»»
n
»
it
»
12 units
1 3 units .
12 dozen .
12 gross, or 144 dozen
20 units .
21 units .
6 score, or 100 .
6 score, or 120 .
24 sheets .
20 sheets .
25 sheets .
20 quires, or 472 sheets
21^ quires, or 516 sheets
2 reams
10 reams, or 200 quires
6 doz., or 60 skins, of parchment
4 pages, or 2 leaves .
8 pages, or 4 leaves .
16 pages, or 8 leaves .
24 pages, or 12 leaves
36 pages, or 18 leaves
72 words in common law
80 words in exchequer
90 words in chancery
make 1 dozen
1 long dozen
1 gross .
1 great gross
1 score
1 long score
1 short hundred
1 long hundred
1 quire of paper or parchment
1 quire of outside
1 printer'^ quire
1 ream of ditto or parchment
1 perfect or printer's ream
1 bundle of ditto
1 bale
1 roll
1 sheet of folio
1 sheet of quarto or 4to.
1 sheet of octavo or 8vo.
1 sheetof duodecimo or 12mo,
1 sheet of eighteens or 18ma
1 sheet
1 sheet
1 sheet
»»
»»
a
»
a
it
a
a
a
ty
a
a
a
a
a
1*
if
a
«>
>f
a
it
METRICAL SYSTEM 01 WEIGHTS AND MEASURES. glB
Sizes anb Contents op Casks.
Sundry Cwks
Lgth.
(llM.)
65~
Diana.
(Int.)
32
Contmti
"lOs
Admiralty Caaka
L«tta.
(In..)
59
IMan.
38
(R*i«.)
164
Marsala pipe .
Leager .
„ hhd. .
41
25
45-5
Butt
53
33
110
Brandy pipe .
62
34
114
Puncheon
41^
30
72
„ hhd. .
40
28
57-5
Hogshead
37
28
54
Port pipe
58
34
113
Barrel .
31J
24-5
36
„ hhd. .
37
30
56-5
Half-hogshead
28
22-5
27
Sherry butt .
50
35
108
Kilderkin
25
19-75
18
yy hhd. .
38
28
54-5
Firkin .
22
17
12
Rum puncheon
42
36
91
Size of Drawing Papers.
Antiquarian
Double elephant
Atlas .
Colombier .
Imperial .
Elephant .
Super royal
Inches
53x31
40x27
34x26
34x23
30x22
28x23
Royal
Medium
Demy
Foolscap •
Tracing papers
Ditto
Ditto
Inobei
24x19
22x17
20x15
17 y 13J
30x20
30x40
60x40
. 27 X 19
Continuous tracing paper, 28, 31, 40, 44, and 56 in. wide by 21 yards long.
ConUnnous tracing linen, 18, 28, 86, 38, and 41 in. wide by 24 yards long.
Conttnuous drawing cartridge, 84, 87, 58, and 60 in. wide by 60 yards long.
METRICAL SYSTEM.
Long Measure (1).
Millimetre .
Metres
Inctaat
-03937
Feflt
Tardi
XUm
- -001
•00328
•00109
Centimetre .
•01
•39370
•03281
01094
•000006
Decimetre .
•1
3-93704
•32809
•10936
•000062
Metre* .
1
39-37043
3-28087
1-09362
•000621
Decametre .
10
393-7043
32-80869
10-93623
•006214
Hectometre .
100
3937-043
32808693
109-36231
•062138
Kilometre .
Myriametre .
1000
looob
39370-43
3*704-3
3280-8693
.32808-693
1093-6231
10936-231
•621377
6^213768
Square Measure.
Milliare
Sq. Mctrot
8q. Inchei
Sq. Feet
Sq. Tardt
Acrot
•1
155
1-076
•119601
-0000247
Centiare
1
1550
10-764
1-19601
•0002471
Declare
10
16500
107-641
11-9601
•0024711
Are* .
100
155003
1076-410
119-601
•0247110
Decare .
1000
1550031
10764104
1196-01
•2471098
Hectare
10000
15500309
107641-04
11960-12
2-4710981
See Long Measure, next page.
The are=the square docametre.
614 METRICAL SYSTEM OF WEI0HT3 AND MEASURES.
Long Measure (2).
Millimetre .
Centimetre
Decimetre .
Metre
Decametre .
Hectometre
Kilometre .
Myiiametre
Inebet tad DccU
m»U of an In
MilM
Fvk. relet
Yard*
Fe«C
Inehca and Fkaetioiir
of an iDeh
» '0394
1 1
•3937
3-9370
39-3704
393-7043
39370432
39370-4320
393704-3196
•
3 X
Q 15 ,
1
1
5
4
4
2
"o"
1
2
1
0
q 5 1 1 1
^ 16 55 51158-
q H 1 J.
Ta • • • Si »
• ••• af Si •" ~
8...-T^...Ygg-
1
1
19
38
28
1
4
6 1
Solid Measure.
Millistere
Cable MMiw
GaUe ItMiM
CaUaFMl
OaMeTaidi
= -001
61025
•03632
•0013(
Centistere
•01
610'254
•35316
•01308
Decistere
•1
6102*639
363166
•13080
Stere* .
. 1
61035887
35-31562
1-30799
Decastere
10
610263-866
35315617
13-07986
Heotostere
100
6102638-659i3531-56172!l30-79868|
Weights.
Milligramme .
Centiscramme .
Decigramme .
Gramme' .
DeeagrMnme
Hectogramme .
Kilogramme
Myriagramme .
Quintal .
Millier.orToniie
Grammec
'^ '-001
•01
•1
1-
10
100
1000^
10000
100000
1000000
Kf. 0«. \ Av. Lb*.
•00004
•00035
•00353
i>3527
•36274
8-5274
86-2739
862-789
8527-39
86273-9
•0000022
-0000221
•0002205
•0022046
•0220J»62
•2204621
2-204^1
22-04621
220-4621
2204-621
Cwtt.
Torn
•0000020
•0000197
•0001968
•0019684;
•0196841
•1968412|
1-968412
19-68412
000001
000010
000098
00098^
009842
098421
984206
Gr^DS Tr.
•015432
•164323
1643235
15-43235
164-32d6
1648-235
16432-35
154828-5
1643235
15432349
Dry AiTD Fluid Measure.
Millilitre
Centilitre
Decilitre
Litre* .
Decalitre
Hectolitre
KUolitre
MyriaJitre
Litre! j Cubic Inehe*
•001
•01
•1
1
10
100
1000
10000
•06102539
•61025387
61026887
61^025387
610-26387
0102-5387
61025-387
610253-87
Cttbie Feet
•0004
•0035
•0353
•3532
3-5316
36-3166
3531662
Gallont
Basheb
•00022
•00003
•0022
•00028
•0220
•00275
•2201
•02751
2-2009
•27511
220091
2-76113
2200906
27-61132
2200-9055
275-11318
* The fttere la a cubic metre, and is used generally for measuring selids.
' Tbe ftramme is tbe wedght In vacuo of a cubic oeal^imetre of disttUed water
at the temperature of 4<^ of the centigrade thermometer.
* The litre is a cubic decimetre.
MILLIMETRES TO INCHES.
615
Tables giying the English Equivalents of 1 Milli-
metre TO 1,000.
MiUi-
motres
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
81
82
83
34
85
36
37
38
Inches aDd
Decimals
of an Inch
0-039370
0078741
0-118111
0157482
0196852
0-236-223
0-275593
0-314963
0-354384
0-393704
0-433075
0-472445
0-511816
0-651186
0-690556
0-629927
0-669297
0-708668
0*748038
0-787409
0-826779
0-866149
0-905520
0-944890
0-984261
1-028631
1-068002
1-102372
1-141742
1-181113
1-220483
1-259854
1-2992^4
1-338555
1-3779^5
1-4173^5
1-456706
1-496076
Milli-
metres
39
40
41
42
43
44
45
46
47
48
49
50
51
52
58
54
55
56
57
58
59
60
61
62
68
64
65
66
67
68
69
70
71
72
73
74
75
76
77
Inches and
Decimals
of an Inch
1-535447
1-574817
1-614188
1-653558
1-692929
1-732299
1-771669
1-811040
1-860410
1-889781
1-929151
1-968522
2-007892
2-047262
2-086683
2-126003
2105iJ74
2-204744
2-244115
2-283485
2-322855
2-362226
2-401596
2-440967
2-480387
2-519708
2-559078
2-598448
2-637819
2-677189
2-716560
2-755930
2-795301
2-834671
2-874041
2-913412
2-9527«2
2-992153
8031523
MilU-
metree
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
Inf'hes and
Ifeciiuals
of an Inch
3-070894
3-110264
3-149635
3-189005
3-2-28375
3-267746
3-307116
3-346487
3-385857
3-425228
3-464698
3-503968
3-548339
3-582709
3-622080
3-661450
3-700821
3-740191
3-779661
3-818932
3-858302
3-897673
8-937043
3-976414
4-015784
4-055156
4-094525
4133895
4-173266
4-212636.
4-252007
4-291377
4-380748
4-370118
4-409488
4-448859
4-488229
4-527600
4-566970
116
MILLIMITBES TO INCHES.
MlUi-
metres
Incbesand
HiUi-
metres
165
Inches and
MilU.
metres
Inches and
Decimals
of an iDch
Decimals
of an Inob
Decimals
of an Inch
117
4*606341
6-496121
213
8-385902
118
4-645711
166
6-535492
214
8-425272
119
4-685081
167
6-574862
215
8*464643
120
4-724452
168
6-614233
216
8-504013
121
4-763822
169
6-653603
217
8-643384
122
4-803193
170
6-692973
218
8-582764
123
4*842563
171
6732344
219
8-622125
124
4-881934
172
6-771714
220
8-661495
125
4-921304
173
6-811086
221
8*700866
126
4*960674
174
6*850456
222
8*740236
127
5000015
175
6-889826
223
8-779606
128
6*039415
176
6-929196
224
8-818977
129
6-078786
177
6-968567
225
8-858347
130
6*118156
178
7*007937
226
8-897718
131
5*157527
179
7047307
227
8-937088
132
5*196897
180
7*086678
228
8*976459
133
5*236267
181
7*126048
229
9-015829
134
6*275638
182
7*166419
230
9-056199
135
5*315008
183
7*204789
231
9094570
136
6*354379
184
7-244160
232
9-133940
137
5*393749
185
7-283530
283
9173311
138
6*433120
186
7*322900
234
9-21-2681
139
6*472490
187
7*362271
236
9-252062
140
6-611861
188
7-401641
236
9*291422
141
6*551231
189
7*441012
287
9-330792
142
5*590601
190
7*480382
238
9*370163
143
6*629972
191
7*519753
239
9-409583
144
6*669342
192
7*559123
240
9*448904
145
5-708713
193
7*598493
241
9-488274
146
6-748083
194
7-637864
242
9*627645
147
5-787454
195
7-677234
248
9-567015
148
6-826824
196
7*716605
244
9*606385
149
5*866194
197
7-755976
246
9-645756
150
5-905565
198
7-795346
246
9*685126
151
5*944935
199
7-834716
247
9*724497
152
5-984306
200
7*874086
248
9-763867
153
6*023676
201
7-913457
249
9-803238
154
6-063047
202
7-952827
250
9-842608
155
6-102417
203
7-992198
251
9-881978
156
6-141787
204
8-031568
252
9*921349
157
6-181158
205
8*070939
253
9*960719
158
6-220528
206
8110309
254
10-000090
159
6-259899
207
8*149679
256
10-039460
160
6-299269
208
8*189050
256
10-078831
161
6-338640
209
8-228420
257
10118201
162
6-378010
210
8*267791
258
10157671
163
6-417380
211
8-807161
259
10-196942
164
6*456751
212
8*346582
260
10-286312
MILLIMETBES TO INCHES.
617
MiUi-
metres
Inches and
Decimals
of an Tnch
MiUi-
metres
Inches and
Decimals
of an Inch
MflH-
metres
Inches uid
Decimals
of an Inch
261
10-276683
309
12165464
357
14-055244 :
262
10-315053
310
12-204834
358
14-094615
263
U)-354424
811
12-244204
359
14133985
264
10-393794
312
12-283575
360
14173356
265
10-433165
313
12-322945
361
14-212726
266
10-472535
314
12-362316
362
14-252096
267
10-511905
315
12-401686
363
14-291467 :
268
10-551276
316
12-441057
364
14-330837
269
10-6?0646
317
12-480427
365
14-870208 i
270
10-630017
318
12-519797
366
14-409578
271
10-669387
319
12-559168
367
14-448949 ,
272
10-708758
320
12-598538
368
14-488ai»
278
10-748128
321
12-687909
369
14*527689 !
274
10-787498
322
12-677279
370
14*567060 ;
275
10-826869
323
12-716660
371
14-606430
276
10-866239
824
12-756200
372
14*646801
277
10-905610
325
12-796390
373
14*685171
278
10-944980
326
12-834761
374
14*724642
279
10-984351
827
12-874181
375
14-763912
280
11-023721
328
12*913602
376
14*803282
281
11-063091
329
12-952872
377
14*842653
282
11-102462
830
12-992248
378
14-882023
283
11-141832
331
13-031618
379
14*921394
284
11-181203
382
13-070983
380
14-960764
285
11-220673
383
13-110354
! 381
15-000135
286
11*259944
334
13-149724
382
15-039605
287
11-299314
385
13*189096
. 383
15-078875
288
11-338684
386
18-228466
384
15-118246
289
11-378065
337
13-267836
385
15167616
290
11-417425
338
18-307206
386
15-196987
291
11-456796
389
13*346576
387
15-286357
292
11496166
340
13-385947
388
15-276728
298
11-535537
341
13*425317
389
15-316098
21f4
11-674907
342
13*464688
390
15-354469
295
11-614277
343
18*504058
391
15*398839
296
11-653648
344
18-543429
392
16*433209
297
11-693018
345
13-582799
393
16-472680
298
11-732889
346
13-622170
394
15-51 1960
299
11-771759
347
13*661540
395
15-651321
300
11-811130
348
13-700910
. 396
15*590691
801
11-850500
349
13*740281
397
15-680062
302
11-889871
350
13-779661
398
16*669432
303
11-929-241
351
13-819022
399
16*708802
304
11-968611
352
13-858392
400
16-748173
305
12-007982
353
18-897763
401
16-787548
306
12-047352
354
13-937133
402
15-826914
307
12-0867-23
365
18-976503
403
15-866284
308
12-126093
356
14016874
404
16-906655
Itt
MII.LIMBTBBS TO INCHES.
[iUi-
etres
Inches aad
Dedmals
of ftn Inch
metres
Inches and
DecimalB
«C an Inch
MilH-
metree
Xncbesand
Dedmab
of an Inch
S05
15*945025
458
17-884806
601
19*724586
(06
15-984395
454
17-874176
502
19-768957
107
16-023766
455
17-913547
603
19-803327
108
16-063186
456
17-952917
504
19*842698
409
16102507
457
17-992287
605
19-882068
410
16-141877
468
18*031658
606
19-921439
411
16181248
459
18-071028
607
19-960809
412
16*220618
460
18-110399
508
20-000179
413
16-259998
461
18-149769
609
20-089550
414
16-299359
462
18-189140
610
20*078920
415
16-338729
468
18-228510
611
20118291
416
16-878100
464
18-267880
612
20-167661
417
16-417470
465
18-307251
613
20*197082
41S
16-456841
466
18-346621
514
20*286402
419
16-496211
467
18-885992
515
20-275778
420
16-535581
468
18-426362
516
20*816143
421
16-574952
469
18-464738
617
20*364513
422
16*614322
470
18-504108
518
20*893884
423
16-653693
471
18-648474
619
20*483254
424
16-693063
472
18-682844
620
20-472625
425
16-732444
478
18-622214
621
20-511995
426
16-771804
474
18-661685
622
20*651366
427
16-811175
475
18-700965
623
20*590736
428
16-850545
476
18-740326
524
20*630106
429
16-889915
477
18-779696
625
20-669477
430
16-929286
478
18-819067
626
20-708847
431
16-968656
479
18-858487
627
20*748218
432
17-008027
480
18-897807
528
20*787588
433
17*047397
481
18-937178
629
20*826959
434
17-086768
• 482
18-976648
680
20*866329
435
17-126188
483
19-015919
631
20-905699
486
17-165508
484
19-056289
532
20*945070
437
17*204879
485
19-094660
533
20*984440
488
17-244249
486
19-134080
634
21*028811
439
17-283620
487
19-178400
635
21*063181
440
17-322990
488
19-212771
636
21*102652
441
17-362361
489
19-262141
687
21*141922
442
17-401731
490
19-291612
688
21*181292
443
17-441101
491
19-380862
689
21-220663
444
17-480472
492
19-370263
640
21-260083
445
17*519842
498
19-409623
641
21-299404
446
17-659218
494
19-448993
642
21-388774
447
17-598588
495
19-488364
643
21-878146
448
17-687954
496
19-627784
644
21-417515
449
17-677324
497
19*567096
645
21-456885
450
17-716694
498
19-606465
646
21*496266
451
17-756065
499
19-645836
647
21-686626
452
17-796435
500
19-685216
548
21-674997
MILLIMETRES TO INCHES.
61S
Mini.
metres
Inches and
DedmalB
ol an Inch
MiUi.
metres
Inches and
Decimals
ci an Inch
Milli-
metres
Inches and
Decimals
of an Inch
549
21-614367
597
23-504148
645
26-393929
550
21-663788
598
23-643518
646
25-433299
551
21-693108
599
23-582889
647
25-472670
552
21-732478
600
28-622259
648
26-512040
553
21-771849
601
23 661630
649
25-551410
554
21-811219
602
23-701000
650
25-590781
555
21-860590
603
23-740371
651
25-630151
556
21*889960
604
23-779741
662
26-669522
557
21-929331
605
23-819111
653
25-708892
558
21-968701
606
23-868482
654
25-748263
559
22008072
607
23-897862
655
25-787633
560
22-047442
608
23-937223
656
25-827003
561
22-086812
609
23-976593
657
25*866874
562
22-126183
610
24-015964
658
25-905744
563
22-165568
611
24055384
659
25-945115
564
•22-204924
612
24-094704
660
25-984486
565
22-244294
618
24-184075
661
26-023856
566
22-283665
614
24-173445
662
26-063226
567
22-323085
615
24-212816
668
26-102596
568
22-362405
616
24-252186
664
26-141967
569
22-401776
617
24-291557
666
26-181387
570
22-441146
618
24-330927
666
26-220708
571
22-480517
619
24-370297
667
26-260078
672
22-619887
620
24-409668
668
26-299449
673
22-559928
621
24-449038
669
26-338819
674
22-598628
622
24-488409
670
26-378189
575
22-687998
623
24-527779
671
26-417560
576
22-677369
624
24-567150
672
26-456930
677
22-716789
625
24-606520
678
26-496301
678
22-766110
626
24-645890
674
26-535671
679
22-795480
627
24-685261
675
26-575042
580
22-834851
628
24-724631
676
26-614412
681
22-874221
629
24-764002
677
26-653782
582
22-913591
680
24-803372
678
26-693163
683
22-952962
631
24-842743
679
26-732623
684
22-99-2382
632
24-882113
680
26-771894
685
23-031703
633
24-921488
681
26-811264
686
28-071073
634
24-960854
682
26-850635
587
23-110444
635
25-000224
683
26-890005
688
23-149814
636
26-039595
684
26-929876
589
23-189184
637
25-078965
685
26-9 8746
690
28-228565
638
25-118336
686
27-008116
691
23-267925
639
25-167706
687
27-047487
692
23-307296
640
26-197077
688
27-086857
693
23-346666
641
26-236447
689
27-126228
694
23-386087
642
26-275817
690
27-165598
695
28-426407
643
26-315188
691
27-204969
596
23-464778
644
26-854558
692
27-244839
120
MILLIMETRES TO IKCHES.
MiUi-
metres
Iiich«B and
Dedmala
of an Inch
Mini-
metres
IncbMand
Decimals
of an Inch
Mmi-
metres
Inchea and
Decimals
of an Inch
698
27-283709
741
29-173490
789
81-063271
694
27-323080
742
29-212861
790
31102641
695
27-362450
743
29-252281
791
31142012
696
27-401821
744
29 291601
792
31-181382
697
27-441191
745
29-330972
793
81-220752
698
27-480562
746
29-370342
794
31-260128
699
27-619982
747
29-409713
796
81-299498
700
27-559302
748
29-449088
796
31-388864
701
27-598673
749
29-488454
797
31-378234
702
27-638043
760
29-527824
798
31-417604
70»
27-677414
761
29-567194
799
81-466975
704
27-716784
752
29-606565
800
31-496346
705
27-756165
763
29-646966
801
81-636716
706
27-795525 .
754
29-685306
802
31-576080
707
27-834895
756
29-724676
808
31-614457
708
27-874266
756
29-764047
' 804
81-663827
709
27-913686
757
29-803417
805
81-693198
710
27-963007
758
29-842787
808
31-782568
711
27-992877
759
29-882168
807
31-771938
712
28031748
760
29-921528
808
31-811309
718
28071118
761
29-960899
809
31-850679
714
28110488
762
80-000269
810
31-890060
716
28-149859
768
30-039640
811
31-929420
716
28-189229
764
80-079010
812
31-968791
717
28-228600
765
30118380
813
82-008161
718
28-267970
766
30-157761
814
32-047532
719
28-307341
767
30-197121
815
82-086902
720
28-346711
768
30-236492
816
32126272
721
28-386081
769
30-275862
817
32-166643
722
28-425452
770
30-315233
818
82-206013
723
28-464822
771
30-354603
819
32-244384
724
28-504193
772
80-393973
820
32-283754
726
28-543563
773
30-433344
821
323231*25
726
28-582934
774
30-472714
822
32-36-2495
727
28-62*2304
776
30-512085
828
32-401866
728
28-661675
776
30-551456
8-24
32-441236
729
28-701045
777
, 30-590825
825
32-480606
730
28-740415
778
30-630196
826
32-519977
731
28-779786
779
30-669568
827
82-559347
732
28-819156
780
80-708937
828
32-598718
733
28-868527
781
30-748307
829
32-638088
731
28-897897
782
30-787678
830
32-677459
735
28-987268
783
80-82704d
831
82-716829
736
28-976638
784
30-866419
832
32-756199
737
29-016008
785
30-905789
833
32-795570
738
29-056379
786
30-945159
884
32-834940
739
29-094749
787
30-984530
835
32-874811
740
29-134120
788
310-23900
836
32-913681
MILLIMETRES TO INCHES.
621
•
unu-
metares
Inches and
Decimals
of an Inch
Mini-
metres
Inches and
Decimals
of an Inch
MIlll-
metm
Inches and
Decimals
of an Inch
8S7
32-953052
885
34-842832
933
36-732613
838
32-992422
886
34-882203
934
36-771984
839
33-031792
887
34-921573
935
86-811364
840
33-071163
888
34-960944
936
86-860724
841
33-110533
889
35-000314
937
86-890095
842
33-149904
890
35-039684
938
36-929465
843
33-189274
891
35-079055
939
36-968836
844
33-228645
892
85-118425
940
37-008206
845
33-268015
893
35-167796
941
37-047576
846
33-307385
894
35-197166
942
37-086947
847
33-346756
895
35-236536
948
87-126317
848
83-3861^
896
85-276907
944
87-165688
849
83-425497
897
86-315277
945
87-205058
850
33-464867
898
85-354648
946
37-244429
851
33-504238
899
35-394018
947
87-283799
852
33-543608
900
35-433389
948
87-323170
853
33-582979
901
85-472759
949
37-362540
854
33-622349
902
S5-512130
950
37-410910
855
33-661719
903
35-551500
951
37-441281
-
856
33-701090
904
85-590971
952
37-480651
857
33-740460
905
35-630241
953
37-520022
858
33-779831
906
35-669611
954
37-659392
859
33-819201
907
35-708982
955
37-698765
860
33-858572
908
35-748352
956
37-638135
861
33-897942
909
85-787723
957
37-677603
862
33-937312
910
35-827093
958
37-716874
863
33-976688
911
85-866464
959
37-756244
864
34-016058
912
35-905834
960
37-795616
865
34055424
913
35-945204
961
37-834985
866
34-094794
914
35-984576
962
87-874356
867
34-134165
915
36-023945
963
37-913726
868
34-173535
916
36-063316
964
37-953096
869
34-212905
917
36-102686
965
37-992467
870
34-252276
918
36-142057
966
38031837
871
34-291646
919
36-181427
967
38-071208
872
34-331017
920
36-220797
968
38-110678
873
34-370387
921
86-260168
969
38-149949
874
34-409758
922
86-299638
970
88-189319
875
34-449128
923
36-338909
971
38-228689
876
34-448498
924
86-378279
972
88-268060
877
34-527869
925
86-417650
973
88-307480
878
34-667239
926
86-457020
974
38-846801
879
34-606610
927
86-496390
975
88-386171
880
34-645980
928
86-535761
976
38-425542
881
34-685351
929
.36-575131
977
38-464912
882
84-724721
930
36-614502
978
38-604288
883
34-764091
931
36-653872
979
88-543658
884
34803462
932
86-693243
980
88-588028
B22
MItLlMETBES AND METRES TO INCHES.
MilU-
metres
Inches and
Dedmals
of an Inch
Milli-
metres
Inches and
DedmaLi
of an Inch
MiUi-
metres
Inches and
Decfmals
of an Inch
981
982 .
983
984
985
986
987
38-622394
38-661764
88-701135
38-740505
38-779876
38-819246
38-858616
988
989
990
991
992
993
994
88-897987
88-937357
88-976728
39-016098
39-055469
39-094839
89-134209
995
997
998
999
1000
39-1785^
89-212950
89-252321
39-291691
39-381062
39-370432
Tabids oiviKa thi
1 ElfeLISH £QXnTAI<BNT8 OF
Metheh in
Ikcitrs
AND Decimals of an Inoh. 1
Inches and
Inches and
Inches and
Mitres
DecimiUs
Metres
Beounala
Metres
Decinuds
of an Inch
of an Inch
of an loch
1
89-370482
34
1338-594687
67
2637-818941
2677-189373
2716-559805
2
78-740864
35
1377-965119
68
8
118111296
36
1417-336561
69
4
157-481728
87
1456-706983
70
2755-930287
5
196-852160
38
1496-076415
71
2795-300669
6
236-222592
39
1535-446846
72
2834-671101
7
275-593024
40
1574-817278
73
2874-041533
8
314-963456
41
1614-187710
74
2913-411965
9
354-333888
42
1653-568142
75
2952-782397
10
393-704820
48
1692-928574
76
2992-152829
11
433-074752
44
1732-299006
77
3031-523261
12
472-445184
45
1771-669438
78
3070-893693
13
611-816616
46
1811-039870
79
3110-264125
14
651-186047
47
1850-410302
80
3149-634557
15
590-666479
48
1889-780734
81
3189004989
16
629-926911
49
1929-151166
82
3228-375421
17
669-297343
60
1968-521698
83
3267-746853
18
708-667775
61
2007-892030
84
8307-116285
19
748-038207
62
2047-262462
85
8346-486717
20
787-408639
63
2086-632894
86
3386-857149
21
826-779071
54
2126-003326
87
3425-227581
22
866-149503
65
2165-37368
88
3464-598013
23
905-519935
66
2204-744190
89
3603-968444
24
944-890367
67
2244-114622
90
3543-338876
25
984-260799
68
2283-485054
91
3582-709308
26
1023-631231
69
2322-855486
92
3622-079740
27
1063-001663
60
2362-225918
93
3661-450172
28
1102-372095
61
2401-596350
94
3700-820604
29
1141-742527
62
2440-966782
95
3740-191036
80
1181-112959
63
2480-337214
96
8779-561468
31
1220-483391
. 64
2519-707645
97
3818-931900
82
1259-853823
65
2559-078077
98
3868-802332
88
1299-224255
66
2598-448509
99
3897-672764
FRACTIONS OF AN INCH TO MILLIMETBESs.
688
Table giving the EairiTALENTs in Millimbtrbs
01" THE Divisions op the Inch.
Bivisioxis of the Inch
•••
• ••
• ••
• • •
•••
• ••
• ••
r
xe
3
Te
3
4
i
i
1
4
i
Si
64
82 64
• •• • • ■
• •• • * •
•»• •••
• • • • • •
[i?
A ••• A
I3f
Millimetares
•198436
•896871
•596307
•793743
•992179
1-190614
1-389060
1-687486
1-785921
1-984357
2-182793
2-385129
2-579664
2-778100
2-976586
3-174972
3-373407
3-571843
3-770279
3-968714
4^167150
4-365586
4-564022
4-762457
4*960893
5-159329
5-357764
5-556200
5-754636
5*968072
6-161508
6-349943
6-548379
6-746814
6-945250
7-143686
7-342122
7-540557
7-738993
7-937429
8-135865
8-334300
Diyisiona of the Inch
5
IS
Iff
1
• • •
32
is
1_
32
V -
Si
1
?
¥
• ••
It •
64 lis
••• •••
•^•- ifc
Si •••
••• ••'•
.- ili
ei ••^»
• • • a • •
••• IM
^ •••
9i T
is
• ft • • • •
••• Tsl
ei •••
A 1S5
• ■ • • I
^
• « •
it
158
• • •
• ••
• • •
• • •
ik
t
• * •
Til
• t •
• *•
• * •
lis
s
• ••
lis
• • •
• ft ft
■ • •
1
Tis
64
• ••
lis
• ■ •
• •4
• • •
S4
128
• ••
lis
« • •
• ••
■ • •
ifc
lis
• • • • •!
MUlimetrts
8-532780
8-731172
8-929007
9-128043
9-826479
9-624916
9'723350
9-921786
10*120222
10-318657
10-617093
10-715629
10*913966
11112400
11*810836
11-509272
11^707707
11-906148
12-104579
12-803015
12-501450
12-699886
12*898322
18096757
18-295193
18-493629
18-692065
18-890600
14-088936
14-287372
14-486808
14*684243
14-882679
15-081115
15-279550
16-477986
16-676422
16-874858
16^073293
16-271729
16*470166
16-668600
24 £QUiVAL£KTS OF CNOUSU AND METRICAL MEASURES.
DiTisioDB of the loch
1
93
JL
C4
i
• ••
1
188
• • •
64 m
i
1
• f •
1
¥
1
i
A
• • •
IS5
• • «
1
T5«
1
• • •
1
• ••
Millimetres
16-867036
17065472
17-263908
17-462343
17-660779
17-859215
18-057660
18-256086
18-454522
18-652968
18-851393
19-049829
19-248266
19-446701
19-646136
19-843672
20042008
20-240443
20-438879
20-637316
20-835761
21-034186
Diviaions of the Inch
^
1
I
« • •
t • •
« • •
i
1
117
^ lb
m
• • • ■ ■ •
oi Is?
• •• • « «
••• lai
r?? •••
1 1
«4 Ifi
••• •••
••• m
A Tiff
• • • • ■ •
'^' Hi
G4 •••
64 199
Millimetre
V li
9
• • •
21-232622
21-431068
21-629493
21-827929
22-026365
22-224801
22-423236
22-621672
22-820108
23018643
23-216979
23-415415
23*613851
23-812286
24-010722
24-209158
24-407694
24-606029
24-804465
25-002901
25*201336
25-399772
Table oivino the Equiyalbnts in Millimetbes
OF THE Divisions of the Foot.
In.
Millimetres
25-39977
In.
10
Millimetres
In.
MUUmetres
In.
MUUmetres
I
253-99772
19
482-59567
28
711-19362
2
50-79954
11
279-39749
20
507*99544
29
736-59339
3
76-19932
12
304-79727
21
633-39521
30
761-99316
4
101-69909
13
33019704
22
658-79499
31
787-39294
5
126-99886
14
356-59681
23
584-19476
32
812-79271
6
162-39863
15
380-99658
24
609*69463
33
838-19248
7
177*79840
16
406-39635
25
634-99430
34
863-59225
8
208-19818
17
431-79613
26
660-39408
35
888*99202
9
228-59796
18 457-19590
27
685*78385
36
914-39180
Table giving the Equivalents of Lineal Feet in
Metbes.
Ft.
Metres
Ft.
6
7-
8
9
10
Metres
Ft.
11
12
13
14
15
Metres
Ft.
16
17
18
19
20
Metres
1
2
3
4
6
•3047973
•6096947
•9143920
1-2191893
1-5239867
1-8287840
2-1335813
2-4383786
2-7431760
30479733
3-3527706
3-6755680
3-9623653
4-2671626
4-5719600
4-8767673
5-1815546
5-4863519
6-7911493
6-0969466
EQUIVALENTS OF ENGLISH AND METRICAL WEIGHTS. 625
Table giving the Equivalents of Avoie.
French Kilogeams.
Oz. IN
Oz.
1
2
3
4
Ejlograms | Oz.
•028349541
•056699082
•085048622
•113398163
5
6
7
8
Kilograms fOz.
•141747704 j~y
•170097245 1 10
•198446785 ill
•226796326 1 12
^Ita
Kilograms
•255145867
•283495408
•311844948
•340194489
Oz.
13'
14
15
16
Kilograms
! -368544030
•396893571
•425243112
•453592652
Table giving the Equivalents op Avoir. Lbs. in
French Kilograms.
1
2
3
4
5
6
7
Kilograms
•45359266
•90718530
1-36077796
1-81437061
2-26796326
2-721555&1
3-17514857
Lbs.
8
9
10
11
12
13
14
Kilograms
3^2~874122
4-08233387
4-53592652
4-98951917
5-44311183
6-89670448
6-35029713
Lbs,
15
16
17
18
19
20
21
Kilograms
6-80388978
7-25748243
7-71107509
8-16466774
8-61826039
9-07185306
9-62644570
bs.
22
23
24
25
26
27
28
Kilograms
9-97903836
10-43263100
10-88622365
11-33981631
11-79340896
12-24700161
12-70059426
Table giving the Equivalents op Quarters in
French Kilograms.
1
KilogEams
12-70059426
Ore.
2
Kilograms I Qrs.
25-40118853|~n38-10178279
Kilograms
Qrs.
4
Kilograms
50-80287705
Cwt
1
2
3
4
5
Table giving the Equivalents op Cwts. in
French Kilograms.
Kilograms
50-80237705
101-60475410
152-40713116
203-20950821
•254'01 188526
Cwt
6
7
8
9
10
Kilograms
1
304-81426231
355-61663936
406-41901642
457-22139347
$108-02377052
Cwt
11
12
13
14
15
Kilograms
558-82614757
609-62862462
660-43090168
711-28327873
762-03565678
Cwt
16
17
18
19
20
Kilograms
812-83803288
868-64 040988
914-44278694
965-24516899
1016*0475411
Table giving the Equivalents op Tons in
French Kilograms.
Tons
1
2
.3
4
:6
.6
7
8
9
10
EUoglums
1016-04754
203209608
304814262
4064-19016
3080-23771:
6090-^8325
7112-33279
8128-38033
9144-42787
10160-4754
Tons
20
30
40
50
.60
70
.80
90
100
200
Kilograms
Tons
20320-.9508 300
30481-4262 400
40641-9016 600
50802-3771 600
60a62:-8626 700
71123-3^9 dOO
81283-8033 dOO
91444-2787 1000
101604-7641100
203209608 1200
Kilograms
304814-262
406419-016
508023-771
609628-626 1^00
71123.3-272 1700
^1^3d033 1800
d; 444^-787 1900
1016047-64 2000
1117652-303000
121925706 4000
Tons
1300
1400
1600
Kilograms
1320861-80
1422466-66
16^071-31
1626676-07
1727280-82
1828885-57
1930490-33
2032096-08
3048142-62
4064190-16
Ss
26
KILOOBAMS TO LBS. AND TONS.
o
H
5
D
M
3
^
4
^
^
^
0
5
^
^
d
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KILOGRAMS TO LBS. AND TONS.
0*^
8
O
O
to
Q
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00
M
o
Ph
P
o
00
o
M
o
IB
M
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'628
DECIMAL EQUITALENTS OP LBS. IN TONS.
Tablb
OP THE Decimal Equivalentb op Pabtb ot a ToyJ
Lbs.
1
Decimals
of a T n
Lbe.
370
Decimala
c»f a Ton
•165179
LbB.
820 '
Decimals
of a Ton
Lbs.
1270
Decimals
of a Ton
•000446
•366071
•566964
2
•000893
380
•169643
830
•370536
1280
•571429
8
•001339
390
•174107
840
•375000
1290
•575893
4
•001786
400
•178571
850
•379464
1300
•580357
6
•002232
410
•183036
860
•383929
1310
•584821
6
•002679
420
•187500
870
•388393
1320
•589286
7
•003125
430
•191964
880
•392857
1330
•593750
8
•003571
440
•196429
890
•397321
1340
•598214
»
•004018
450
•200893
900
•401786
1550
•602679
10
•004464
460
•205367
910
•406250
1360
•607143
20
•008929
470
•209821
920
•410714
1370
•611607
30
•013393
480
•214286
930
•415179
1380
•616071
40
•017851
490
•218760
940
-419643
1390
•620536
50
•022321
500
•223214
950
•424107
1400
•625000
m
•026786
510
•227679
960
•428571
1410
•629464
70
•031250
520
•232143
970
•433086
1420
•633929
80
•035714
530
•236607
980
•437600
1430
•638393
90
•040179
540
•241071
990
•4419«4
1440
•642857
100
•044643
550
•246536
1000
•446429
1460
•647321
110
•049107
560
•250000
1010
•460893
1460
•651786
120
•063571
670
•264464
1020
•466367
1470
•656250
130
•058036
580
•268929
1030
•459821
1480
•660714
140
•062500
690
•263393
1040
•464286
1490
665179
150
•066964
600
•267867
1050
•468750
1600
•669643
160
•071429
610
•272321
1060
•473214
1510
•674107
170
•075893
620
•276786
1070
•477679
1520
•678571
180
•080857
630
•281260
1080
•482143
1530
•683036
190
•084821
640
•285714
1090
•486607
1540
•687500
200
•089286
650
•290179
1100
•491071
1550
•691964
210
•093760
660
•294643
1110
•496686
1660
•696429
220
•098214
670
•299107
1120
•600000
1670
•700893
280
•102679
680
•303571
1130
•604464
1680
•705357
240.
•107143
690
•308036
1140
•608929
1^0
•709821
250
•111607;
700
•312500
1160
•513393
1600
•714286
260
•116071
710
•316964
1160
•617887
1610
•718750
270
•120536
720
•321429
1170
•622321
1620
•7232J4
^0
•125060
730
•326.893 .
1180
•526786
1630
•727679
290
•129464-
740
•3303^
1190
•531250
1640
•73^143
aoe
•133929
760
•334B2i
1.200
•6a67l4
1660
'736607
810
•1^8393
760
•339^86
12i0
•640179
16d0
•741071
feo
•142857'
770
•343750
1.220
•544643
1670
•745536
330
•147321
780
•348214
1230
•649107
16^0
•760000
340
•151786
790.
•352679
1240
•653571
1690
•764464
3.50
•156250
800
•357143
1250
•558036
1700
•758929
360
•160714.
-810-
•361607:
1260
•662600 .
1710
•763393
DECIMAL EQUIVALENTS OP ENGLISH WEIGHTS.
629
Table of the Dbcimal Equivalents of Pabts of
A Ton (concluded).
Lbs.
1720
1730
1740
1750
1760
1770
1780
1790
1800
1810
1820
1830
1840
Decimals
of a Ton
•767857
•772321
•776786
•781250
•785714
•790179
•794643
•799107
•803671
•808036
•812600
•816964
•821429
Lbs.
1850
1860
1870
1880
1890
1900
1910
1920
1^30
1940
1950
1960
1970
Decimals
of a Ton
•825893
•830357
•834821
•839286
•843750
•848214
•852679
•857143
•861607
•866071
•870536
•875000
•879464
Lbs.
1980
1990
2000
2010
2020
2030
2040
2050
2060
2070
2080
2090
2100
Decimals
of a Ton
•883929
•888393
•892857
•897321
•901786
•906250
•910714
•915179
•919643
•924107
•928571
•933036
•937500
Lbs.
2110
2120
2130
2140
2150
2160
2170
2180
2190
2200
2210
2220
2230
Decimals
of a Ton
•941964
•946429
•950893
•956857
•959821
•964286
•968750
•973214
•977679
•982143
•986607
•991071
•995636
2240 lbs. = 1 ton
T
^mm
Ozs.
Decimals
of a Lb.
OZB.
Decimals
of a Lb.
Ozs.
Decimals
of a Lb.
Ozs.
Decimals
of a Lb.
i
•015625
H
•266625
H
•615626
12i
•765626
i
•031260
H
•281250
Si
•531250
12*
•781250
1
•046876
4i
•296876
8|
•546876
12f
•796876
1
•062600
6
•312600
9
•562600
13
•812500
u
•078125
^\
•
•828126
H
•678125
13J
•828126
H
•093750
5]
■
•343750
n
•593750
13J
•843750
If
•109375
5\
•
•359376
n
•609375
m
•859876
2
•125000
6
•875000
10
•625000
14
•875000
2i
•140625
6;.
•390626
lOi
•640625
14i
•890626
24
•156250
H
•406260
104
•666260
IH
•906260
21
•171876
6i "
•421876
lot
•671875
14i
•921876
3
•187600
7
•437500
11
•687600
16
•937500
3i
•203125
H
•453125
Hi
•703126
15;
16
•953126
H
•218750
7*
•468760
lU
•718750
•968750
3f
•234376
7:
•484375
11
•734376
15}
•984875
4
•260000
8
•600000
12
•750000
16
1-000000
Qr.
Decimals
of a Ton
rN^ Decimals
^^^ of a Ton
Qrs.
Decimals
of a Ton
Qrs.
Decimals
of a Ton
1
•012500
2 -025000
3
•087600
4
•050000
Cwts.
D«cimalt
of a Ton
Cwts.
6
6
7
8
Decimak
of a Ton
9
10
11
12
Decimal:
of a To!
•460
•500
•650
•600
'wts.
13
14
15
16
DecinmL
of a Tor
Cwts
17
18
19
20
Decimals
of a Ton
1
2
3
4
•050
•100
•160
•200
•260
•300
•360
•400
•650
•700
•750
•800
•850
•900
•950
1-000
580
DECIMAL EQUIVALENTS OF PAETS OP THE FOOT.
d
•
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Hn
1
M
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0'«*<t«-©'^t<b-02t't:»pHJt2
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«OQCO?COCO«OOCO<0O25
rHOia0«^eQrHOS0O«O'«*«
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Hoo
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I
DECIMAL EQUIVALENTS OF PARTS OF THE INCH. 68 ^
Table op the Fractional Parts of the Inch, with
THEIR Corresponding Decimals.
Decimals Fractions
0078125
0156260
1-0234375
0312500
0390625
0468760
0546876
0625000
1-0703125
0781250
0859376
0937600
1015625
1093750
11171875
1250000
1328125
11406250
1484375
1562600
1640625
H718750
1796875
I- 1875000
1953126
2031260
1-2109375
2187600
12266625
2343750
2421S75
2600000
2678126
12656260
2734375
2812500
2890625
2968760
3046875
3125000
3203125
3281250
1
• • •
•••us
04 •••
• • •
L 1
• • •
64 158
is
• • • « • •
1
1
¥
•••128
32
64 •••
1
1 1
1
m
64 li28
• • •
• • • • • •
1
s
• • •
• t •
•••128
84 •••
1
1 1
\f^
• • •
1
64 128
V
?
• •• • • •
1
\8
••• l28
1
18
• • •
• •ft
«4 128
• • • • • •
•••123
J
• • •
8
• • •
»tTS»
V
• • m • • •
L
1
1
H
32
••• 128
X
1
I
fl
32
ST •••
jL
1
1 1
8
55
04 128
?
• • •
• • • • • •
I
15
• • •
•••TtFS
JL
1
16
• • •
64 •••
3
1 1
S
• • •
1
M
«4 1^8
• • • • • •
Jl
1
1
16
32
••• 12H
3
1
1
16
32
64 •••
3
1
1 1
lo
33
64 I5H
• • •
• • •
• • • •• •
••• 128
• • B
1 n:
64 128
• • •
• • • • • •
I
32
1
•••155
1
1
52
84 ♦••
6
T??
A
64 159
• • •
• • • • • •
6
1
I
•• •
•••TaS
TiA •••
« • ■
Decimals Fractions
•3359375
•3437600
3615625
3693750
3671875
3760000
3828125
3906260
3984375
4062500
4140625
4218750
4296875
4375000
4453125
4531260
4609375
4687600
4765625
4843750
4921875
5000000
5078126
5156250
5234376
5312500
5390625
-5468750
-5646875
5625000
5703125
•5781260
•6859375
•5937500
•6016625
•6093750
•6171875
•6250000
•6328126
•6406250
•6484375
•6562500
-6640625
1 1
S4 158
1
151
55 ••
¥ 'i
55 64 ••
55 65 l88
1
158
•••64 ••
••• 64 I5S
A
32 •••158
32 ^'i • • •
5a 6 4 128
I28
•••64 •••
J- 1
•••64 I5S
aA • • • • • •
1 _i
55 •••128
55 64 •♦•
32 "it T28
1
l55
••• 5? •••
• ' • 34 i5ff
3? • • • • • •
WS •••]59
32 64 •••
l JL .1_
55 64 128
1
i5i
... jfj ...
•••9? T5f
55
32 •••T5I
¥¥ V
55 5i 155
l25
... Tgi ...
•_• • Si 155
• • •
85
5g '••Ta
Decimals
Fractions
•6718760
•6796875
•6875000
•6953125
•7031250
•7109376
•7187500
•7265625
•7343750
•7421875
•7600000
•7578125
•7656250
•7734375
•7812500
•7890626
•7968760
•8046875
•8125000
•8203126
•8281260
•8359375
•8437500
•8516625
•8593760
•8671875
•8750000
•8828126
•8906250
•8984375
•9062500
•9140625
•9218750
•9296875
•9375000
•9453125
•9531250
•9609375li|
•9687500
•9765626
•9843750
•9921875
1^0000000
I 1
55 64 •••
82 64 158
1
155
• • • 64 • • •
••• ei lai
HA • • • • • •
1 _i
55 ••• i55
55 54 •••
35 64 l28
128
1
'** w* •••
i 1
••• 64 l58
sx « * • • • •
55 • • • 125
55 64 •••
111
5a 5t 155
... •••jfg
... ^ ...
1 1
'•• 64 128
OQ • • • • • •
52 •••155
55 ih •••
55 6i 155
ihi
\*
11
15
"6
8
16
• • •
1_
••• 64 128
vs • • • • • •
55 •••155
M 6? •••
55 64 128
• •• ••• 128
1
... 5j ...
532
FOKEION MONET, WEIGHTS, AND MEASURES.
Table op Foreign Money, Weights, and Measures,
WITH THEiB English Value.
CountrieB
Austria
Bombay
ChlziA
Denmark
France*
Germany
Greece
Holland
Madras
Portugal
Russia
Spain
Sweden
MONET
Gold Coins
8 florins
Mohnr
SOkrondalerjl
20 francs
30 reichs-
mark
SOdraobina
Ryder
Mohur
dmilreas
10 roubles
20 pesetas
30 krondaler
£ s. d.
16 10
1 9 2
15 10
0 0
15 10
6 1
9 2
3 4
12 2i
1ft 10
1 11^
SIlTer Coins
Value
i. d.
3 florins
8 lU
1 10}
Rupee
Tael
6 8
4 krondaler
4 6i
5 francs
3 11
6 reichs-
mark
• 0
ftd,rachma
9 10
Guilder
1 8
Rupee
110|
fiOOreas
3 2
Rouble
3 U
6 pesetas
4 krondaler
3 111
4 fit
} florin
t rupee
Mace
Krondaler
]frano
30pfennlge
Drachma
2S cents
imped
SOreaa
26oopect
Peseta
Daler
Value
s, d.
Countries
liBNGTH
Austria
Bombay
Chiaa
Denmark
France
Germany
Greece
Holland
Madras
Portugal
Russia
Spain
Sweden
Measure
Fuss
Hath
Chil^
Fod
Mdtre
Fuss
Attic foot
Palm
Covid
Palmo
Archine
Pie
Fot
Length
Inches
12'445
18
14-1
12-357
39*3704
12-357
1210
3*93704
18*6
8-656
28
11'128
11-6904
Measure
Klafter
Guz
Yan
Altt
D^camtoe
Rutbe
Stadiom
EUe
Vara
Sachine
Vara
Famn
Length
Feet
6*3226
8-35
U7'6 ■
2*0595
32-809.
12-367
6U0
8*2809
8*6067
7
2-783
68462
Measure
MeUe
Li
MiU
Myriam^tre
Postmeila
Mijle
Mil
Verst
Legnia
MU
X^ength
Miles
4*7143
■3458
4*6807
6*3138
4-6807
•6314
1-8786
•6639
4-2163
6-6428
Countries
Attitria
Bombay
China
Denmark
France
Germany
Greece
Holland
Madras
Portui^
Russia
Spain
Sweden
liquid capacity
Measures
Gallons
Eanne
•1667
Adoulie
1-616
Shingtsoug
•12
Pott
-2126
Litre
•2301
Quartier
•262
Kan
-2201
Puddy
•338
Canada
•3034
Vedro
2*7049
Qnartillo
•1105
Stop
•2878
Measures
Viertel
Para
Tau
Viertel
Decalitre
Anker
Matretes
Marcal
Pote
Anker
Azumbre
Xanna
Gallons
3*1143
34-24
1*2
1-7008
2-2009
7*669
8-488
2-704
1-8*202
81147
•4422
•6766
Measures
Eimer
Candy
Hwtth
Anker
Hectolitre
Eimer
Vat
Parah
Almfide
Sarokowaja
Arroba
Tunna
-Gallons
12-4572
198 93
13
8-3914
230097
15*118
33-0097
13-52
3*6405
334*688
8^6380
27-6288
* France, Italy, Belgium, aud Switzerland have perfect reciprocity in their
arrency. ...«■••'
FOREIGN MONEY, WEIGHTS, AND MEASURES. Q^^
Tabli
1 OF Foreign Money, Weights, and Measure j
WITH THBIR English Valitb (concluded).
Couotriea
DRY CAPACITY 1
1
Measure
Contents
Measure
Contentsl Measure
Contents
Bushels
Bushels
Quarters
Austria
Viertel
•4230
Metae
1-6918
Muth
6-3446
Bombay
Adoolie
•1898
Parah
303
Candy
8-8
China
Shingt^ong
•02
Tau
•2
Hwiih
•25
Denmark
Fjerdfng
•9667
Tonne
8*8268
Last
10-6235
France
Decalitre
•2761
Hectolitre
2-7611
Kilolitre
8-664
Germany
Viertel
•3780
acheffel
1-6121
Wispel
8-4022
Greece
Bachel
•763
Kila
•9162
Staro
•2824
Holland
Schepel
•2761
Mudde
2-7511
Last
10«317
Madras
Puddy
•0423
Parah
1-69
Garoe
16-9
Portugal
Alqueire
•878
F&nga
l-487a
Moio
2-79
Russia
Pajak
1^4426
Osmin
3-8862
Tschetwert
•7218
Spain
Almude
•1292
Fanega
15603
Cahic
2-8254
Sweden
Eanna
•0720
SiMum
2-015
Tunna
•50876
Conntries
1
WillG
HT
Name
Weight
Name
Weight
Name
Weight
Lbs.
Lbs.
Tons
Austria
Pfund
1-2852
Centner
12^d62
_
-i.
Bombay
Seer
•7
Maund
28
Candy
•25
China
Tael
•0838
Catty
1^883
Pecul
•0691
Denmark
Mark
•6614
Fund
11029
Skippund
•1575
France
Kilogmmme
2^2046
Quintal
220-46
Tonne
•9842
(Germany
Pfund
10311
Centner
113-426
Schiffpfund
•1619
Greece
Pound
•8811
Oke
2-8
Cantaro
•06
Holland
Pond
2-2046
-~
^^
Madras
SeRT
•626 •
Maund
25
Candy
•2232
Portugal
Airatel
10119
Arrobit
82-3795
QnintAl
•0578
Russia
Funt
■> -90264
Pud
36-1066
Packen.
•4836
Spain
Marco
•6072
Tiibra
1-0144
Quintal
•0463
1 Sweden
SkiUpund
•9876
Lispund
18-752
Skeppund
•1674
XHOLISH C0IK9.
Pound Stebung.
Pare gold in sovereign = 113*001 Troy grains.
Copper alloy in sovereign = 10*273 „
Fineness of sovereign = 22 caraU = '9161.
Total weight of sovereign = 123*273 Troy grains.
Silver.
Weight of pure silver in half-crown = 201*8 Troy grains,
„ „ shilling =80-7
„ „ sixpence — 40*3
Total weight of shilling »= 87*273
A pound Avoirdupois of copper is coined in 2i pence or 48
halfpennies.
S84
Discouirr table.
Table showino Bates or Discoukt at Yabiovs Fer-
CSNIA.QB&
Amount
£6
£7i
i
eio
£12^
£1»
£S0
£85
ofAooonnt
percent.
percent
£ »7 d.
perCt.
per Cent.
per Cent.
percent.
percent
£ «. d.
£ s, d.
£
i. d.
£ «.
d.
£ «. d.
£ 9. d.
£ $. d,
0 2 6
0 0 1^0 0 2i
0
0 3
0 0
H
0 0 4^
0 0 6
0 0 71
0 5 0
0 0 8
0 0 4^0
0 6
0 0
7*
0 0 9
0 10
0 13
0 10 0
0 0 6
0 0 9 0
1 0
0 1
8
0 16
0 2 0
0 2 6
0 15 0
0 0 9
0 1 u
0
1 6
0 1
m
0 2 3
0 3 0
0 3 9
10 0
0 10
0 16
0
2 0
0 2
6
0 8 0
0 4 0
0 5 0
1 10 0
0 16
0 2 3
0
8 0
0 8
9
0 4 6
0 6 0
0 7 6
1 15 0
0 19
0 2 7i
0
8 6
0 4
4^
0 68
0 7 0
0 8 9
2 0 0
0 2 0
0 3 0
0
4 0
0 6
0
0 6 0
0 8 0
0 10 0
2 10 0
0 2 6
0 8 9
0
6 0
0 6
8
0 7 6
0 10 0
0 12 6
2 16 0
0 2 9
0 4 U
0
6 6
0 6
lOi
0 8 3
0 11 0
0 13 9
8 0 0
0 8 0
0 4 6
0
6 0
0 7
6
0 9 0
0 12 0
0 15 0
8 10 0
0 8 6
0 6 8
0
7 0
0 8
9
0 10 6
0 14 0
0 17 6
8 15 0
0 8 9
0 6 7i6
7 6
0 9
^
0 11 8
0 15 0
0 18 9
4 0 0
0 4 0
0 6 0
0
8 0
0 10
0
0 12 0
0 16 0
10 0
4 10 0
0 4 6
0 6 9
0
9 0
0 11
3
0 13 6
0 18 0
12 6
4 15 0
0 4 9
0 7 li
0
9 6
0 11
10^
0 14 8
0 19 0
13 9
5 0 0
0 6 0
0 7 6
0
10 0
0 12
6
0 16 0
10 0
15 0
5 10 0
0 5 6
0 8 8
0
11 0
0 13
9
0 16 6
12 0
1 7 C
5 15 0
0 6 9
0 8 7i
0
11 6
0 14
4i
0 17 8
18 0
1 8 9
6 0 0
0 6 0
0 9 0
0
12 0
0 15
0
0 18 0
1 4 0
1 10 0
6 10 0
0 6 6
0 9 9
0
13 0
0 16
8
0 19 6
16 0
1 12 6
6 15 0
0 6 9
0 10 li
0
13 6
0 16
10^
10 8
17 0
1 13 9
7 0 0
0 7 0
0 10 6
0
14 0
0 17
6
1 1 0
1 8 0
1 15 0
7 10 0
0 7 6
0 11 3
0
16 0
0 18
9
12 6
1 10 0
1 17 6
8 0 0
0 8 0
0 12 0
0 16 0
1 0
0
14 0
1 12 0
2 0 0
8 10 0
0 8 6
0 12 9
0 17 0
1 1
3
15 6
.1 14 0
2 2 6
9 0 0
0 9 0
0 13 6
0 18 0
1 2
6
1 7 0
1 16 0
2 5 0
9 10 0
0 9 6
0 14 8
0 19 0
1 8
9
18 6
1 18 0
2 7 6
10 0 0
0 10 0
0 15 0
0 0
1 6
0
1 10 0
2 0 0
2 10 0
10 10 0
0 10 6
0 15 9
1 0
1 6
8
1 11 6
2 2 0
2 12 6
11 0 0
0 11 0
0 16 6
2 0
1 7
6
1 13 0
2 4 0
2 16 0
11 10 0
0 11 6
0 17 3
'^
8 0
1 8
9
1 14 6
2 6 0
2 17 6
12 0 0
0 12 0
0 18 0
4 0
1 10
0
1 16 0
2 8 0
3 0 0
12 10 0
0 12 6
0 18 9
6 0
1 U
8
1 17 6
2 10 0
3 2 6
18 0 0
0 13 0
0 19 6
6 0
1 12
6
1 19 0
2 12 0
3 5 0
13 10 0
0 13 6
I 0 3
7 0
1 13
9
2 0 6
2 14 0
8 7 6
14 0 0
0 14 0
I 1 0
8 0
1 15
0
2 2 0
2 16 0
8 10 0
14 10 0
0 14 6
1 1 9
9 0
1 16
8
2 3 6
2 18 0
8 12 6
16 0 0
0 15 0
12 6
1 10 0
1 17
6
2 6 0
8 0 0
8 15 0
20 0 0
1 0 0
1 10 0
2
0 0
2 10
0
3 0 0
4 0 0
6 0 0
30 0 0
I 10 0
2 5 0
3
0 0
3 15
0
4 10 0
6 0 0
7 10 0
40 0 0
2 0 0
8 0 0
4
0 0
5 0
0
6 0 0
8 0 0
10 0 0
50 0 0
2 10 0
3 16 0
5
0 0
6 6
0
7 10 0
10 0 0
12 10 0
60 0 0
3 0 0
4 10 0 6
0 0
7 10
0
9 0 0
12 0 0
16 0 0
70 0 0
3 10 0
5 6 0 7
0 0
8 15
0
10 10 0
14 0 0
17 10 0
80 0 0
4 0 0
6 0 0 8
0 OilO 0
0
12 0 0 16 0 C
20 0 0
90 0 0
4 10 0
6 15 0 19
0 O'U 6
0
18 10 0 18 0 C
►22 10 0
PRlCEit PKG I.R, QB., CW't.
636
TIMBER LOADS.
TlMUUI LOIDB.
One ton of Ebony b 26-30 o. feet
Oak = 3S-iO
Hahogany » 32-SO
Ash = 84Mf
Beeoh e 48-SO
Maple = 46-49
m
n
n
n
n
»
**
n
ft
M
Walnut
50-64
w
f»
n
M
Sand .
Gravel
Hud .
Harl
Clay .
Ohalk
about SO cwL
» 30 »
■ *• ••
» 36 „
One ton of Battio Fir
b:60 68e
.feit
*• >i
Elm
B 63-66
»
w t»
Pine
«s6»-60
n
n n
Deak
= 66-66
n
H f>
lime-tree
:^ 66-66
n
»» n
Scotch Fir
= 60-66
»
CB, STC., PBB CUBIC
TABD.
iandstono
• • •
about 39 ewt
Shato .
• •
» M
M
Qoarts
• • «
.. 41
»
Granite .
• •
n «
1)
Trap .
• • •
.. «
«
Slate
• «
^ «
(1
Table of the Points of the Compass and their
Angles with the Meridian.
Hortii
N. by E.
NNE.
N. by W.
NNW.
NE. by N. NW. hy N.
NE*
NE. by E.
ENK.
E. by N.
East
NW.
NW.bvW.
WNW.
W. by N.
W«8t
Points
1
1
1
1
2
II
2J
B
3i
8
3
4
4*
5
6
6|
7
f
« I H
2 48 45
6 37 8Q
8 26 15
11 16 0
14 8 45
16 52 30
19 41 15
22 80 a
25 18 45
28 7 80
80 56 15
88 45 0
86 88 45
89 22 80
42 11 15
45 0 0
47 48 45
50 37 80
53 26 15
56 15 0
59 8 45
61 52 80
64 41 15
67 80 0
70 18 45
78 7 30
75 56 15
78 45 0
81 88 45
84 22 30
87 11 15
90 0 0
Points]
?^
Oi
Of
1
li
II
2
if
2|
3
8i
H \
4
South
5
bl
51
6
6}
7
n
7f
8
S. by E.
SSE.
SE. by S.
SE.
SE. by E.
E8E.
E. by S.
East
aby W.
SSW.
SW. by S.
SW.
SW. by W.
WSW.
W. by R
West
nsuruL NOUBERa,
UKMIMAU HyUlYALiCXIlS %JW mXJUiBil, t.L\j.
Table of Incohe,
Wages, ob Expenses.
Per
Per
Per
Per
Per
Per
Per
Per
Tear
Mouth
£ «. d.
Week
Day
Year
£
Month
Wee
d.
Day
e «.
£ ». d. £ *. d.
£ t. d.£ i.
£
«. d.
1 0
0 1
8
0 0 4A 0 0 Of
13 0
118 0 5
0
0
0 8^
1 10
0 2
6
0 0 7 0 0 1
13 13
12 9 0 5
3
0
0 9
2 0
0 3
4
0 0 94 0 0 U
0 0 9} 0 0 U
0 0 lU 0 0 ll
0 1 l| 0 0 2
14 0
1 3 4' 0 5
4i
0
0 9|
2 2
0 3
6
14 14
1 4 6; 0 5
8
0
0 9|
2 10
0 4
2
15 0
15 0 0 5
9
0
d 10
8 0
0 5
0
15 15
1 6 3 0 6
0*
0
0 10^
8 3
0 5
3
0 1 2^ 0 0 2
16 0
16 8 0 6
2
0
0 10^
3 10
0 5
10
0 1 4^ 0 0 24
0 1 eX 0 0 21
16 16
18 006
H
0
0 11
4 0
0 6
8
17 0
1 8 4! 0 6
6
0
0 \n
4 4
0 7
0
0 1 7i 0 0 2|
17 17
1 9 9; 0 6
10
0
0 111
4 10
0 7
6
0 1 8}; 0 0 8
18 0
1 10 0, 0 6
11
0
0 111
5 0
0 8
4
0 1 11 |0 0 84
18 18
1 11 6 0 7
8
0
1 ^
6 6
0 8
9
0 2 04 0 0 3|
0 2 ll 0 0 3|
19 0
1 11 8 0 7
H
•0
1 Oi
6 10
0 9
2
20 0
1 13 4 0 7
8
0
1 4
6 0
0 10
0
0 2 3 0 0 4
80 0
2 10 0 0 11
6
0
1 7|
6 6
0 10
6
0 2 5
0 0 44
40 0
3 6 8 0 15
^
0
2 2^
6 10
0 10 10
0 2 6
0 0 fj
60 0
4 3 4 0 19
3
0
2 9
7 0
0 11
8
0 2 84
0 0 4k
0 0 41
60 0
5 0 013
Of
0
3 di
7 7
0 12
3
0 2 10
70 0
5 16 8 1 6
11
0
3 10
7 10
0 12
6
0 2 10^
0 0 5
80 0
6 13 4 1 10
9
0
4 4i
8 0
0 18
4
0 8 1
0 0 54
90 0
7 10 0; 1 14
^
0
4 11
8 8
0 14
0
0 8 2| 0 0 5i
100 0
8 6 8; 1 18
5
0
5 51
8 10
0 14
2
0 8 84 0 0 5i
200 0
16 13 4 3 16 11 1
0 10 nil
9 0
0 15
0
0 3 5l 0 0 6
800 0
25 0 0, 5 15
4i
0 16 5II
9 9 0 15
9
0 8 7}; 0 0 64
0 8 10 0 0 6i
400 0
33 6 8 ,7 13 lO'l
1
1 11 1
10 0
0 16
8
500 0
41 13 4 9 12
H
1
7 4||
0 10
0 17
6
0 4 04 0 0 7
600 0
50 0 Oil 10
9
1
12 loJ
.1 0
0 18
4
04 8
0 0 74
700 0
58 6 8 13 9
2f
1
18 4i
.1 11
0 19
3
0 4 54
0 4 7i
0 0 7^
800 0
66 18 4 15 7
?l
2
3 10
2 0
1 0
0
0 0 8
900 0
75 0 017 6
2
§ 34
2 12jl 1
0 0 4 10'
0 0 8i
1,000 083 6 819 4
2 U 9i|
Table of the Pecihal Equivalents op Pence asb
Shillings.
Pence
I
1
11
2
II
3
Shillings
*02083dd
•0416666
•0625000
•0833333
•1041666
•1260000
•1468333
•1666666
•1875000
•2083833
•229^666
•2500000
Pence
Shillings
•2708883
•2916666
•3126000
•3333833
•8541666
•3760000
•3958333
•4166666
•4375000
•4583333
•4791666
•5000000
Pence! Shillingg
1^08333
5416666
•5625000
•5833333
•6041666
•6250000
•6468338
•6666666
•6875000
•7088333
•7291666
•7^00000
Pence
l\
10
lOi
104
10|
11
?1J
12
Shillings
7706383
•7916666
•8125000
•8333338
•8541666
•8760000
•8958383
•9166666
•9376000
•9583333
•9791666
l^OOOOOOO
TABLE OF GIBOniAR HEAI^GBB.
6»9
Tabus of the Oircular Meisxjbe, ob Length of Oir-
CTJLAR AeC STTBTENDING ANY ANGLE, RADIUS BEING UNITT.
To calculate the circular measure of any angle, see * Tri-
gonometry ' (pp. $ and 9 )•
Use op the Table.— JJr. : Required to find the length
of the circular arc subtending an angle of iO^ IV 15" on a
circle of 660 feet radius.
Tabular No. for 40°= -698131701
ir = 003199770
15''= 000072722
tt
if
»
Length of arc = (660 x -701404193) = 392*78634808 ft.
Seconds.
Sec.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Circ. Meas.
0000048481
0000096963
0000146444
0000193926
0000242407
0000290888
0000339369
•0000387860
-0000436332
-0000484814
-0000633296
-0000681776
•0000630268
■0000678739
■0000727221
Sec.
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Circ. Meas.
•0000775701
•0000824183
•0000872665
•0000921146
•0000969627
•0001018109
•0001066691
•0001115071
•0001168663
•0001212034
•0001260616
•0001308997
•0001367478
•0001406960
•0001464441
Sec.
31
32
33
34
36
36
37
38
39
40
41
42
43
44
46
Circ. Meas.
•0001602922
•0001561404
•0001699886
•0001648367
•0001696848
•0001746329
•0001793811
•0001842291
•0001890773
•0001939256
•0001987736
•0002036217
•0002084699
•0002133180
•0002181662
occ»
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Circ. Mens.
0002230143
0002278624
0002327106
0002376587
0002424068
0002472550
0002521031
0002669513
0002617994
0002666476
0002714967
0002763487
0002811919
0002860401
0002908882
Minutes.
M.
1
2
3
4
6
6
7
8
9
10
U
12
13
14
16
Circ. Meas.
0002908882
0005817764
0008726646
0011636528
0014544410
M.
16
17
18
19
20
0017453293 21
0020362176
002327i057
0026179939
0029088821
0031997703
0034906686
0037816467
0040724349
0043633231
22
23
24
25
26
27
28
29
30
Olrc. Meas.
0046642113 31
0049460995 32
0062359878 33
0056268760 34
0068177642 35
0061086624 36
0063995406 37
•0066904288 38
•0069813170 39
•0072722062 40
•0076630934 41
•0078639816 42
•0081448698 43
•008436758l| 44
•00872664631 46
M.
Circ. Meas. M.l Circ. Meas.
•0090176346 46
•0093084227 47
-0096993109 48
0098901991 49
0101810873 60
0104719765 51
0107628637 62
0110537519 63
0113446401 54
0116356283 55
0119264166 56
0122173048 67
0126081921 68
0127990812 59
0130899694 60
0133808576
•0136717458
01396263401
0142635222
0146444104
0148352986
0151261869
0164170761
0167079633
0159988615
0162897397
0166806279
0168715161
0171624043
0174532926
40
TABLE OF CIRCITLAR lifEASURB.
Table of th» Cibcplab Mb'abpbe of aity Angle (oontinoedlj
DEOREBS.
Cire. H«M.
•017463293
•034906586
•052359878
•069813170
•087266463
•104719766
•122173048
•139626340
•157079639
•174532926
•191986218
•209439510
•226892803
•244346095
•261799388
•279262680
•2916705973
•314169265
•331612558
•349066850
•366619143
•388972435
•401426728
•418879020
•436332813
•468785606
•471238898
•488692191
•506145483
•523598776
•541062068
•558506361
•675968653
•593411946
•610865238
•628318531
•645771823
•663226116
•680678408
•6ddl31701
•715584^93
•733038286
•760491578
•767944871
•786398.16.3
Dcg
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
Clrc. Mcai.
•802851456
•820304748
•837758041
•855211333
•872664626
■890117919
•907571211
925024504
•942477796
•969931089
•977384381
•994837674
1-012290966
Clrc Heaa.
Deg.
"9lll-6882496XI
92 1-60570291:
1-029744259 104
1-047197561
1064660844
1*082104136 107
108
109
1-09955742^
1-117010721
1-134464014|110
M51917806 111
1-169370599112
•186823891
1-204277184 114
1-221730476
1-289183769
1266637061
1^274090354 118
1-2^1648646
1-308996939
1-326450232
1'343903624
1-361356817
1^378810109I124
1^396263402 125
1^413716694
1-431169987
r448623279
120
121
122
123
126
127
128
466076572 129
1^48352d664 130
V500983J57 l3l
1-518436449 132
1-535889742 133
1-553343034 134
1:6.70796327
93
94
95
d6
97
98
99
100
101
102
103
105
106
113
115
116
117
16231 66204
1-640609497
1-658062789
1-675516082
1-692969371
1^710422667
1 •727876969
1-745329252
1 •762782545
1^780235837
1-797689130
1-815142422
1-832596716
1^850049007
1 •867502300
1-884955692
1^902408885
1 •919862177
1-937315470
1-954768762
1-972222055
1-989676347
2-007128640
2-024681932
2-042036225
2-059488517
1192-07e941«K 164
2-0943951021 165
2-111848396 166
2-129301687 167
2-146754980 168
2-164208272 169
2-181661566 1^0
2^1991 14868 171
2-216568160 172
2-234021443 173
2-261474735 174
2^268d2$028 17i5
^•266381320 176
2-303S34613
2-321287906
2-338741198
JL3512-35619449(
Deg.
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
163
164
155
156
157
158
159
160
161
162
163
177
178
179
180
Clrc. Veal.
2-373647783
2-391101076
2^408554368
2^426007660
2^443460953
2-460914245
2-478367538
2-496820830
2-513274123
2-530727415
2-548180708
2-66563400
2^6830«7293
2600540685
2*617993878
2-635447170
2-652900463
2-670358766
2-687807048
2-7O526O340
2-722713633
2-740166926
2-767620218
2-775078511
2-792526803
2-809980096
282743S388
2^844886681
2-86283^73
2^879793266
2^897246558
2-914699851
2-932168143
2-949606486
2-967069728
2-984513021
8-001966313
3019419606
.8-086872898
3054d2619l
3'07i77d484
8^089232776
3^ 106686069
3124139361
3141692654
TABLE OF CIRCULAR MEASURE.
6U
Table of the Cibculab Measure of any Angle (concluded).
DEaaEES.
Dcg.
Cire. Meat.
181 1315904694^226
182|317e499239 227
183,3193952631 228
184 3-211405824 229
185!3-228859116 230
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200 3-490668604 245
201
202
203
204
206
206
207
208
209
3 246312409 231
3-263766701 232
3-281218994 233
3-298672286 234
3-316126679 236
3-333578871 236
3-361032164 237
3-368486466 238
3-386938749 239
3-403392041 240
3-420846334 241
3-4S8298626 242
3-465751919 243
3-473205211 244
3-608111797 246
3-625665089 247
3-543018382 248
3-560471674 249
3-677924967 260
3-595378269 251
3-612831562 252
3-630284S44 263
3-647738137 264
210 3-665191429
211
212
213
214
215
216
217
218
219
220
221
222
223
224
226
3-682644722 266
3-700098014 267
3-717651307 258
3-735004599 269
3-762467892 260
3-769911184 261
3-787364477 262
3-804817769 263
3-822271062 264
3-839724354 266
3-867177647 266
3-874630939 267
3-892084232 268
3-909637624 269
3-926990817 270
Circ. Uev.
peg.
01271
255
3-94444411
3-961897402 272
3-979350695 273
3-996803987 274
4-014257280 275
4031710572 276
4049163865 277
4 066617157 278
4084070450 279
4101523742 280
4-118977035 281
4136430327 282
4153883620 283
4-171336912 284
4-188790205 286
4 -206243497 286
4 -223696790 287
4-241150082 288
4 -268603376 289
4 -276056667 290
4-293609960 291
4-310963252 292
4-328416546 293
4-345869837 294
4-363323130 295
4-380776423 296
4-398229716 297
4-416683008 298
4-4831363O0 299
4-450689593 300
4-468042886 301
4-485496178 302
4-502949470 303
4-520402763 304
4-537866055 305
4-565309348 306
4-672762640 307
4-590215933 308
4-607669225 309J5
4-626122618 310
4-642576810 311
4-660029103 3i2
4-677482896 313
4-694935688 314
4-712388980 316
CIre. Meat.
Deg.]
4-7298422731316
4-747295565 317
4-764748868 318
4-782202160 319
4-799665443 320
4-817108736 321
4-834562028 322
4-862015321 323
4-869468613 324
4-886921906 326
4-904376198 326
4-921828491 327
4-939281783 328
4-966736076 329
4-974188368 330
4-991641661 331
6009094963 332
6-026648246 333
6-044001538 334
5-061464831 336
5-078908123 336
6096361416 337
6113814708 338
5-131268001 339
5-148721293 340
5-166174586 341
5-183627878 342
5-201081171 343
5-218534463 344
5-235987766 345
5'26d441049 346
5-270894341 347
6-288347633 348
5-306800926 349
5-323264219 360
6-340707611 351
6-358160804 362
6;37&614096 363
393067389 354
6-410520681 356
5427973974 366
6-4464272661367
5-462880669|358
5-48033385l'36i
6-497787144.360
Circ. MeM.
5-616240436
5-532698729
5-660147021
6-667600314
6-686063606
6-602606899
5-619960191
5-637413484
6-664866776
5-672320069
6-689778362
6-707226664
5-724679947
6-742138239
6-759686532
5-777039824
5-794498117
5-81 1946409
5-829399702
6-846852904
5-864306287
6-881769679
5-899212872
5-916666164
6-934119467
5-951572749
5-969026042
6-986479334
6-003932627
6021386919
6-038839212
6-066292504
6073746797
6-091199089
6-108662382
6-126106676
6-143568967
6-161012260
6-178466662
6195918846
6-21337^137
6-230826430
6-248278722
6-265732015
6-283186307
Tt
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8
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>52
AREAS OF SEGMENTS OF CIRCLES.
Table ov xhs Akbas of the Seomentb of a Circle,
THE Diameter being Unity.
To find tlie area of the segment of antf circle from t?ie follonnng
tablet.
Rule.— Divide the height of the segment by the diameter,
take out the corresponding tabular area, which multiply by
the square of the diameter for the result.
•001
•002
•003
•004
•005
•006
•007
•008
•ooy
•010
•on
•012
•013
•014
•016
•016
•017
•018
•019
•0201
•021
•022
•023
•024
•025
•026
•027
•028
•029
•030
•031
•032
•033
•034
•035
•036
•037
Aiva
000042
000119
000219
000337
000470
000618
000779
000951
001135
001329
001533
001746
001968
002199
002438
002685
002940
003202
003471
003748
001031
004322
004618
004921
005230
005546
005867
006194
006527
006866
007209
007558
007913
008273
008638
009008
009383
H
D
•038
•039
•040
•041,
•042
•048
•044
•046
•046
•047
•048
•049
•050
•051
•052
•053
•054
•056
•056
•057
•058
•059
•060
•061
•062
•063
•064
•06S
•066
•067
•068
•069
•070
•071
•072
•073
•074
Area
•009763
•010148
•010537
•010931
•011380
•011734
•012142
•012564
•012971
•013392
•013818
•014247
•014681
•015119
•015661
•016007
•016467
•016911
•017369
•017831
•018296
•018766
•019239
•019716
•020196
•020680
•021168
•021659
•022154
•022652
•023154
•023659
•024168
•024680
•025196
•025714
•026236
D
•075
•076
•077
•078
•079
•080
•081
•082
•083
•084
•086
•086
•087
•088
•089
•090
•091
•092
•093
•094
•095
•096
•097
•098
•099
•100
•101
•102
•103
•104
•105
•106
•107
•108
•109
•110
•111
Area
H
•026761
•027289
•027821
•028356
•028894
•029436
•029979
•030526
•031076
•031629
•032186
•032745
•033307
•033872
•034441
•035011
•035685
•036162
•036741
•037323
•037909
•038496
•039087
•039680
•040276
•040876
•041476
. -042080
•042687
•043296
•043908
•044622
•045139
•045759
•046381
•047005
•047632
112
118
114
115
116
117
118
119
120
121
122
123
124
126
126
127
128
129
130
131
132
133
134
136
136
137
138
139
140
141
142
143
144
146
146
147
148
Area
•048262
•048894
•049628
•060165
•050804
•061446
•062090
•062736
•053886
•064036
•064689
•066345
•066003
•066663
•067826
•057991
•068658
•059827
•059999
•060672
•061848
•062026
•062707
•063389
•064074
•064760
•065449
•066140
•066833
•067628
•068225
•068924
•069625
•070328
•071033
•071741
•072460
AREAS OP SEGMENTS OF CIReLES.
Q5i
Table of the Abeab op the Segments of a
Circle,
the Diameter being Unity (continued)
•
•149
Area
H
D
Area
•106261
H
D
Area
H
1)
Area
•073161
•193
•237
•142387
•281
•180918
•160
•073874
•194
•107051
•238
•143238
•282
•181817
•161
•074689
•195
•107842
•239
•144091
•283
•182718
•162
•075306
•196
•108636
•240
•144944
•284
•183619
•163
•076026
•197
•109430
•241
•145799
•285
•184521
•164
•076747
•198
•110226
•242
•146655
•286
•185425
•156
•077469
•199
•111024
•243
•147512
•287
•186329
•166
•078194
•200
•111823
•244
•148371
•288
•187234
•167
•078921
•201
•112624
•245
•149230
•289
•188140
•168
•079649
•202
•113426
•246
•160091
•290
•189047
•169
•080380
•203
•114230
•247
•160963
•291
•189956
•160
•081112
•204
•115035
•248
•161816
•292
•190864
•161
•081846
•205
•115842
•249
•162680
•293
•191776
•162
•082682
•206
•116650
•260
•153646
•294
•192684
•163
•083320
•207
•117460
•251
•164412
•296
•193696
•164
•084059
•208
•118271
•252
•165280
•296
•194509
•165
•084801
•209
•119083
•253
•166149
•297
•195422
•166
•085644
•210
•119897
•254
•167019
•298
•196337
•167
•086289
•211
•120712
•265
•167890
•299
•197262
•168
•087036
•212
•121529
•256
•158762
•300
•198168
•169
•087785
•213
•122347
•257
•169636
•301
•199085
•170
•088536
•214
•123167
•268
•160510
•302
•200003
•171
•089287
•215
•123988
•259
•161386
•303
•200922
•172
•090041
•216
•124810
•260
•162263
•304
•201841
•173
•090797
•217
•125634
•261
•163140
•306
•202761
•174
•091654
•218
•126459
•262
•164019
•306
•203683
•175
•092313
•219
•127286
•263
•164899
•307
•204606
•176
•093074
•220
•128113
•264
•165780
•308
•205527
•177
•093836
•221
•128942
•266
•166663
•309
•206451
•178
•094601
•222
•129773
•266
•167646
•310
•207376
•179
•095366
•223
•180605
•267
•168430
•311
•208301
•180
•096134
•224
•131438
•268
•169315
•312
•209227
•181
•096903
•226
•132272
•269
•170202
•313
•210154
•182
•097674
•226
•133108
•270
•171089
•314
:211082
•183
•098447
•227
•133945
•271
•171978
•316
•212011
•184
•099221
•228
•134784
•272
•172867
•316
•212940
•186
:099997
•229
•135624
•273
•173768
•317
•213871
•186
•100774
•230
•136465
•274
•174649
•318
••214802
•1.87
•101663
•231
•137307
•276
•176542
•319
•21^733
•188
•102334
•232
•138150
•276
•176435
•320
•216666
•189
•103116
•'233
•138996
•277
•177330
•321
•217699
•190
•103900
•234
•139841
•278
•178225
•322
•218533
•191
•104685
•235
•140688
•279
•179122
•323
•219468
•192
•105472
•236
•141637
•280
•180019
•324
•220404
54
AREAS OF SEGUBIITS OF CIRCIiBS.
Tablb of the Abbas of the Segments of a Circle,
THE DiAMETEB BEING UNITY (concluded).
H
D
Area
H
D
Area
H
D
•413
- 1 F
Area
•326
•221340
•369
•263213
•306140
•467
•349752
•326
•222277
•370
•264178
•414
•307125
•458
•360748
•327
•223216
•371
•265144
•415
•308110
•459
•351745
•328
•224154
•372
•266111
•416
•309095
•460
•352742
•329
•226093
•373
•267078
•417
•310081
•461
•363739
•330
•226033
•374
•268045
•418
•311068
•462
•354736
•331
•226974
•376
•269013
•419
•312054
•463
•366732
•332
•227915
•376
•269982
•420
•313041
•464
•366730
•333
•228868
•377
•270951
•421
•314029
•465
•357727
•334
•229801
•378
•271920
•422
■315016
•466
•368725
•336
•230746
•379
•272890
•423
•316004
•467
•359723
•336
•231689
•380
•273861
•424
•316992
•468
•360721
•337
•232634
•381
•274832
•425
•317981
•469
•361719
•338
•233580
•382
•276803
•426
•318970
•470
•362717
•339
•234526
•383
•276775
•427
•319959
•471
•363716
•340
•236473
•384
•277748
•428
•320948
•472
•364713
•341
•236421
•385
•278721
•429
•321938
•473
•365712
•342
•237369
•386
'279694
•430
•322928
•474
•366710
•343
•238318
•387
•280668
•431
•323918
•476
•367709
•344
•239268
•388
•281642
•432
•324909
•476
•368708
•345
•240218
•389
•282617
•433
•325900
•477
•369707
•346
•241169
•390
•283692
•434
•326892
•478
•370706
•347
•242121
•391
•284568
•436
•327882
•479
•371705
•348
•243074
•392
•286544
•436
•328874
•480
•372704
•349
•244026
•393
•286521
•437
•329866
•481
•373703
•350
•244980
•394
•287498
•438
•330858
•482
•374702
•351
•246934
•396
•288476
•439
•331860
•483
•375702
•362
•246889
•396
•289463
•440
•332843
•484
•376702
•363
•247845
•397
•290432
•441
•333836
•485
•377701
•354
•248801
•398
•291411
•442
•334829
•486
•378701
•366
•249757
•399
•292390
•443
•335822
•487
•379700
•366
•260715
•400
•293369
•444
•336816
•488
•380700
•367
•261673
•401
•294349
•445
•337810
•489
■381699
•368
•252631
•402
•296330
•446
•338804
•490
•382699
•369
•253690
•403
•296311
•447
•339798
•491
-383699
•360
•254560
•404
•297292
•448
•340793
•492
•384699
•361
•265610
•405
•298273
•449
•341787
•493
•386699
•362
•256471
•406
•299265
•450
•342782
•494
•386699
•363
•257433
•407
•300238
•461
•343777
•496
•387699
•364
•258395
•408
^301220
•462
•344772
•496
•388699
•366
•269357
•409
'302203
•463
•345768
•497
•389699
•366
•260320
•410
•303187
•464
•346764
•498
•390699
•367
•261284
•411
•304171
•465
•347769
•499
•391699
•868
•262248
•412
•305155
•466
•348766
•500
•392699
B^VAXBS, CUBES, BOOTS, AND SECIPROCALS. 65
Tablb of SavABis, Citbbs, SavABB Roots, Oubb Roots, ait
Recipbooals <
9P ALL InTEOBB NuJIBERS FBOK ]
L to 2200
No.
Square
Cube
Square Root
Cube Root
Reciprocal
1
1
1
l-OOOOOOO
1-0000000
l-00000000<
2
4
8
1-41421S6
1-2599210
•60000000<
8
9
27
1-7320508
1-4422496
-333833331
4
16
64
2-0000000
1-5874011
-25000000(
5
25
125
2-2360680
1-7099759
•20000000(
6
86
216
2-4494897
1-8171206
•16666666:
7
49
848
2-6467518
1-9129312
-142857141
8
64
612
2-8284271
2-0000000
-12600000(
9
81
729
3-0000000
2-0800887
-iiiiiui:
10
100
1000
3-1622777
2-1544347
•lOOCOOOOi
11
121
1831
3-3166248
2-2239801
•09090909;
12
144
1728
3-4641016
2-2894286
•08333333J
18
169
2 197
3-6065513
2-3513347
•07692307:
14
196
2 744
3-7416574
2-4101422
•07142857:
15
2 25
8 875
3-8729833
2-4662121
-06666666:
16
2 56
4 096
4-0000000
2-5198421
•06250000(
17
2 69
4 913
4-1231056
2-6712816
•06882352J
18
8 24
6 882
4-2426407
2-6207414
•05556666(
19
8 61
6 859
4-3588989
2-6684016
•062631571
20
4 00
8 000
4-4721360
2-7144177
•06000000(
21
4 41.
9 261
4-5825757
2-7589243
•04761904J
22
4 84
10 648
4-6904168
2-8020393
•04546454^
28
6 29
12 167
4-7958315
2-8438670
•04347826J
24
6 76
18 824
4-8989795
2-8844991
•04166666^
26
6 25
15 625
5-0000000
2-9240177
•04000000(
26
6 76
17 676
5-0990195
2-9624960
•03846153^
27
7 29
19 688
5-1961524
3-0000000
•03708703^
28
7 84
21952
5-2915026
3-0365889
•03571428(
29
8 41
24 889
5-3851648
8-0723168
•034482751
80
9 00
27 000
5-4772256
3-1072325
•03338333(
8r
9 61
29 791
5-5677644
8-1413806
•03225806^
82
10 24
82 768
6-6568642
3-1748021
•08125000(
88
10 69
85 987
5-7445626
3-2075848
•08030303(
84
1166
89 804
5-8309519
3-2396118
•02941176^
86
12 25
42 875
5-9160798
8-2710663
-028671421
86
12 96
46 650
60000000
8-3019272
•02777777^
87
18 69
60 658
6-0827625
8-3322218
•02702702^
88
14 44
64 672
6-1644140
3-3619754
-026316781
89
15 21
69 819
6-2449980
3-3912114
•02664102(
40
16 00
64 000
6-8246658
3-4199519
*02500000<
41
16 61
68 921
6'4081242
3-4482172
•02439024^
42
17 64
74 088
6-4807407
8-4760266
•02380962^
48
18 49
79 607
6-5574385
3-5038981
•02326681^
44
19 86
85 184
6-6332496
8-5303483
•022727271
45
20 25
91125
6-7082089
d*666893d
•022222225
ede
8QUABE8, CUBSS, BOCXTO, AKD BSGIPBOCAXM.
46
Sqaura
Onba
Square Root
CnbeBdot
Reciprocid
2116
97 886
6-7823300
3-5830479
•021739130
47
22 09
108 828
6-8556546
3-6088261
•021276600
48
28 04
110 592
6-9282032
3-6342411
•02O8S3333
49
24 01
117 649
7-0000000
3-6593057
•020408163
60
25 00
125 000
7-0710678
36840314
•020000000
51
26 01
182 651
71414284
3-7084298
•019607843
52
27 04
140 608
7-2111026
37325111
•019-280769
58
28 09
148 877
7-2801099
3-7562858
•018867925
54
29 16
157 464
7-3484692
3-7797631
•018618619
55
80 25
166 375
7-4161985
3-8029625
•018181818
56
8186
175 616
7-4833148
3-8258624
•017867143
57
82 49
186 108
7-5498344
3-8485011
•017543860
58
88 64
195 112
7-6157731
3-8708766
•017241379
59
84 81
205 879
7-6811457
3-8929965
-016949163
60
86 00
216 000
7-7459667
3-9148676
•016666667
61
87 21
226 981
7-8102497
3-9364972
'016?93443
62
88 44
288 828
7-8740079
3-9678915
•016129032
68
89 60
250 047
7-9872639
3-9790571
•016873016
64
40 96
262 144
8-OOOOOCO
4-0000000
•015625000
65
42 25
274 625
8-0622577
40207256
•016384615
66
48 56
287 496
8-1240384
4-0412401
•016161515
67
44 89
800 768
8-1853528
4-0616480
•014926373
68
46 24
814 482
8-2462113
4-0816551
•014705882
69
47 61
828 509
8-3066239
4-1015661
•014492764
70
49 00
843 000
8-3666003
4-1212863
•014286714
71
50 41
857 911
8-4261498
4-1408178
•014084507
72
5184
878 248
8-4852814
4-1601676
•018888889
78
58 29
889 017
8-5440037
4-1793892
•018698630
74
54 76
405 224
8-6023263
4'1983864
•013613514
76
56 25
421875
8-6602540
4-2171633
•018383333
76
57 76
488 970
8-7177979
4-2358236
•013167895
77
59 29
456 588
8-7749644
4-2543210
•012987013
78
60 84
474 552
8-8317609
4-2726586
•012820513
79
62 41
493 089
8-8881944
4-2908404
•012658228
80
64 00
512 000
8-9442719
4-3088695
•012600000
81
65 61
581 441
9-0000000
4-3267487
•012345679
82
67 24
551 868
90563851
4-3444816
•012195122
88
68 89
571 787
91104336
4-3620707
•012048193
84
70 56
592 704
9-1661514
4-3796191
•011904762
86
72 25
614 126
9-2195446
4-3968296
•(U1764706
86
78 96
686 056
9-2786186
4-4140049
•011627907
87
75 69
658 508
9-3273791
4-4310476
•011494253
88
77 44
681472
9-3808316
4-4479602
•011363636
89
79 21
704 969
9-4839811
4-4647451
•011286955
90
8100
729 000
9-4868330
4-4814047
•oiiiuui
91
82 81
768 571
9-6393920
4-4979414
•010989011
92
84 64
778 688
9-6916630
4'6143674
•010869665
98
86 49
804 857
9-6486508
4-6306549
•010762688
94
88 86
880 584
9*6963597
4-6468369
•010638298
SQUARES, CUBES, ROOTS, AND RECIPROCALS. 657
No.
Sqntra
Oabe
Square Root
Cube Root
Redptrocal
95
90 26
857 876
9-7467948
4-6629026
•010526316
96
92 16
884 786
9-7979690
4-6788570
-010416667
97
94 09
912 678
9-8488678
4-5947009
-010309278
98
96 04
941 192
9-8994949
4-6104363
•010204082
99
98 01
970 299
9-9498744
4-6260660
•010101010
100
100 00
1 000 000
10-0000000
4-6415888
•010000000
101
102 01
1 030 801
10-0498766
4-6670095
•009900990
102
104 04
1 061 208
10-0995049
4-6723287
•009803922
108
106 09
1 092 727
10-1488916
4-6876482
•009708738
104
108 16
1 124 864
10-1980390
4-7026(194
-009615385
106
110 25
1 157 625
10-2469608
4-7176940
-009523810
100
1 12 86
1 191 016
10-2966801
4-7326235
•009433962
107
114 49
1 225 048
10-8440804
4-7474694
-009345794
108
116 64
1 259 712
10-3923048
4-7622032
-009269259
109
118 81
1 295 029
10-4403065
4-7768.')62
-009174312
110
12100
1 381 000
10-4880885
4-7914199
•009090909
111
128 21
1 867 681
10-5356638
4-8058955
•009009009
112
125 44
1 404 928
10-5830062
4-8202846
-008928571
118
127 69
1 442 897
10-6301468
4-8345881
•008849558
114
129 96
1 481 544
10-6770783
4-8488076
-008771930
116
1 82 25
1 520 875
10-7238053
4-8629442
-00«695652
116
184 56
1 560 896
10-7703296
4-8769990
•008620690
117
186 89
1 601 618
10-8166538
4-8909732
•008547009
118
189 24
1 648 082
10-8627805
4-9048681
-008474576
119
14161
1 685 159
109087121
4-9186847
-008403361
120
144 00
1728 000
10-9644512
4-9324242
-008333333
121
146 41
1 771 661
11-0000000
4-9460874
•008264463
122
148 84
1 815 848
11-0463610
4-9696757
-008196721
128
15129
1 860 867
11-0905865
4-9731898
-008130081
124
153 76
1 906 624
11-1355287
4-9866310
•008064516
125
156 25
1 958 126
11-1803399
5-0000000
•008000000
126
158 76
2 000 876
11-2249722
50132979
•007936508
127
16129
2 048 888
11-2694277
60265257
•007874016
128
168 84
2 097 152
11-3137085
5-0396842
•007812600
129
166 41
2 146 689
ll-a578167
50527743
•007751938
180
169 00
2 197 000
11-4017543
5-0667970
•007692308
181
17161
2 248 091
11-4455231
50787531
•007633688
182
174 24
2 299 968
11-4891263
60916434
•007676758
188
176 89
2 852 687
11-6325626
6-1044687
-007618797
184
179 56
2 406 104
11-5758369
5-1172299
•007462687
185
182 25
2 460 876
11-6189500
5-1299278
•007407407
186
184 96
2 515 456
11-6619038
51425632
•007352941
187
187 69
2 571 858
11-7046999
5-1551367
•007299270
188
190 44
a 628 072
11-7473401
5-1676493
•007246377
189
198 21
2 685 619
11-7898-261
5-1801016
-007194245
140
196 00
a 744 000
11-8321596
5-19249 U
•007142857
141
198 81
a 808 221
U-8743422
5-2048279
•007092199
142
2 0164
a 868 288
11-9163753
5-2171034
•007042254
148
2 04 49
2 924 207
11-9682607 6-2293215
•006993007
u u
658
SQUAftBS, CUBES, ROOTS, AND REClPHOCAIiS.
Ha
8(|a«rt
Onte
SqtiAreRool
Cube Boot
Beciprocal
144
2 07 80
ft 986 984
12-0000000
5-2414828
'006944444
146
210 25
8 048 620
120415946
5-2635879
•006896552
146
21816
S lis 186
12-0830460
5-2656874
•006849315
147
216 09
8 176 628
12-1243567
5-2776321
•006802721
148
2 19 04
8 241 792
121656261
5-2895726
•006756757
149
2 22 01
•8 807 949
l-2'2065556
5-3014592
•006711409
150
2 25 00
8 875 000
12-2474487
6-3132928
•006666667
151
2 28 01
8 442 951
12-288-2057
5-3250740
•006622517
163
2 8104
8 511808
12-3288280
5-3368033
-006578947
168
2 84 09
8 681 677
12-3693169
6-3484812
-006535948
154
2 87 16
8 652 264
12-4096786
5-3601084
•006493506
155
2 40 25
8 728 875
12-4498996
5-3716854
-006451613
156
2 48 86
8 796 416
12-4899960
6-3832126
-006410256
157
2 46 49
8 869 898
12-6299641
5-3946907
-006369427
168
2 49 64
8 944 812
12-5698051
5-4061202
-006329114
159
2 52 81
4 019 679
12-6095202
5-4175015
-006289308
160
2 56 00
4096 000
12-6491106
5-4288852
-006250000
161
2 59 21
4 178 281
12-6885775
6-4401218
-006211180
162
2 62 44
4 251 528
12-7279221
5-4513618
-006172840
163
2 65 69
4 880 747
12-7671453
6-4625556
-006134969
164
2 68 96
4 410 944
12-806-2485
6-4737037
-006097661
165
2 72 25
4 492 126
12-845-2826
5*4848066
006060606
166
2 75 56
4 574 296
12-8840e87
6-4958647
-006024096
167
2 78 89
4 657 468
12-9228480
5-5068784
-006988024
168
2 82 24
4 741 682
12-9614814
6-6178484
•005952381
169
2 85 61
4 826 809
13-0000000
5-5287748
•0(te917160
170
2 89 00
4 918 000
180384048
5-6396583
-0Q5882368
171
2 92 41
5 000 211
18-0766968
5-5504991
•005847953
179
2 95 84
6 088 448
18-1148770
6-5612978
•005813953
173
2 99 29
6 177 717
18-1529464
5-5720546
-006780347
174
8 02 76
6 2( 8 024
18-1909066
5-5827702
-006747126
175
8 06 25
6 859 875
18-2287566
6-5934447
•005714286
176
8 09 76
6 451 776
13-2664992
5-6040787
-005681818
177
8 18 29
6 545 238
18-3041847
6-6146724
•006649718
178
816 84
6 689 752
18-3416641
5-6252263
-006617978
179
8 20 41
5 785 889
13-3790882
5-6357408
-006586592
180
8 24 00
5 832 OOO
13-4164079
5-6462162
-006555556
181
8 27 61
6 929 741
13-4536240
5-6566528
-005524862
182
8 8124
6 028 568
13-4907876
5-6670511
•006494505
183
8 84 89
6 128 487
13-5277498
6-6774114
•006464481
184
8 38 56
6 229 604
13-5646600
6-6877340
-005434783
185
8 42 25
6 881 62S
18-6014705
66980192
-005405405
186
8 45 96
6 484 856
13-6381817
6-7t)82675
•006376344
187
8 49 69
6 689 208
13-6747943
6-7184791
-005347594
188
8 68 44
6 641 672
13-7113092
5-7286543
•005319149
189
8 57 21
6 751 269
13-7477271
5-7887936
-006291005
190
8 61 00
6 859 000
13-7840488
6-7488971
•005263158
191
8 64 81
6 967 871
13-8202760
5-7589652
•006235602
192
8 68 64
7 077 888
13-8564065
5-7689982
-005208333
SQVABESy OUBES, ROOTS, AND :
REGXPROCAVSU OQ^
K<N
Square
OcdMi
Square Rook
Cnbe Root
' IteciproMi}
198
8 78 491
7 18&Q5T
18$924440
5-7789966
-005181847
194
8 76 86
7 801884
18-9288883
6-7889604
•005154689
19d
8 80 85
7 414 875
18-9642400
5-7988900
00612820$
196
88416
7 689 586
140000000
5-8087867
-006102041
197
888 09
7 645 878
14-0856688
5-8186479
-006076148
198
8 98 04
7 768 898
140712478
5-8284767
•005050505
199
8 96 01
7 880 599
14-1067860
5-8882725
-00602512&
800
4 00 00
frOOOOOQ
141421866
5*8480355
•006000000^
aoi
4 04 01
8180 601
141774469
5-8577660
•004975124
902
4 08 04
8 848 408
14-2126704
6-8«7464d
*00495049ft
808
418 09
8 865 487
14-2478068
5-8771307
*00492610»
804
4 16 16
8 480 664
14-2828569
5-8867658
*004901961
806
480 85
8 615 185
14-8178211
5-8963685
-004878049-
806
484 86
8 741 816
14-8527001
5-9059406
*004854869
807
4 88 49
8 86C 748
14*8874946
5-9154817
-004830918
808
4 88 64
8 996 913
14-4222061
5-9249921
•004807692
809
4 86 81
918I9 888
14-4568828
5-9344721
•004784689
810
441 OO
»8C10Q0
14-4913767
5-H3922a
*004761905
811
4 45 21
» 808 981
14-5258890
5-9533418
*004f739d86
818
4 49 44
» 688 188
14-560'^198
5-9627320r
*004716981
818
4 68 69
»668 97
14-5945195
5-9720926
•004694886
814
4 57 96
9 800 844
14-6287888
5-9814240.
-004672897
815
4 68 25,
9^988^5
14 8628788
5-9907264
*00465116S
8ie
4 66 66
10 077 606
14-8969885
6-0000000
•004629680
817
4 70 89
10 818 818
14-7309199
6-0092450
•004608295
818
4 75 24
10 860888
14-7648281
6-0184617
•004587156
819
4 79 61
10 608 469
14-7986486
6-0276502^
*00456621O
880
4 84 00
10 648 000
14-8323970
6-08t68107
*004545455
881
4 88 41
10 798 861
14-8660687
6^0459485.
*004624887
888
4 98 84
10 941048
14*8996644
6-0550489
*004504505
288
4 97 29
11 089 567
14-9331845
6-0641270
*004484305
1 — ■
884
5 0176
11 889 484
14-9666295
60781779
•004464286
885
6 06 85
11 890 686.
16-0000000
6*0822020.
-004444444
886
5 10 76
11 548 17a
ld-0632964
6-0911994
•004424779
887
516 89t
11 697 088
16-0665192
6-1001702
•004405286
888
519 64
U 868 858
15-0096689
6-1091147
•004885965
QQ»
5 84 41
18 008 989
lfi-1827460
6-11^0332
•004366812
280
5 89 00
18 167 000
15-1657509
6-1269257
•004847826
281
5 88 61
18 886 891
15*1986842
6-1357924
•004829004
282
5 88 84
18 467 168
15^2815462
6*1446387
•004810345
288
5 48 89
18 649 887
15^2643875
6-1584495
•004291845
284
5 47 66
18 813 904
15r2970585
6*1622401
•004273604
285
5 68 85
18 977 876
I6t8297097
6-1710058
•004255819
286
6 66 96
18 144 866
15'3622915
6*1797466
•004237288
287
6 6169
18 813 068
15-3948048
6*1884628
•004219409
288
6 66 44
18 46t 878
15*4272486
6*1971544
1004201681
289
6 7131
18 651 019
15^4596248
6*2058218
•004184100
240
6 76 00
18 884 000
15:4919834
6-2144650
•004166667
241
5^80 81
18.997 681
15*5241747
6*2280843
•00414937g
)60 SQUARES, CUBES, ROOTS, AND RECIPROCALS.
Ho.
Sqaare
Gate
Square Root
Cube Root
Recipix)cal
242
5 85 64
14 172 488
16-5568492
6-2316797
•004182281
248
6 90 49
14 848 907
15-5884578
6-2402615
^04115226
244
6 95 86
14 526 784
15-6204994
6-2487998
-004098861
245
6 00 25
14 706 125
16-6624758
6-2573248
•004081633
246
605 16
14 886 986
15-6848871
6-2658266
•0O4066041
247
6 10 09
15 069 228
15-7162886
6-2743054
-004048583
248
615 04
15 252 992
16-7480167
6-28-27613
-004032258
249
6 20 01
15 488 249
16-7797888
6-2911946
•004016064
250
6 25 00
15 625 000
15-8118883
6-2996053
-004000000
261
6 80 01
15 818 251
15-8429795
6-8079935
•008984064
262
6 85 04
16 008 008
16-8745079
6-8163696
•008968254
258
6 40 09
16 194 277
16-9069737
6-8247035
-008952669
264
6 46 16
16 887 064
15-9378776
6-8380256
•008937008
256
6 60 25
16 581 875
15-9687194
6-8413267
-003921669
260
6 65 86
16 777 216
160000000
6-8496042
-003906250
267
6 60 49
16 974 598
160312195
6-8578611
•003891051
258
6 65 64
17 178 512
16-0628784
6-3660968
•008876969
269
6 70 81
17 878 979
16-0934769
6-8743111
-008861004
260
6 76 00
17 576 000
16-1245166
6-3825043
•008846154
261
6 8121
17 779 581
16-1554944
6-8906765
•003831418
262
6 86 44
17 984 728
16-1864141
6-8988279
•003816794
268
6 9169
18 191 447
16-2172747
6-4069585
-008802281
264
6 96 96
18 899 744
16-2480768
6-4160687
•003787879
265
7 02 25
18 609 625
16-2788206
6-4231583
■003778585
266
7 07 56
18 821 096
16-3095064
6-4812276
-008759398
267
7 12 89
19 084 168
16-3401846
6-4892767
-003745318
268
7 18 24
19 248 882
16-3707056
6-4478057
•008731843
269
7 28 61
19 465 109
16-4012195
6-4653148
•008717472
270
7 29 DO
19 688 000
16-4316767
6-4633041
•003703704
271
7 84 41
19 902 511
16-4620776
6-4712736
-008690037
272
7 89 84
20 123 648
16-4924225
6-4792236
•003676471
278
7 45 29
20 846 417
16-6227116
6-4871541
•003668004
274
7 50 76
20 670 824
16-5529464
6-4950653
•003649635
275
7 56 25
20 796 €75
16-5831240
6-6029672
•003636364
276
7 6176
21024 576
16-6132477
6-6108300
•003623188
277
7 67 29
21253 988
16-6433170
6-5186839
•003610108
J278
7 72 84
21 484 952
16-6733320
6-5265189
•003597122
279
7 78 41
21 717 689
16-7032981
6-5343351
•003684229
280
7 84 10
21 952 000
16-7832005
6-5421326
•003671429
281
7 89 61
22 188 041
16-7680546
6-6499116
•003568719
282
7 95 24
22 425 768
16-7928556
6-5576722
•003646099
288
6 00 89
22 665 187
16-8226088
6*5654144
•003633569
284
8 06 56
22 906 804
16-^22995
6*5731385
-003621127
285
6 12 25
28 149 125
16-8819480
6-5808443
•003508772
286
8 17 96
28 898 666
16-9116846
6-5885323
•003496503
287
8 28 69
28 639 908
16-9410748
6-6962023
-003484321
288
8 29 44
28 887 872
16-9705627
6-6088545
•003472222
289
8 86 21
24 187 569
17-0000000
6-6114890
•003460 08
290
8 4100
24 889 000
17-0293864
, 6-6191060
•003448276
SQUASES^ CUBBS, BOOTS, AND BECIFBOCALS. 661
No.
291
293
293
294
295
296
297
298
299
800
801
802
808
804
805
806
807
808
809
810
811
812
818
814
815
816
817
818
819
820
821
822
828
824
825
826
827
828
829
880
881
882
888
834
885
836
887
888
880
Bqaarft
8 46 81
8 52 64
8 58 49
664 86
8 70 25
6 76 16
8 82 09
8 88 04
8 94 01
900 00
9 06 01
9 12 04
918 09
9 24 16
9 80 26
9 86 86
9 42 49
9 48 64
9 64 81
9 6100
9 67 21
9 78 44
9 79 69
9 65 96
9 92 25
9 98 56
10 04 89
10 11 24
10 17 61
10 24 00
10 80 41
10 86 84
10 48 20
10 49 76
10 66 26
10 62 76
10 69 29
10 75 84
10 82 41
10 89 00
10 95 61
11 02 24
11 08 89
11 15 56
11 22 26
11 28 96
11 85 69
11 42 44
11 49 21
OolM
24 648 171
24 897 088
25 158 757
25 412 184
25 672 876
85 984 886
26 198 078
26 463 592
26 780 899
27 000 000
27 270 901
87 548 608
87 818 127
88 094 464
89 872 626
28 652 616
86 984 448
89 218 112
89 608 629
29 791 000
80 080 231
80 871828
80 664 297
80 959 144
81 255 875
81 564 496
81 865 018
82 157 482
82 461 759
82 768 000
as 076 161
88 886 248
88 698 267
84 012 224
84 828 126
84 645 976
84 965 788
85 287 662
85 611 289
86 987 000
86 264 691
86 694 868
86 926 087
87 259 704
87 595 875
87 983 966
88 272 758
88 614 472
88 958 219
SqnareBoot
17-0587221
17*0880075
17-1172428
17-1464282
171755640
17-2046605
17-2386879
17-2626766
17-2916166
17-3206081
17-3498616
17-3781472
17-4068952
17-4366958
17*4642492
174928557
17-5214165
17'6499288
17-6783958
17-6068169
17-6351921
17-6635217
17-6918060
17-7200461
17-7482393
17-7763888
17-8044938
17-832664d
17-8605711
17-8885438
17-9164729
17-9443584
17-9722008
18-0000000
18-0277564
18-0554701
18-0831413
18-1107703
18-1383571
18*1659021
18*1934054
18*2208672
18*2482876
18*2756669
18-3030052
18-8303028
18-3575598
18-8847763
18-41 1952G
Cube Root
6*6267054
6-6342874
6*6418522
6-6493998
6-6569302
6-6644437
6*6719403
6-6794200
6*6868881
6-6943295
6-7017593
6-7091729
6*7165700
6*7239508
6-7313155
6-7386641
6-7459967
6*7533184
6*7006143
6-7678995
6-7751690
6*78-24229
6*7896613
6*7968844
6-8040921
6*8112847
6*8184020
6-8256242
6*83*27714
6-8399037
6-8470213
6*8541240
6*8612120
6*8082855
6-8753443
0-8823888
0-8894188
0-8904345
0-9034359
0-9104232
6-9173964
6-9243556
0-9313008
6-9382321
6-9451496
6-9520533
6-9589434
6-9658198
6-9726826
Reciprocal
008486426
008424658
003412969
008401861
008389831
003878378
008367003
003355705
003344482
003338333
•003322259
•008311258
003300330
•003289474
•003278689
•003267974
1-006257329
•008246753
-003236246
-008225806
-003215434
-003205128
•003194888
•003184713
•003174603
-003164557
-003154674
-003144654
•003134796
*0U8I25G00
•003115265
•003105590
*003095975
*003080420
•00307C923
•003067485
•003058104
•003048780
•003039514
•003030303
•003021148
-003012048
•003003003
•002994012
•002985075
•002970190
•0029G7359
•002958580
•002949853
662
SQUABBI, OVBSS, BOOCS, ASD SmStPttOOAU.
K^
Sqiiwa
Onto
8q«areB«ol
GabetUxA 1 Bedgnwal
840
118BM
89 804 000
18-4890889
6*9795321
•002941176
841
U6B81
89 661821
18*46611858
6*9868681
•002982651
842
U60 84
40 001688
18*4932420
6*9931966
-002923977
848
11 76 49
40 868 607
18'6202592
7*0000000
•002915462
844
Ut» 86
40 707 684
18*5472870
7-00679«
-002906977
845
U90 25
41068 625
18*5741756
7-0185791
•002898551
846
1197 16
41421786
18*6010752
70203490
•002890178
847
12 04 09
41781928
18*6279860
7*0271058
•002881844
848
1211 04
42 144 192
18*6547681
7-6888497
•002873563
849
12 18 01
42 608 549
18*6816147
7-0405806
■OO2866330
850
12 25 00
42 875 000
18*7082869
7-0472987
•002857143
851
12 82 01
43 248 561
18*7349940
70540041
•002849003
852
12 89 04
48 614 208
18*7616680
7*6606967
002840909
858
12 46 09
48 986 977
18*7882942
7-0678767
•062832861
854
12 68 16
44 861864
18-8148877
7-0740440
•002824859
856
12 60 25
44 7B8 875
18^14487
7-0806988
•002816901
856
12 67 86
46 118 016
18«679e28
7-6878411
•002808989
867
12 74 49
46 499 298
18-8944186
70939709
•002801120
868
18 8184
46 882 712
18*9208879
7^1006886
•002793296
859
18 88 81
46 268 279
18*9478958
7^1071937
•002785515
860
12 96 60
46 666 000
18-9736660
7*1137806
•002777778
861
18 08^
47 045 881
19'0000000
7-1803674
002770088
862
18 10 44
47 487 928
190262976
7'1269d60
002762431
868
18 17 69
47 882 147
190526569
7*1834925
•002754821
864
18 24 96
48 228 644
190787840
7*1400370
-002747253
865
18 82 26
48627 125
19*1049782
7*1465695
•002789726
866
18 89 56
49 027 896
19*1311266
7*1630901
-002732240
867
18 46 89
49 430 868
19*1572441
7-1695988
•002724796
868
18 54 24
49 886 082
19-1888261
7*1660957
-002717391
869
18 61 61
50 248 409
19*2098727
7-1725809
*002710027
870
18 69 00
50 658 000
19*2368841
7*1790544
•002702703
871
18 76 41
51 064 811
19*2618608
7*1855162
•002695418
872
18 8$ 84
51 478 848
19*2878015
7*1919668
•0O2688I72
878
18 91 29
51 895 117
19*3182079
7*1984050
-002680965
874
13 98 76
62 813 624
19*3390796
7*2048822
•002673797
876
14 06 25
52 784 876
19*8649167
7*2112479
-002666667
876
14 1$ 76
68 167 876
19«907194
7*2176522
•002659574
877
14 21 20
63 582 688
194164878
7*2240460
*OO2652520
878
14 28 84
54 010 162
19-4422221
7*2304268
002645503
879
14 86 41
64 489 980
19*4679223
7*2867972
•002688523
860
14 44 00
64 872 000
19*4935887
7*2431565
•002681579
881
14 51 61
56 806 841
19-5192213
7*2495045
-002624678
882
14 59 24
65 742 968
19*5448203
7*2658415
•002617801
888
14 66 8d
66 181 887
19*5708858
7*2621675
O0261096S
884
14 74 56
66 623 104
19*6959179
7*2684824
•002604167
886
14 82 25
67 066 626
19*6214169
7*2747864
002597403
866
14 89 98
67 512 466
19*6468827
7*2810794
002590674
887
14 97 69
67 960 608
19-6726156
7*2878617
•002683971
888
15 06 44
58 411 072
19-6977166
7*2936830
O02577320
SQITABES) GUBBS, ROOTS, AND RECIPROCALS.
66^
No.
Square
Oabe
Square Root
Cube Root
Reciprocal
889
15 18 21
58 668 869
197280829
7-2998936
-002670694
890
15 21 00
59 819 000
19-7484177
7-3061436
-002664103
891
15 28 81
69 776 471
19-7737199
7-3123828
-002557646
892
15 86 64
60 236 288
19-7989899
7-3186114
-002651020
898
15 44 49
60 698 457
19-8242276
7-3248295
•002544629
894
15 52 86
61 162 964
19-8494332
7-8310369
•002538071
895
15 60 25
61 629 875
19-8746069
7-3372339
•002531646
396
15 68 16
62 099 186
19-8997487
7-3434205
•002525253
897
15 76 09
62 570 778
19-9248588
7-3495966
•002518892
898
15 84 04
68044 792
19-9499373
7-3557624
-002512663
899
15 92 01
68 521 199
19-9749844
7-3619178
-002606266
400
16 00 00
64i»00 000
20-0000000
7-3680630
-002600000
401
16 08 01
64 481 201
20-0249844
7-3741979
-002493766
402
16 16 04
64 964 806
20-0499377
7-3803227
•002487662
408
16 24 09
65 450 827
20-0748599
7-3864373
•002481890
404
16 82 16
65 989 264
200997512
7-3925418
•002476248
405
16 40 25
66 430 125
20-1246118
7-3986868
-002469186
406
16 48 86
66 923 416
20-1494417
7-4047206
•002463054
407
16 56 40
67 419 148
20-1742410
7-4107950
-002457002
408
16 64 64
67 917 812
20-1990099
7-4168696
-002450980
409
16 72 81
68 417 929
20-2237484
7-4229142
•002444988
410
16 8100
68 921000
20-2484567
7-4289589
-002439024
411
16 89 21
69 426 581
20-2781349
7-4349938
■002433090
412
16 97 44
69 934 528
20-2977831
7-4410189
■002427184
418
17 05 69
70 444 997
203224014
7-4470342
•002421808
414
17 13 90
70 957 944
20-8469899
7-4530399
-002415459
415
17 22 25
71 478 875
20-8716488
7-4690859
■002409639
416
17 80 56
71 991 296
20-8960781
7-4650223
-002403846
417
17 88 89
72 611 718
20-4205779
7-4709991
-002398082
418
17 47 24
78 034 682
20-4450483
7-4769664
•002392844
419
17 56 61
78 560 059
20-4694896
7-4829242
-002386635
420
17 64 00
74 088 000
20-4939016
7-48887-24
■002380952
421
17 72 41
74 618 461
20-6182846
7-4948113
•002375297
422
17 80 84
75 151 448
20-6426386
7-5007406
-002369668
428
17 89 29
75 686 967
20-6669638
7-5066607
•002364066
424
17 97 76
76 225 024
20-6912603
7-5125715
■002368491
425
18 06 26
76 765 625
20-6155281
7-5184730
■002362941
426
18 14 76
77 808 776
20-6397674
7-5243662
-002347418
427
18 28 29
77 854 483
20-6639783
7-5302482
-002341920
428
18 81 84
78 402 752
20-6881609
7-5361221
•002336449
429
18 40 41
78 958 689
20-7123162
7-5419867
-002331002
480
18 49 00
79 507 000
20-7364414
7-5478423
•002325581
481
18 57 61
80 062 991
20»7605395
7-5536888
•0023201 80
482
18 66 24
80 621 568
20-7846097
7-5596263
-002314815
488
18 74 89
81 182 787
20-8086520
7-5663548
-002309469
484
18 88 56
81 746 604
20-83266G7
7-5711743
-002304147
435
18 92 25
82 312 875
20-8566536
7-5769849
-00229P8 1
486
19 00 96
82 881 856
20-8806130
7-5827866
•002293578
487
19 09 69
88 458 458
20-9045450
7-6886793
-002288330
364 SQUARES, CUBES, ROOTS, AND RECIPBOCALS.
No.
Bqaare
Cube
Squftre Boot
GabeBoot
Bedprocal
488
19 18 44
84 027 672
20-9284495
7-5943633
•002283105
439
19 27 21
84 604 519
20-9523268
7-6001385
•002277904
440
19 86 00
85 184 000
20-9761770
7-6059049
•002272727
441
19 44 81
85 766 121
21-0000000
7-6116626
•002267674
442
19 53 64
86 850 888
21-0237960
7-6174116
■002262443
443
19 62 49
86 988 807
21-0476652
7-6231519
•002257386
444
19 71 86
87 528 884
21-0713075
7-6288837
-002252252
445
19 80 26
88 121 126
210950231
7-6846067
•002247191
446
10 89 16
88 716 686
211187121
7-6403213
•002242152
447
19 98 09
89 814 628
21-1423745
7-6460272
•002237136
448
20 07 74
89 916 892
21-1660105
7-6617247
•002232143
449
20 10 01
90 618 849
21-1896201
7-6674138
•002227171
460
20 25 00
91 126 000
21-2132034
7-6680943
•0022222-22
451
20 84 01
91 788 861
21-8367606
7-6687666
•002217296
452
20 48 04
92 846 408
21-2602916
7-6744303
•002212889
458
20 52 09
92 969 677
21-2837967
7-6800867
•002207506
454
20 61 16
98 676 664
21-3072758
7-6857828
•002202643
455
20 70 26
94 196 875
21-3307290
7-6913717
•002197802
456
20 79 86
94 818 816
21-8541666
7-6970023
-002192982
457
20 88 49
96 448 998
21-3776683
7-7026246
•002188184
458
20 97 64
96 071 012
21-4009346
7-7082388
-002183406
459
21 03 81
96 702 679
21-4242853
7-7138448
•002178649
460
2116 00
97 886 000
21-4476106
7-7194426
•002173913
461
21 25 21
97 972 181
21-4709106
7-7260326
•002169197
402
21 84 44
98 611 128
21-4941853
7-7806141
•002164502
468
21 43 60
99 252 847
21-6174848
7-7361877
•002159827
464
21 C2 96
99 897 844
21-6406592
7-7417532
•002155172
465
21 62 25
100 644 625
21-6638587
7-7473109
-002150538
4G6
21 71 56
101 194 696
21-6870381
7-7628606
•0021459-23
467
21 80 89
101 847 568
21-61018-28
7-7684023
-002141828
468
2ir0 24
102 503 232
21-6333077
7-7639361
•002136752
469
21 99 61
103 161 709
21-6564078
7-7694620
•002132196
470
22 09 00
103 823 000
21-6794884
7-7749801
•002127660
471
22 18 41
104 487 111
21-7026344
7-7804904
•002123142
472
22 27 84
105 154 048
21-7256610
7-7859928
•002118644
478
22 87 29
105 823 817
21-7485682
7-7914875
•002114165
474
22 46 76
106 496 424
21-7715411
7-7969745
•002109705
475
22 56 26
107 171 876
21-7944947
7-8024538
•002105263
470
22 65 76
107 850 176
21-8174242
7-8079254
•002100840
477
22 76 29
108 581 888
21-8403297
7-8133892
•002096436
478
22 84 84
109 215 852
21-8632111
7-8188466
•002092050
479
25 94 41
109 902 289
21-8860686
7-8242942
-002087683
480
28 04 00
110 592 000
21-9089023
7-8297353
•002083838
481
28 18 61
111 284 641
21-9317122
7-8351688
-002079002
482
28 28 24
111 980 168
21-9544984
7-8405949
•002074689
483
23 82 89
112 678 587
21-9772610
7-8460184
-002070393
i84
23 42 66
118 879 904
22-0000000
7-8514244
•002066116
485
23 52 25
114 084 125
22-0227155
7*8568281
002061856
486
28 61 96
114 791 256
22-0454077
7-8622242
002057613
SQUARES, CUBES, ROOTS, AND RECIPROCALS. 665
SqtiAre
23 71 69
23 81 44
23 91 21
24 01 00
24 10 81
24 20 64
24 80 49
24 40 36
24 50 25
24 6016
24 70 09
24 80 04
24 90 01
25 00 00
25 10 01
25 20 04
25 80 09
25 4016
25 50 25
25 60 86
25 70 49
25 80 64
25 9081
26 0100
26 11 21
26 21 44
28 31 69
26 4196
26 52 25
26 62 56
26 72 89
26 83 24
26 93 61
27 04 00
27 14 41
27 24 84
27 85 29
27 45 76
27 56 26
27 66 76
27 77 29
27 87 84
27 98 41
28 09 00
28 19 61
28 80 24
28 40 89
28 51 56
28 62 26
Cube
116 601 308
116 214 272
116 930 169
117 649 000
118 370 771
119 095 488
119 823 157
120 653 784
121 287 375
122 023 936
122 768 473
128 505 992
124 251 499
125 000 000
125 751 501
126 506 0O8
127 268 527
128 024 064
128 787 625
129 554 216
130 828 843
131 096 512
131 872 229
182 651 000
133 432 881
134 217 728
185 005 697
186 796 744
136 590 875
187 888 096
188 188 418
138 991 882
139 798 859
140 608 000
141 420 761
142 236 648
143 055 667
148 877 824
144 708 126
145 531 576
146 863 188
147 197 952
148 035 889
148 877 000
149 721 291
150 668 768
151 419 487
152 273 304
163 130 375
Square Itoot
22-0680765
22-0907220
221133444
22-1359436
22-1585198
221810730
22*2036033
22-2261108
22-2485955
22-2710575
22-2934968
22-3159136
22-3383079
22-3606798
22-3830293
22-4053565
22-4276615
22-4499443
22-4722051
22-4944438
22-5166605
22-5388563
22-5610283
22-5831796
22-6053091
22-6274170
22-6495038
22-6715681
22-6936114
22-7156334
22-7376340
22-7596134
22-7815716
22-8035085
22-8264244
22-8473193
22-8691933
22-8910463
22-9128785
22-9846899
22-9564806
22-9782506
23-0000000
23-0217289
23-0434372
23-0651252
23-0867928
231084400
23-1300670
Cube Root
7-8676130
7-8729944
7-8783684
7-8837352
7-8890946
7-8944468
7-8997917
7-9051294
7-9104599
7-9157832
7-9210994
7-9264086
7-9317104
7-9370053
79422931
7-9475739
7-9528477
7-9681144
7-9633743
7-9686271
7-9738731
7-9791122
7-9843444
7-9895697
7-9947883
8-0000000
80052049
8-0104032
8-0155946
8-0207794
8-0259574
8-0311287
80362935
8-0414515
8-0466080
8-0517479
8-0568862
8-0620180
8-0671432
8-0722620
8-0773743
8082480O
8-0875794
8-0926723
8-0977589
8-1028390
8-1079128
8-1129803
8-1180414
Reciprocal
•002053388
•002049180
•002044990
-002040816
-002036660
-002032520
•002028398
•002024291
•002020202
-002016129
-002012072
-002008032
-002004008
-002000000
•001996008
■001992032
•001988072
-001984127
•001980198
-001976285
•001972387
•001968504
•001964637
•001960784
-001956947
•001953125
•001949318
•001945525
•001941748
•001937984
•001934286
•001930602
-001926782
-001923077
-001919386
•001915709
•001912046
-001908397
•001904762
•001901141
•001897633
•001893939
•001890359
•0"1886792
•001883239
•001879699
•001876173
•001872659
•001869159
16
6QUABE8, CUBES, ROOTS, AND RSCIPBOGALS.
.^0.
Bqaare
Oube
SqaafeBoot
Cube Boot
Beciprocal 1
»86
■
28 72 06
158 990 656
23-151 6788
8-1280962
•001865672
►37
28 88 69
154 854 153
23-1 782605
81281447
•001862197
i86
28 94 44
155 720 872
23-1948270
8-1331870
•001858736
>d9
29 05 21
156 590 819
23-2163735
8-1382230
•001855288
)40
29 16 00
167 464 000
23-2379001
81432529
•001851852
Ul
29 26 81
158 840 421
23-2594067
8-1482765
•001848429
>42
29 87 64
169 220 088
23-2808935
8-1532939
•001845018
>48
29 48 49
160108 007
23-3023604
8-1583051
•001841621
Hi
29 59 86
160 989 184
23-3238076
8-1683102
•001888235
>46
29 70 25
161 878 626
23-3452351
8-1683092
•001834862
)46
29 81 16
162 771 886
23-3666429
81783020
•001881502
»47
29 02 09
163 667 828
23-3880311
8-1782888
•001828154
>48
80 03 04
164 566 692
28-4003998
81832695
•001824818
>49
80 14 01
165 469 149
23-4807490
8-1882441
•001821494
>50
80 25 00
166 876 000
23-4620788
8-1982127
•001818182
>51
80 86 01
167 284 151
23-4733892
8-1981753
-001814882
>5a
80 47 04
168 196 606
23-4946802
8-2031319
•001811594
V53
80 58 09
169 112 877
23-5159520
8-2080825
•001808818
S54
80 69 16
170 081 464
23-5872046
«-2130271
H)01805054
>55
80 80 25
170 953 875
23-5584380
8-2179657
•001801802
$66
80 0186
171 879 616
23-5796522
8-2228985
•001798561
557
81 02 49
172 808 098
23-6008474
8-2278254
•001795332
558
81 13 64
178 741 112
23-6220236
8-2327463
•001792115
»59
81 24 81
174 676 879
23-6431808
8-2376614
-001788909
560
8186 00
175 616 000
23-6643191
8-2425706
•001786714
561
81 47 21
176 658 481
23-6854386
8-2474740
•0017S2631
SC2
81 58 44
177 604 828
28-7065392
8-2523715
•001779359
563
81 69 69
178 453 647
23-7276210
82572683
•001776199
S64
8180 96
179 406 144
23-7486842
8-2621492
•001773050
S65
3192 25
180 802 125
23-7697286
8-2670294
•001769912
S66
82 03 56
181 821 496
23-7907.545
8-2719039
•0017667W
567
82 14 89
182 284 268
28-8117618
8-2767726
•001763668
568
82 26 24
183 260 432
23-8327506
8-2816355
•001760563
569
82 87 61
184 220 009
28-8537209
8-2864928
•001757469
570
82 49 00
185 198 000
23-8746728
8-2913444
•001754386
571
82 60 41
186 160 411
28-8956063
8-2961903
•001751313
572
82 71 84
187 149 248
28-9165215
8-3010304
•001748252
573
82 83 29
188 132 617
23-9374184
8-3058651
•001745201
574
82 94 76
189 119 224
23-9582971
8-3106941
•001742160
576
88 06 25
190 109 876
23-9791576
8-3155175
•001 739130
576
88 17 76
191 102 976
24-0000000
8-3203353
•001736111
577
88 29 29
192100 088
24-0208243
8-3251475
•001733102
578
83 40 84
198 100 562
24-0416306
8-3299542
•001730104
579
83 62 41
194 104 589
24-0624188
8-3347553
•001727116
580
83 64 00
195 112 000
24-0831891
8-3395509
•001724138
581
83 75 61
196 122 941
24-1039416
8-3443410
•001721170
582
88 87 24
197 187 868
24-1246762
8-3491256
•001718213
588
88 96 89
196 165 287
24-1453929
8-3539047
•001715266
584
84 10 56
199 176 704
24-1660919
8-3586784
•001712829
8QUAKES, GUBBS, ROOTS, AND EECIPROGALS. 667
fiqoare
^85 I
686
687
688
689
690
691
699
698
694
696
696
697
698
609
600
601
609
608
604
605
606
607
608
609
610
611
613
613
614
615
616
617
618
619
620
621
622
628
624
626
626
627
628
629
680
681
632
638
Ji
84 23 26
84 83 96
84 45 69
84 57 44
84 69 21
84 8100
84 92 81
86 04 64
86 16 49
86 28 86
86 40 25
86 52 16
8^ 64 09
85 76 04
85 88 01
86 00 00
86 12 01
86 24 04
86 86 09
86 48 16
86 60 26
86 7-2 86
86 84 49
86 96 64
87 08 81
87 21 00
87 88 21
87 45 44
87 57 69
87 69 06
87 82 25
87 94 66
38 06 89
38 19 24
88 81 61
88 44 00
88 66 41
88 68 84
88 8129
88 93 76
89 06 25
89 18 76
89 81 29
89 48 84
89 56 41
89 69 00
89 8161
89 94 24
40 06 89
Cube
300 201
301230
802 362
303 397
204 886
306 879
206 426
307 474
308 627
209 684
210 644
311 708
313 776
318847
314 921
216 000
317 081
218 167
219 266
220 848
221446
322 646
328 648
324 756
226 866
826 981
228 099
229 220
830 846
831 476
332 608
288 744
234 886
236 029
237 176
238 828
239 488
340 641
341 804
242 970
244 140
246 814
246 491
347 678
248 858
250 047
251289
252 486
253 686
686
066
008
472
469
000
071
688
8^7
684
876
796
1^
192
700
000
801
208
227
864
125
016
548
712
589
000
1^1
9i28
897
644
876
8^6
118
032
659
000
061
848
867
624
686
376
888
162
189
000
501
068
187
ScpnzeBoot
24»18677B2
24-2074369
24-2280829
24'2487113
24-2693222
24'2899166
24-3104916
24-8310501
24-3515918
24-3721162
24-8926218
24-4131112
24-4836834
24-4540385
24-4744765
24^948074
24*5158018
24*5356888
24*5560583
24-5764115
24-6967478
24-6170673
24-6378700
24-6576560
24-6779254
24-6981781
24-7184142
24-7386338
24*7588368
24-7790234
24-7991936
24-8198473
24-8894847
24-8596058
24-8797106
24-8907992
24-0108716
24-9800278
24-9500679
24-0700020
25-0000000
25-0109020
25-0300681
25-0500282
25-0708724
25-0098008
25-1107184
25-1306102
25-1594018
Oube Root
8-8684466
8-8682005
8-3720668
8-8777188
8-8824653
8-8872065
8-3910423
8-8066720
8-4013081
8-4061180
8-4108326
8-4155410
8-42(^460
8-4240448
8-4296383
8-4348267
8-480000d
8-4436877
8-4488605
8-4530281
8-4576906
8*4628470
8-4670000
8-4716471
8-4762892
8-4800261
8-4855570
8-4001848
8-4048065
8-4004233
8-5040350
8-9086417
8-5132435
8-5178403
8-5224321
8-5270180
8-5316000
^5361780
8-5407601
8-5453173
8-O408707
8-5544372
8-5680800
8-5635377
8-5680807
8-5726180
8-57715^3
8-5816800
8*5862047
Reciprooal
L
•001700402
-001706485
^01703578
•006700680
•001697703
•001604015
•001692047
•001680180
•001686841
•001683502
•001680672
•001677852
•001675042
^1672241
•001669440
-001666667
•001668804
•001661130
•001658375
•001655620
•001652803
•001650165
-001647446
•001644737
•001642036
•001630344
•001636661
•001633087
•001631321
-001628664
•001626016
-001623377
-001620746
•001618123
•001615500
-001612903
•001610306
•001607717
•001605136
•001602564
•001600000
•001597444
•001594806
•001592367
•001689825
•001687302
•001584786
•001582278
001579779
168
SQUARES) CUBES, ROOTS, AND RBCIPBOCAL8.
Square
40 19 56
40 82 25
40 44 96
40 57 69
40 70 44
40 88 21
40 96 00
41 08 61
41 21 64
41 84 49
41 47 86
4160 25
41 78 16
41 86 09
41 99 04
42 12 01
42 25 00
42 86 01
42 5104
42 64 09
42 77 16
42 90 25
43 08 86
43 16 49
48 29 64
48 42 81
48 66 00
48 69 21
48 82 44
43 95 60
44 06 06
44 22 25
44 86 56
44 46 89
44 62 24
44 76 61
44 89 00
45 02 41
45 15 84
45 29 29
45 42 76
45 66 25
45 69 76
45 88 29
45 96 84
46 10 41
46 24 00
46 87 61
46 51 24
Onbe
864 840 104
256 047 876
257 259 466
258 474 868
259 694 072
260 917 119
262 144 000
268 874 721
264 609 288
266 847 707
267 089 984
268 886 126
269 (86 186
270 840 088
272 097 702
278 859 449
274 626 000
275 894 461
277 167 808
278 445 077
879 726 264
881 Oil 876
282 900 416
288 698 898
284 890 812
286 191 179
287 496 000
288 804 781
290 117 588
291 484 247
292 754 944
294 079 625
295 408 296
296 740 968
298 077 688
299 418 809
800 768 000
802 111 711
808 464 448
804 821 217
806 182 024
807 546 876
808 916 776
810 288 783
811 665 758
818 046 889
814 482 000
815 821 241
817 214 668
Bquftre Boot
Cube Boot
Bedprocal
1
25-1793566
25*1992063
25-2190404
25-2388589
26-2586619
25-2784493
25-2982213
25-8179778
25-3377189
25-3574447
25-3771551
25-3968502
25-4165301
25-4361947
25-4558441
25-4754784
25-4950976
26-5147016
26-5342907
25-5538647
25-5734237
26-5929678
25-6124969
25-6320112
25-6515107
25-6709953
25-6904652
25-7099203
25-7293607
25-7487864
25-7681975
25-7876939
25-8069758
25-8268431
25-8456960
25-8650343
25-8848582
25-9036677
25-9229628
25-9422435
26-9616100
25-9807621
26-0000000
26-0192237
26-0384831
26-0576284
28-0768096
26-0959767
26-1151297
8-6907-238
8-6952380
8-5997476
8'6042526
8-6087526
8-6132480
8-6177388
8-6222248
8-6267063
8*6311830
8-6356561
8*6401226
8-6445865
8-6490437
8*6534974
8-6679465
8-6623911
8-6668310
8-6712665
8-6766974
8-6801237
8-6846466
8-6889630
8*6933769
8*6977843
8*7021882
8*7066877
8*7109827
8*7153734
8*7197596
8*7241414
8*7285187
8*7328918
8*7372604
8*7416246
8*7469846
8*7503401
8-7646913
8*7690383
8-7633809
8*7677192
8*7720632
8*7763830
8*7807084
8*7850296
8*7893466
8*7936593
8*7979679
8*8022721
•001577287
-001574803
•001572327
•001569859
•001567398
•001564940
•001562500
•001560062
•001557632
•OO1555210
•001552795
•001550388
•001547988
•001545595
•001543210
•001540832
•001538462
•001536098
•001533742
•001531394
•001529062
•001526718
•001524390
•001522070
•001519767
•001517461
•001515152
•001512859
•001510674
•001608296
-001506024
•001603759
-001501602
•OO1499250
•001497006
-001494768
•001492537
-001490318
-001488095
•001485884
-001483680
•001481481
-001479290
•OO1477105
•001474926
•001472754
-001470588
•001468429
•0014662
!l
SQUABBS, CUB£8, ROOTS, AND BECIPROCALS.
OGf
Sqnaft
46 64 89
46 W S6
46 92 25
47 06 96
47 19 69
47 88 44
47 47 21
47 6100
47 74 81
47 88 64
48 02 49
4816 86
48 80 25
48 44 16
48 68 09
48 72 04
48 86 01
49 00 00
49 14 01
49 28 04
40 42 09
49 66 16
49 70 25
49 84 86
49 98 49
60 12 64
50 26 81
50 4100
50 65 21
50 60 44
50 88 69
50 97 96
61 12 25
51 26 66
51 40 89
61 56 34
51 69 61
61 84 00
51 98 41
62 12 84
62 27 29
52 41 76
52 66 26
52 70 76
52 85 29
62 99 84
58 14 41
53 29 00
68 48 61
CulM
818 611 987
820 013 604
821 419 126
822 828 856
824 242 708
825 660 672
827 082 769
828 509 000
829 989 871
881 878 888
882 812 557
834 255 884
836 702 876
887 158 586
888 608 878
840 068 892
841 632 090
848 000 000
844 472 101
845 948 408
847 428 927
848 918 664
850 402 625
851 895 816
858 898 248
854 894 912
856 400 829
957 911 000
859 425 481
860 944 128
862 467 097
868 994 844
865 525 875
867 061 696
868 601 813
870 146 282
871 694 959
878 248 000
874 805 861
876 867 048
877 988 067
870 508 424
881 078 125
882 657 176
884 240 688
886 828 862
887 420 489
889 017 000
890 617 891
Square Hoot
26-1342687
26'1633937
26-1725047
26-1 916017
26-2106848
26-2297541
26-2488096
26-2678511
26-2868789
26-8068929
26-8248932
26-3438797
26-3628627
26-3818119
26-4007676
26-4196896
26-4386081
26-4576131
26-4764046
26-4952826
26-6141472
26-5329988
26-5618361
26-5706605
26*5894716
26-6082694
26-6270539
26-6458252
26-6645833
26-6833281
26-7020598
26-7207784
26-7394839
26-7581763
26*7768657
26-7955220
26-8141754
26*8328157
26*8514432
26-8700577
26*8886593
26*9072481
26*9258240
26*9443872
26-9629375
26-98U751
27-0000000
27-0185122
27*0370117
Cube Root
8-8066722
8-8108681
8*8161598
8*8194474
8*8237307
8*8280099
8*8822860
8-8365559
8-8408227
8-8450854
8-8498440
8*8535985
8-8578489
8*8620952
8*8663375
8-8705757
8-8748099
8-8790400
8-8832661
8-8874882
8-8917063
8-8959204
8-9001304
8-9043366
8-9085387
8*9127369
»*9169311
8-9211214
8*9253078
8-9294902
8-9336687
8-9378438
8*9420140
8*9461809
8-9503438
8-9545029
8*9586581
8-9628095
8-9669570
8-9711007
8-9752406
8-9793766
8-9835089
8-9876373
8-9917620
8-9958829
9-0000000
9*0041134
9*0082229
Reciprocal
•001464129
-001461988
-001469854
•001457726
•001455604
-001453488
•001451379
•001449276
•001447178
•001445087
•001443001
•001440922
•001438849
-001436782
•001434720
-001432665
*001430615
•001428571
•001426534
•001424501
•001422476
•001420455
•001418440
•001416481
•001414427
•001412429
*001410437
•001408451
•001406470
•0014Q4494
•001402525
•001400560
•001398601
•001896648
•001394700
-001392758
•001390821
•001888889
•001386963
•001385042
•001383126
-001381215
-001879310
-001877410
•001375516
•001378626
•001371742
-001369863
•001367989
ro
SQUARBS, 0UUX8, BOOIfi, AKD BECI^BOCALS.
N©.
Sviaite
Qttbe
Mawr^Root
Cube Root
ftecipiooftl
782
68 56 24
892 228 1<(8
27-0554985
9<»23288
H)01866120
7d8
58 72 69
898 882 887
27-0789727
9^X164309
•001864266
784
68 87 66
896 446 9Q4
27-0924344
9-0205293
-001362898
786
68 02 26
897 065 875
27-1108834
9-0246239
•001360544
786
54 16 06
898 688 266
27-1293199
9H)287149
-001358696
787
64 81 69
400 815 5£t8
27-1477439
9-08280^1
•001856852
788
64 46 44
461 947 272
27-1661664
9-0868857
•00185$014
789
64 6121
408 688 419
27-1845644
9-0409655
-00185?180
740
64 76 00
406 284 000
27-2029410
9H)4§0417
•001351361
741
64 96 81
406 869 021
27-22iai52
90491142
•001849528
742
66 05 64
408 618 488
27-2890769
9^631831
•001847709
748
66 26 49
410 172 407
27-2580263
9-0672482
O01846895
744
66 86 96
411 860 784
27-2763634
9-0618098
<M>1S44086
745
65 50 25
418 498 626
27-2946881
9^668677
•001342282
746
65 65 16
416 160 986
27-8180006
9H)694220
-001840483
747
66 80 99
416 882 728
27-3313007
9-0784726
•001338688
748
56 96 04
418 508 992
27-8495887
9-0775197
•001336898
749
6610 01
420 189 740
27 8678644
9-0815631
•001335113
760
66 26 00
421 875 000
27-3861279
9-0866030
•001833383
751
56 40 01
428 664 761
27-4048792
9*0896392
•001831558
752
66 55 04
425 269 006
27-4226184
9-0936719
-001829787
763
56 70 09
426 967 777
27-440^66
9-0977010
-001828021
754
66 86 16
428 661 064
27-4690604
9-1017266
•001326260
755
67 00 25
480 868 876
27-4772639
9-1057485
•001824608
756
67 16 86
482 081 21«
27-4964642
9-1097669
-001822751
757
67 80 49
488 798 098
27-6186330
9-1137818
•001321004
758
57 46 64
485 649 512
27-6317998
9-1177931
-001319261
759
67 60 81
487 245 479
27-6499546
9-1218010
•001317523
760
57 76 00
488 976 000
27-6680975
9-1258053
-001315789
761
57 91 1)1
440 711 081
27-6862284
9-1298061
•001314060
762
68 06 44
442 460 726
27-6043476
9-1888034
•001812336
763
68 21 Q9
444 194 947
27-6224646
9-1877971
•001310616
764
68 86 96
445 948 744
27-6406499
9-1417874
-001308901
765
68 51^25
447 697 126
.27-6686834
9-1467742
•001307190
766
68 67-^
449 465 096
27-6767060
9-1497676
•001305483
767
68 82 39
461 217 668
27-6947648
9-1637876
•001^3781
768
6a 98 24
462 984 882
27-7128129
9-1577189
•00 302083
769
6918 61
464 756 609
27-7808492
9-1616869
H)0ta00390
770
6929^00
466 688 000
27-7488739
9-1656566
•001298701
771
69 44 41
468 814 011
27-7668868
9-1696226
•001297017
772
69 69B4
460 090 648
27-7848880
9-1735862
-001295387
773
69 76 29
461 889 917
27*8628775
9-1775445
-001293661
774
59 9076
468 684 824
27-8208555
9-1815003
•001291990
775
60 00 25
465 484 876
27-8888218
9-1854627
•001290323
776
60 2176
467 288 676
27-866776^
9-1894018
•001288660
777
60 87 29
469 097 488
27-8747197
9-1933474
-001287001
778
60 62 84
470 910 969
27-8926614
9-1972897
•001285847
779
60 68 41
47^ 729 189
27-9106715
9-2012286
-001283697
780
6a 84 00
474 552 000
27-9284801
9-2051641
-001282051
SQVARiaS, CUBES, ROOTS, AND BEOIPROOALS. 671
No.
Square
Oabt
Square Boot
Cube Boot
Beclprocal
781
60 99 61
476 379 541
27-94e3772
9-2090962
-001280410
782
61 15 24
478 211 768
27-9642629
9-2130260
-001278772
783
61 80 89
480 048 687
27-9821372
9-2169506
-001277139
784
61 46 56
481 890 804
28-0000000
9-2208726
-001276510
785
61 62 25
488 786 625
28-0178515
9-2247914
-001273885
786
61 Y7 96
485 587 656
28-0356915
9-2287068
-001272266
787
61 98 69
487 448 408
28-0536203
9-2326189
-001270648
788
62 09 44
489 808 872
28-0713377
9-2365277
-001269036
789
62 25 21
491 169 069
28-0891438
9-2404333
•001267427
790
62 41 00
498 039 000
28-1069386
9-2443366
•001266823
791
62 56 81
494 913 671
28-1247222
9-2482344
•001264223
792
62 72 64
496 798 088
28-1424946
9-2621300
•001262626
798
62 88 49
498 677 257
28-1602567
9-2660224
•001261034
794
63 04 86
500 566 184
28-1780056
9-2699114
•001269446
795
63 20 25
502 459 875
28-1967444
9-2637973
•001267862
796
63 86 16
504 858 886
28-2134720
9-2676798
•001266281
797
68 52 09
506 261 678
28-2311884
9-2715692
•001264705
798
63 63 04
508 169 692
28-2488988
9-2764362
•001263133
799
68 84 01
510 082 899
28-2665881
9-2793081
•001261564
800
64 00 00
512 000 000
28-2842712
9-2831777
-001260000
801
64 16 01
618 922 401
28-3019434
9-2870440
•001248439
802
64 82 04
615 849 608
28-3196045
9-2909072
-001246883
803
64 48 09
617 781 627
28-3372646
9-2947671
•001245330
804
64 64 16
519 718 464
28-3648938
9-2986289
•001243781
805
64 80 25
521 660 125
28-3725219
9-3024775
•001242236
800
64 96 86
528 606 616
28-3901391
9-3063278
•001240695
807
65 12 49
625 557 948
28-4077464
9-3101760
•001239157
808
65 28 64
527 514 112
28-4268408
9-3140190
•001237624
809
65 44 81
629 475 129
28-4429253
9-3178699
•001236094
810
65 6100
681 441 000
28-4604989
9-3216975
•001234668
8n
65 77 21
588 411 781
28-4780617
9-3265320
•001233046
812
65 98 44
585 887 828
28-4966137
9-3293634
•001231627
813
66 09 69
687 867 797
28-513^549
9-3331916
•001230012
814
66 25 96
689 858 144
28-6306862
9-3370167
-001228501
815
66 42 25
541 843 875
28-6482048
9-3408386
-001226994
816
66 68 56
548 838 496
28-5667137
9-3446575
•001225490
817
66 74 89
545 838 518
28-5832119
9-8484731
-001223990
818
66 91 24
547 848 432
28-6006993
9-3522857
-001222494
819
67 07 61
649 853 259
28-6181760
9-3660962
-001221001
820
67 24 00
551 868 000
28-6366421
9-3699016
-001219612
821
67 40 41
558 887 661
28-6530976
9-3637049
-001218027
822
67 56 84
555 412 248
28-6706424
9-3675061
-001216546
823
67 78 29
557 441 767
28-6879766
9-3713022
•001216067
824
67 89 76
559 476 224
28-7054002
9-3750963
-001213692
825
68 06 25
561 515 625
28-7228132
9-3788873
-001212121
826
68 22 76
568 559 976
28-7402157
9-3826752
-001210664
827
68 89 29
565 609 288
28-7576077
9-3864600
-001209190
828
68 55 84
667 668 652
28-7749891
9-3902419
•001207729
829
68 72 41
669 722 789
28-7923601
9-3940206
-001206273
72
SQUARES, CUBBS, ROOTS, AND RECIPROGAU.
1
No.
880
881
882
883
834
836
886
887
888
889
840
841
842
843
844
846
846
847
848
849
850
851
852
858
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
i 870
871
872
873
874
875
876
877
878
Square
68 89 00
69 06 61
69 22 24
69 88 89
69 55 56
60 72 25
69 88 96
70 05 69
70 22 44
70 89 21
70 56 00
70 72 81
70 89 64
71 06 49
71 28 86
71 40 25
71 57 16
71 74 09
71 91 04
72 OS 01
72 26 00
72 42 01
72 59 04
72 76 09
72 98 16
78 10 25
78 27 86
78 44 49
78 61 64
78 78 81
73 90 00
74 18 21
74 80 44
74 47 69
74 64 96
74 82 25
74 99 56
76 16 89
75 84 24
75 51 61
75 69 00
75 86 41
76 03 84
76 21 29
76 38 76
76 66 25
76 73 76
76 91 29
77 08 84
Cabe
671 787 000
678 856 191
575 980 368
578 009 587
680 098 704
582 182 875
584 277 056
586 876 258
588 480 472
590 589 719
592 704 000
694 823 821
696 947 688
599 077 107
601 211 584
603 851 125
605 495 736
607 646 423
609 800 192
611 960 049
614 125 000
616 295 051
618 470 208
620 650 477
622 835 864
625 026 875
627 222 016
629 422 793
631 628 712
638 889 779
636 056 000
638 277 881
640 503 928
642 736 647
644 972 544
647 214 625
649 461 896
651 714 863
653 972 032
656 234 909
658 503 000
660 776 811
663 054 848
665 338 617
667 627 624
669 921 875
672 221 876
674 526 138
676 686 152
SqaareBoot
28-8097206
28-8270706
28-8444102
28-8617394
28-8790582
28-8968666
28-9136646
28-9309523
28-9482297
28-9664967
28-9827535
29-0000000
29-0172363
29-0344623
29-0516781
29-0688837
29-0860791
29-1032644
29-1204396
29-1376040
29-1547595
29-1719043
29-1890390
29-2061637
29-2232784
29-2403830
29-2574777
29-2746623
29-2916370
29-3087018
29-3257566
29-3428015
29-3598365
29-3768616
29-3938769
29-4108823
29-4278779
29-4448637
29-4618397
29-4788059
29-4957624
29-5127091
29-5296461
29-5465734
29-5634910
29-5803989
29-5972972
29-6141858
29-6310648
Cube Boot
9-3977964
9-4015691
9-4053387
9-4091054
9-4128690
9-4166297
9-4203873
9-4241420
9-4278936
9-4316423
9-4353880
9-4391307
9-4428704
9-4466072
9-4503410
9-4540719
9-4577999
9-4615249
9-4652470
9-4689661
9-4726824
9-4763957
9-4801061
9-4838136
9-4875182
9-4912200
9-4949188
9-4986147
9-5023078
9-5059980
9-5096854
9-5133699
9-5170515
9-5207303
9-5244063
9-5280794
9-5317497
9-5354172
9-5390818
9-5427437
9-5464027
9-5500589
9-5537123
9-5573630
9-5610108
9-5646559
9-5682982
9-5719377
9-5755745
RedintKal
•001204819
•001203369
•001201923
•001200480
•001199041
•OO1197605
•001196172
•001194743
•001193317
•001191895
•001190476
•001189061
•001187648
•001186240
•001184834
•001183432
•001182033
•001180638
•001179245
•001177856
•001176471
•001175088
•001173709
•001172333
•001170960
•001169691
•001168224
•001166861
•001165501
•001164144
•001162791
•001161440
•001160093
•001158749
•001167407
•001166069
•001164734
•001 163403
-001152074
•001160748
•001149425
•001148106
•001146789
•001145475
•001144165
•001142857
•001141653
•001140261
•001138952
9QUAItX8, CUBES, BOOTS, ASD BBGIPBOCALS. 67E
Sigpian
77 06 41
77 44 00
77 W 61
77 79 24
77 96 89
78 14 56
78 82 25
78 49 96
78 67 69
78 85 44
79 08 21
79 2100
79 88 81
79 66 64
79 74 49
79 92 86
80 10 25
80 28 16
80 46 09
SO 64 04
80 82 01
8100 00
81 18 01
81 86 04
81 64 09
81 72 16
81 90 25
82 08 86
82 26 49
82 44 64
82 62 81
82 81 00
82 99 21
88 17 44
88 85 69
88 53 96
88 72 25
88 90 66
84 08 89
84 27 24
84 45 61
84 64 00
84 8^ 41
65 OD 84
.85 19 29
65 S7 76
85 66 25
85 74 76
85 9B 29
Cube
679 151 489
681 472 000
638 797 841
686 128 968
688 465 887
690 807 104
698 154 195
696 506 466
697 864 108
700 227 072
702 595 869
704 969 000
707 847 971
709 782 288
712 121 967
714 516 984
716 917 876
719 823 196
721 784 278
724 160 792
726 572 699
729 000 000
781 482 701
788 870 808
786 814 827
788 763 264
741 217 626
748 677 416
746 142 648
748 618 812
751 089 ^9
758 571 000
756 068 081
768 660 628
761 048 497
768 661 944
766 060 875
768 675 296
771 095 218
778 620 682
776 161 669
778 688 000
781 2^9 961
788 777 448
786 880 467
788 889 024
791 453 125
794 022 776
796 597 988
MMMtfll
^|g|g^
f^pisre Boot Oube Boot Beciproctd
29*6479342
29-6647939
29*6816442
29-6984848
29-7163159
29-7321376
29-7489496
29-7667521
29*7825462
29-7993289
29-8161080
29-«328678
29-^96231
29-8663690
29*8831056
29*8898328
29*9165506
29-9832591
29*9499583
29*9666481
29*9833287
80*0000000
30-0166620
30^)338148
30*0499684
30*0665928
30*0832179
30'09983B9
30*1164407
80*1830383
80-1496269
30-1662063
30*1827766
80'1 993377
80'21d8899
80-2324329
80-2489669
80*2654919
30*2820079
80-2986148
30*3150158
30*3315018
30*8479818
10-364452^
30-3809151
30-8973683
304138127
30-4302481
30*4466747
9*6792085
9-5828397
9-5864682
9-6900939
9-5937169
9-6973373
9*6009548
9*6045696
9*6081817
9*6117911
9*6163977
9-6190017
9*6226030
9*6262016
9*6297976
9*6833907
9*6869812
9*6405690
9*6441542
9-6477367
9*6513166
9*6648938
9*6584684
9*6620403
9*6666096
9*6691762
9*6727403
9*6763017
9*6798604
9*6834166
9*6869701
9-6905211
9*6940694
9-6976161
9*7011583
9-7046989
9*7082369
9*7117723
9-7153061
9*7188354
9-7223631
9-7258883
9-72W169
9-7^29309
9-7364484
9-7399634
9*7434758
9*7469857
9*7504930
•001137656
*0011 36364
*001135074
-001138^7
*001 132503
*001131222
*001129944
*001 128668
*001 127396
*00n 26126
*001 124859
*001123596
•001122334
*001121076
•001119821
*0011 18668
*001 117318
*0011 16071
•001114827
*001113586
*001112347
-OOlUllU
*001109878
*001108647
*001107420
•001106195
*001104972
•001108753
•001102536
•001101322
•001100110
•001098901
•001097696
•001096491
•001095290
•001094092
•001092896
•001091703
•001090513
•001089325
*001088139
*001086967
*0ai086776
-001084599
•001083424
•001082251
•001081081
•001079914
•001078749
XX
74
/SQUARES, CUBBS, ROOTS, AND RBCIPBOCAIA.
No.
Square
Cube
SqaareBoot
Cube Root
Redprocml
928
86 11 84
799 178 762
30-4630924
9-7539979
•001077586
929
86 80 41
801 765 089
30-4795018
9-7576002
•001076426
930
86 49 00
804 867 000
80-4959014
9-7610001
•0O1075269
981
86 67 61
806 964 491
80*5122926
9-7644974
•001074114
982
86 86 24
809 667 668
30-6286760
9-7679922
•001072961
983
87 04 89
812 166 287
80-5450487
9-7714846
•001071811
984
87 28 56
814 780 604
30-5614136
9-7749743
•001070664
985
87 42 25
817 400 876
30-57VV697
9-7784616
•001069519
986
87 60 96
820 025 856
30-5941171
9-7819466
•001068376
987
87 79 69
822 666 968
30-6104567
9-7854288
•001067236
988
87 96 44
826 298 672
827 986 019
30-6267857
9-7889087
•001066098
989
88 17 21
80-6431069
9-7923861
•001064963
940
88 86 00
880 584 000
80^594194
9-7958611
-001068830
941
88 54 81
888 287 621
80-67572S3
9-7993336
•001062699
942
88 78 64
886 896 888
30-6920186
9-8028036
•001061571
948
88 92 49
888 561 807
80-7083051
9-8062711
•001060445
944
89 11 86
841 282 884
80-7246830
9*8097862
•001059322
945
89 80 26
848 908 626
30-7408528
9-8181989
-001058201
946
89 49 16
846 690 586
80-7571180
9-8166591
•001057082
947
89 68 09
849 278 128
30-7733651
9-8201169
•001055966
948
89 87 04
861 671 892
30-7896086
9-8235723
•001064852
949
90 06 01
864 670 849
30-8058436
9-8270252
•001058741
960
90 25 00
867 875 000
30-8220700
9-8304757
•001052632
951
90 44 01
860 085 861
30'8382879
9-8339238
•001051625
952
90 68 04
862 801 408
30*8544972
9-8378695
•001050420
958
90 82 09
865 528 177
30-8706981
9-8408127
•001049318
954
91 01 16
868 260 664
30-8868904
9-8442536
•001048218
955
9120 25
870 988 876
80-9030748
9-8476920
•001047120
956
9189 86
878 722 816
30-9192497
9-8511280
•001046025
957
9168 49
876 467 498
30-9854166
9-8546^17
•001044932
958
9177 64
879 217 912
30-9516761
9-8579929
•001043841
959
9196 81
881 974 079
30^677261
9-8614218
•001042753
960
9216 00
884 786 000
30-9838668
9-8648483
•001041667
961
92 86 21
887 608 681
31-0000000
9-8682724
-001040583
962
92 64 44
890 277 128
81-0161248
9-8716941
•001039501
968
92 78 69
898 056 847
31-0322418
9-8761135
•001038422
964
92 92 96
896 841 844
31-0488494
9-8785305
•001037344
965
98 12 25
898 682126
31-0644491
9-8819461
•001036269
966
98 81 66
901 428 606
31-0805406
9-8853574
•001036197
967
98 60 89
904 281 068
81-0966236
9-8887673
•001034126
968
98 70 24
907 089 282
81*1126984
9-8921749
•001038058
969
98 89 61
909 858 209
31-1287648
9-8955801
•001031992
970
94 09 00
912 678 000
31-1448230
9-8989880
•001030928
971
94 28 41
916 498 611
81-1608729
9-9028835
•001029866
972
94 47 84
918 880 048
8 1-1769 145
9-9057817
•OO1028807
978
94 67 29
921 167 817
81-1929479
9-9091776
•001027749
974
94 86 76
924 010 424
31-2089731
9-9125712
•001026694
975
95 06 25
926 869 875
31-2249900
9-9159624
•001025641
976
95 25 76
929 714 176
31-2409987
9-9193513
•OO1O24590 1
SQtJABES, CUBES, BOOTS, AND BEOIPBOOALS. 6?^
No.
977
978
979
980
981
982
988
984
985
986
987
988
989
990
991
992
998
994
996
996
997
998
999
1000
1001
1002
1008
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1016
1016
1017
1018
1019
1020
1021
1022
1023
1024
1026
Sqaaxe
95 46 29
96 64 84
95 84 41
96 04 00
96 23 61
96 43 24
96 62 89
96 82 66
97 02 26
97 2196
97 41 69
97 6144
97 8121
98 0100
98 20 81
98 40 64
98 60 49
98 8086
99 00 25
99 2016
99 40 09
99 60 04
99 80 01
00 00 00
00 20 01
00 40 04
00 60 09
1 00 80 16
1 01.00 26
1 01 20 86
1 01 40 49
1 01 60 64
1 01 80 81
1 02 01 00
1
1
1
1
02 2121
02 41 44
02 61 69
02 8196
03 02 26
03 22 66
03 42 89
03 63 24
03 88 61
04 04 00
04 24 41
04 44 84
1 04 66 29
1 04 86 7C
1 06 06 26
Oabe
982
935
988
941
944
946
949
952
966
968
961
964
967
970
973
976
979
982
986
988
991
994
997
1000
1003
1006
1009
012
016
018
021
024
027
080
033
036
039
042
045
048
061
064
1068
1061
1064
1067
1070
1073
1076
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
674 838
441 862
813 789
192 000
076 141
966168
862 087
768 904
671 626
685 266
604 808
480 272
861669
299 000
242 271
191488
146 667
107 784
074 876
047 986
026 978
011992
002 999
000 000
008 001
012 008
027 027
048 064
076126
108 216
147 848
192 612
243 729
801000
864 831
488 728
609197
590 744
678 875
772 096
871 918
977 832
089 869
208 000
882 261
462 648
699167
741 824
890 625
SqtiareRoot
31-2569992
81»2729915
31*2889757
81-S049517
31-8209195
31-3368792
31-3528308
31-3687743
31-3847097
31-4006369
31-4165561
31-4324673
31-4483704
31-4642654
31-4801525
31*4960315
81-5119025
31-5277655
31-5436206
31-5594677
31-5768068
31-6911380
81-6069618
81-6227766
31*6385840
31-6543886
31*6701752
31-6859590
31«7017849
31*7176030
dl*7832683
81-7490157
81-7647603
81*7804972
81*7962262
31*8119474
31-8276609
31*8483666
31*8590646
31-8747549
31*8904874
81*9061123
31-9217794
31-9874388
81-9530906
81-9687847
81-9843712
320000000
320156212
Cube Root
9-9227379
9-9261222
9*9295042
9-9328889
9-9862613
9-9396363
9-9430092
9-9463797
9-9497479
9-9531138
9-9564775
9-9593389
9*9631981
9-9665549
9*9699095
9-9732619
9-9766120
9*9799699
9 9833055
9-9866'188
9-9899)00
9-9933289
9-9966656
10-0000000
10-0033322
10-0066622
10*0099899
10-0183165
10*0166389
10-0199601
10-0232791
10*0265958
10-0299104
10-0332228
10*0365330
10*0398410
10*0431469
10*0464606
10-0497621
10-0580514
10*0568485
10*0596435
10-0629364
10-0662271
10-0695156
10*0728020
10-0760863
10-0793684
10-0826484
Bedpiocal
-001023641
•001022495
•001021450
•001020408
•001019368
•001018330
-001017294
•001016260
-001015228
-001014199
-001013171
-001012146
-001011122
-001010101
•001009082
•001008065
-001007049
•001006036
-001006025
-001004016
-001003009
•001002004
•001001001
•001000000^
•000999001^
•000998004^
•000997009^
-000996015!
-000995024J
•000994035J
•000993048:
•0009920631
•0009910801
•000990099(
*000989119:
*000988142i
*000987166j
•0009861981
•000985221:
*000984252<
*000983284i
*0009823i8j
•0009818545
*000980392!
*000979431J
*000978473(
•000977517
•00(»976562
•0009766091
r6 ciQUAOGS, 0UBS8, ROOTS, AND RBCIPROOALS.
la
Square
1 05 26 76
Oobe
Square Root
Cube Root
Bedprocal i
)26
1 080 045 576
32-0812348
10-0859262
•0009746689
}27 i 1 06 47 29
1 083 206 683
32-0468407
10-0892019
•0009737098
928 ; 1 05 67 84
1 086 873 952
32-0624391
10-0924755
-0009727626
320 1 05 88 41
1 080 547 889
32-0780298
10-0957469
-000971817a
D30 : 1 06 09 00
1 092 727 000
32-0936131
10-0990163
-000970873S
OSl ' 1 06 29 61
1 095 912 791
32-1091887
10-1022835
-0009699321
032 ; 1 06 50 24
1 099 104 768
32-1247568
10-1055487
•0009689922
OSS ! 1 06 70 89
1 102 S02 937
32-1403173
10-1088117
-0009680542
034 1 1 06 91 66
1106507 804
32-1558704
10-1120726
•0009671180
035 1 07 12 25
1 108 717 875
32-1714169
10-1153314
-0009661836
036 107 32 96
1 111 934 656
32-1869539
10-1185882
-0009652510
037 i 1 07 58 69
1 115 157 658
32-2024844
10-1218428
-0009643202
038 1 07 74 44
1 118 S86 872
32-2180074
10-1260953
-0009633911
039 1 1 07 95 21
1 121 622 819
32-2336229
10-1283467
-0009624639
040 1 08 16 00
1 124 864 000
32-2490310
10-1315941
-0009615385
041 108 86 81
1 128 111 921
32-2645316
10-1348403
-0009606148
042 108 57 64
1 181 866 088
32-2800248
10-1380845
-0009596929
043 , 1 08 78 49
1 134 626 507
32-2955105
10-1413266
•0009587728
044 1 08 99 86
1 137 898 184
32-3109888
10-1445667
-0009578544
045 ' 1 09 20 25
1 141 166 125
32-3264598
10-1478047
-0009669378
046 : 1 09 41 16
1 144 446 886
32-3419233
10-1510406
•0009560229
047 109 62 09
1147 730 828
32-3573794
10-1542744
•0009551098
048 109 88 04
1 151 022 592
32-3728281
10-1575062
-0009541985
049 , 1 10 04 01
1 154 820 649
32-8882695
10-1607859
•0009532888
050 110 25 00
1 157 625 000
32-4037035
10-1639636
-0009523810
051 110 46 01
1 160 985 651
32-4191301
10-1671893
•0009514748
052 110 67 04
1164 252 608
32-4345495
10-1704129
•0009505703
053 110 88 09
1 167 575 877
32-4499615
10-1736844
•0009496676
054 11109 16
1 170 906 464
32-4658662
10-1768539
•0009487666
055 11180 25
1 174 241 675
32-4807635
10-1800714
•0009478673
056 1115186
1 177 588 616
32-49ol536
101832868
•0009469697
057 11172 49
1 180 982 198
3-2-5115364
10-1865002
•0009460738
058 1 11 98 64
1 184 287 112
32-5269119
101897116
•0009451796
059 ; 1 12 14 81
1 167 648 879
32-5422802
10-1929209
-0009442871
060 j 1 12 86 00
1 191 016 000
32-5576412
101961283
-0009433962
061 1 1 12 57 21
1 194 889 981
32 6729949
10-1993386
•0009425071
062 , 1 12 78 44
1 197 770 828
32-5888415
10-2025369
•0009416196
063 11299 69
1 201 167 047
32-6036807
10-2057382
•0009407338
064 1 1 18 20 96
1 204 550 144
32-6190129
10-2089375
-0009398496
065 ! 1 18 42 25
1 207 949 625
32-6848377
10-2121347
-0009389671
066 1 18 63 56
1 211 856 4C6
32-6496554
10-2163300
•0009380863
067 118 84 89
1 214 767 768
32-6649659
10-2185283
•0009372071
068 114 06 24
1216186 462
82-6802693
10-2217146
-0009363296
069 114 27 61
1221611509
82-6956654
10-2249039
'0009354687
070 1114 49 00
1 225 043 000
32-7108544
10-2280912
•0009345794
071 114 70 41
1 228 480 911
32-7261363
10-23127^6
•0009337068
072 114 9184
1 281 925 848
32-7414111
10-2344599
•0009328858
073
1 15 13 29
1 285 876 017
32-7566787
10-2376413
•0009319664
074
1 15 84 76
1 288 888 224
32-7719392
10-2408207
•0009310987
SQUA&ES, CUBES, ROOTS, AND RECIPROCALS. 67'
No.
1076
1076
1077
1078
1070
1080
1081
loss
1088
1084
1088
1086
1087
1088
1089
1090
1091
1093
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1W3
Squaue
15 56 25
15 77 76
16 99 29
16 20 84
16 42 41
16 64 00
16 85 61
17 07 24
17 28 89
17 50 56
17 72 25
17 93 96
18 IB 69
18 87 44
18 69 21
18 81 00
19 02 81
19 24 64
19 46 49
19 68 86
19 90 26
20 12 16
20 84 09
20 56 04
20 78 01
2i 00 00
2122 01
21 44 04
2166 09
21 88 16
22 10 25
22 82 86
22 54 49
22 76 64
22 98 81
23 21 00
23 13 21
23 65 44
23 87 69
24 09 96
24 82 25
24 54 56
24 76 89
24 99 24
25 21 61
25 44 00
25 66 41
25 88 84
26 11 29
Cabe
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
242
245
24»
252
256
259
263
266
270
273
277
280
284
287
291
295
298
302
305
309
312
816
320
823
327
331
384
338
341
345
349
852
856
360
863
867
371
875
878
B82
386
889
893
897
401
404
408
412
416
296 875
766 976
243 533
726 552
216 089
712 000
214 441
723 368
238 787
760 704
289 125
824 056
865 503
918 472
467 969
029 000
596 571
170 688
751 357
838 584
932 875
532 786
189 673
753 192
373 299
000 000
638 301
278 208
919 727
572 864
232 625
899 016
572 043
251 712
938 029
631 000
830 631
036 928
749 897
460 544
195 875
928 896
668 618
415 032
168 159
928 000
694 561
467 848
247 867
Sqoare Root
82-7871926
82-8024389
32-8176782
32-8329103
82-848I354
d2*863d535
32-8785644
32-8937084
32-9089653
32-9241553
32-9398382
32-9645141
32-9696830
32-9848450
33-0000000
33-0151480
33-0302891
33-0454233
33-0605505
33-0756708
330907842
33-1068907
38-1209903
33-1360830
33*1511689
83-1662479
33-1818200
33-1968853
38-2114438
83-2264955
33-2415408
88-2566783
33*2716095
33-2866339
39-301 6516
83-31'66626
33-38l$666
33*3461640
83-3616546
33-3766385
33-391$157
33-46(65862
33'4215499
83-4365070
33-4514573
33-4664011
33-4818381
33-4962684
33-5111921
CAbeRoot
10-2439981
10-2471735
10-2503470
10-2535186
10-2566881
10-2598667
10-2630213
10-2661850
10-2698467
10-2726065
10-2756644
10-2788203
10-2819743
10-2851264
10-2882765
10-2914247
10-2946709
10-2977158
10-3008577
10-3039982
10-3071368
10-3102735
10*3134083
10-3166411
10-3196721
10-3228012
10-3259284
10-3290537
10-3321770
10-3362985
10-3384181
10-3416358
10'3446517
10-3477657
10-3608778
10-3539880
10-3570964
10-3602029
10-3633076
10-3664103
10-3695113
10-3726103
10-3767076
10-3788030
10-3818965
10-8849882
10-3880781
10:8911661
10-3942523
Reciprocal
•0009302326
•0009293680
•0009285061
-000927643^
-0009267841
-000926925S
-0009260694
-0009242144
-0009233610
•0009226092
•000921 6590
•0009208103
•0009199632
•000919117G
•0009182730
•0009174312
•0009165903
•0009167603
-0009149131
•0009140768
•0009132420
<)009124088
•0009115770
•0009107468
•0009099181
•0009090909
•0009082662
•0009074410
•0009066183
-0009067971
•0009049774
•0009041691
•0009033424
•0009025271
-0009017133
•0009009009
•000900090Q
•000899280G
•0008984726
-0008976661
•0008968610
•0008960673
•0008952561
•0008944644
•0008936550
-0008928571
•0008920607
•0008912656
•0008904720
SQUARES, CUBES, BOOTS, AfiTD RECIPROCALS.
Bqnan
86 88 76
86 56 26
86 78 76
87 0129
27 23 84
27 46 41
27 69 00
27 91 61
2814 24
28 86 89
88 69 66
88 82 25
89 04 96
89 27 69
29 60 44
29 73 21
29 96 00
80 18 81
80 4164
80 64 49
80 87 86
81 10 25
81 83 16
81 56 09
81 79 04
82 02 01
82 25 00
82 48 01
82 71 04
82 04 09
88 17 16
83 40 26
83 63 86
83 86 49
84 09 64
84 82 81
84 56 00
84 79 21
85 02 44
35 25 69
85 48 96
35 72 25
35 95 56
36 18 89
36 42 24
36 65 61
36 89 00
37 12 41
87 85 84
0ab6
1480 084 684
1 428 888 186
1 487 688 876
1 481 485 888
1 486 849 168
1 489 069 689
1 448 897 000
1 446 781 091
1 450 671 068
1 454 419 687
1 458 874 104
1 468 186 876
1 466 008 466
1 469 878 868
1 478 760 072
1 477 648 619
1 481 544 000
1 485 446 221
1 489 856 888
1 493 271 807
1 497 198 984
1 501 128 625
1 505 060 186
1 509 008 628
1 512 958 792
1 516 910 949
1 520 875 000
1 524 845 951
1 528 828 808
1 682 808 577
1 536 800 264
1 640 798 875
1 544 804 416
1 548 816 898
1 652 836 812
1 556 862 679
1 560 896 000
1 564 936 281
1 568 988 628
1 578 037 747
1 577 098 944
1 581 167 126
1 585 242 296
1 589 824 468
1 593 418 688
1 597 609 809
1 601 618 000
1 605 728 811
1 609 840 448
8<IinreRoot
33'5261092
d8-5410196
38-d5592d4
88*5708206
88*5857112
38*6005952
38-6154726
88-6808434
88-6452077
88*6600658
88*6749165
88*6897610
88*7045991
38*7194306
88*7842556
38*7490741
88*7688860
38*7786915
38*7934905
83*8082880
83*8230691
83*8378486
83*8526218
33*8673884
83*8821487
83*8969025
83*9116499
83*9268909
38*9411255
33*9558537
38-9705756
83-9852910
84-0000000
840147027
84*0293990
84*0440890
84*0587727
34*0734501
34*0881211
84*1027858
84*1174442
84*1320963
84*1467422
34*1613817
34*1760150
34*1906420
34*2052627
84*2198773
34*2344855
CnbeBoot
10*d978366
10*4004192
10-4084999
10^065787
10*4096557
10-4127310
10*4158044
10*4188760
10*4219458
10*4250138
10-4280800
10-4811443
10*4342069
10*4872677
10-4408267
10*4433839
10-4464398
10*4494929
10*4525448
10*4555948
10-4586431
10*4616896
10*4647343
10*4677773
10-4708185
10*4738579
10*4768955
10-4799314
10*4829656
10*4859980
10*4890286
10-4920576
10*4950847
10-4981101
10*5011337
10-5041556
10*5071757
10*5101942
10*6132109
10*5162269
10*5192391
10*6222506
10-5252604
10*5282686
10-5312749
10*6342795
10*5372825
10-5402837
10*5432832
Beciproctl
1
•0008896797
-0008888889
-0008880995
•0008873114
•0008865248
•0008857396
•0008849568
•0008841783
•000883392*2
•0008826125
•0008818342
•0008810573
•0008802817
•0008796076
•0008787346
•0008779631
•0008771930
•0008764242
•0008766667
•000874890a
•0008741259
*0008733624
•0008726003
•0008718396
•0008710801
*0O08703220
*0008695662
*0008688097
*0008680556
•0008673027
*0008665511
•0008658009
*0008650619
-0008643042
*0008635579
•0008628128
•0008620690
•000861326i
•0008605862
•00085984'i2
•000869.065
•0008583691
•0008576329
•0008568980
•0008561644
•0008654380
•0008547009
•OOO85S9710
•0008532423
SQUARES, CUBES, ROOrS, AND RECIPBOOALS. Qfji
Ha
fl^inore
1178
1174
1175
1176
1177
1178
1179
1180
1181
1182
1188
1184
1188
1186
1187
1188
1189
1190
1191
1192
1198
U94
1196
1196
1197
1198
1190
1200
1201
1202
1208
1204
1206
1206
1207
1206
1200
1210
1211
1212
1218
1214
1216
1216
1217
1218
1219
1220
1221
87 69 29
87 82 76
88 06 25
88 29 76
88 68 29
88 76 84
89 00 41
89 24 00
89 47 61
89 71 24
89 94 89
4018 66
40 42 25
40 66 96
40 89 69
41 13 44
41 87 21
41 61 00
41 84 81
42 08 64
42 82 49
42 56 86
42 80 25
48 0416
48 28 09
43 62 04
43 7601
44 00 00
44 24 01
44 48 04
44 72 09
44 96 16
146 21025
14544 86
1 45 60 49
1 45 92 64
1 46 16 ^1
1 46 41 00
1 46 65 21
46 89 44
47 18 69
47 87 96
47 62 25
47 86 66
48 10 89
48 85 24
48 69 61
48 84 00
49 08 41
Cube
1618
1
1
1
1
1
1
618
622
626
680
684
638
1648
1C47
1651
1666
1659
1064
1668
1 673
1076
1680
1686
1689
ld98
1697
1702
1706
1710
1715
1719
723
728
732
786
740
745
749
754
758
762
767
771
775
780
784
789
1798
1798
1802
1806
1811
1815
1820
964 717
096 024
284 375
879 776
682 233
691 752
868 839
082 000
212 741
400 568
595 487
797 504
006 625
222 856
446 203
•76 672
914 269
159 000
410 871
669 888
936 057
209 384
489 876
777 636
072 373
374 392
688 699
000 000
328 601
654 408
992 427
887 664
690126
049 816
416 743
790 912
172 829
601000
966 981
860128
770 597
188 844
618 875
045 696
485 318
982 282
886 459
848 000
816 861
Square Hoot
34-2490875
34-2636834
34-2782730
84-2928564
34-3074336
34-3220046
34-3365694
34-8511281
34-3656805
34-3802268
34-3947670
34-4093011
34-4238289
34-4383507
34-4628663
34-4673769
34-4818793
34-4963766
34-6108678
34-6263630
34-6398321
34-554S061
34-6687720
34-6832329
34-6976879
34-6121366
34-6265794
34-6410162
34-6564469
84-6698716
34-6842904
34-6987031
34-7131099
34-7276107
34-7419056
34-7562944
84-7706773
34-7850643
34-7994253
84-8137904
84-8281496
84-8425028
34-8568501
84-8711915
34-8865271
34-8998567
34-9141805
84-9284984
34-9428104
Cube Boot
10-8462810
10-5492771
10-6522715
10-5552642
10-5582662
10-5612446
10-5642322
10-5672181
10-5702024
10-5781849
10-5761658
10-5791449
10-6821225
10-5860983
10-5880725
10-5910450
10-6940168
10'5969860
10-6999625
10-6029184
10-6068826
10-6088461
10-6118060
10-6147662
10-6177228
10-6206788
10-6236331
10-6265867
10-6296367
10-6324860
10-6864338
10-6383799
10-6413244
10-6442672
10-6472086
10-6601480
10-6630860
10-6660223
10-6589570
10-6618902
10-6648217
10-6677516
10-6706799
10-6736066
10-6765317
10-6794662
10-6823771
10-6852973
10-6882160
lleciprocal
-0008525149
•0008517888
•0008610638
-0008503401
-0008496177
-0008488964
•0008481764
•0008474676
•0008467401
-0008460237
•0008463085
■0008446946
•0008438819
•0008431703
•0008424600
-0008417608
•0008410429
•0008403861
-0008396806
•0008389262
•0008382230
•0008375209
•0008368201
-0008361204
-0008364219
•0008347246
•0008340284
-0008333333
•0008326396
-0008319468
•0008312562
•0008305648
-0008298765
-0008291874
-0008285004
•0008278146
•0008271299
-0008264463
•0008257638
-0008250825
•0008244023
•0008237232
•0008230453
-0008223684
•0008216927
•0008210181
•0008203445
•0008196721
0008190008
B0OX8, ANB RECIPBOCALS.
Sqoan
L 49 82 84
I 49 67 23
L 49 81 7e
L 60 06 25
L 60 80 76
L 60 65 29
L 50 79 84
I 61 04 41
L 61 29 GO
L 61 63 61
L 61 78 24
L 52 02 89
L 62 27 56
L 62 62 25
I 52 7,6 C6
L 58 01 69
L 53 26 44
L 63 51 21
L 53 76 00
L 54 00 81
L 64 25 64
L 64 50 49
L 64 76 86
L 65 00 25
I 55 25 16
L 55 50 09
. 55 75 04
. 56 00 01
. 56 25 00
66 60 01
66 75 04
57 00 09
57 25 16
57 60 25
57 75 86
68 00 49
58 25 64
58 50 81
58 76 00
69 01 21
59 26 44
59 51 69
59 76 96
60 02 25
60 27 56
60 52 89
60 78 24
61 03 61
6129 00
Oabo
1M4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
829
838
888
842
847
851
866
860
865
869
874
8^9
883
888
892
897
902
006
911
916
926
929
984
989
948
948
953
057
962
967
971
976
981
986
990
995
2 000
2 005
2 009
2 014
2 019
2 024
2 029
2 038
2 038
2 043
2048
T98M8
276 667
7B7 424
9G5 625
771 176
284 068
804 852
83X989
867 000
409 891
959 168
616 387
080 904
652 876
2-J2 266
819 063
418 272
014 919
624 000
240 521
864 488
406 907
184 784
781126
434 986
006 228
764 902
441 240
125 000
816 251
615 006
221277
935 064
656 375
885 216
121 698
865 512
616 979
876 000
142 581
916 728
698 447
487 744
284 625
089 096
901168
720 882
648 109
883 000
Sqnitn Boot
OobeRoot
34*9571166
34*9714169
84-9857114
35-0000000
85-0142828
35-0285598
85-0428309
35*0570968
85*0713558
85*0856096
85-0998575
35*1140997
36*1283361
85*1425668
35-1667917
35*1710108
35*1852242
85*1994318
85*2186387
35-2278299
do*2420204
85'2562051
85*2708842
85*2845575
85-2987252
35*3128872
35*3270435
35*3411941
36*3558391
35*3694784
85*3836120
85*8977400
35*4118624
35*4259792
35*4400903
85-4541958
85-4682957
35-4828900
35-4964787
35-5105618
85*5246393
85-5687118
35-5627777
35-5668385
35-5808937
85*5949434
35*6089876
85*6230262
35*6870593
10^911381
10*6940486
10*6969625
10*6998748
10*7027855
10*7056947
10-7086023
10*7115083
10*7144127
10*7178165
10*7202168
10*7231165
10-7260146
10-7289112
16i7818062
l«b7846997
10-7876916
10*7404819
10*7438707
10-7468579
10-7491436
10*7620277
10-8549103
10:7677913
10*7606708
10-7685488
10*7664252
10*7698001
10*7721735
10*7750453
10*7779156
10*7807843
10*7836516
10*7865173
10*7898815
10*7922441
10-7951053
10*7979649
10-8008230
10^8086797
10-8065348
10*8093884
10*8122404
10*8150909
10*8179400
10-8207876
10-8236836
10-8264782
10-8293213
BeoiiiroGpl
-0008183306
•0008176615
*0008I 69935
-0008163265
-0008156007
-0008149959
-0008143322
-0008186696
-0008180081
-0008123477
•0008116888
*0008110300
•0008103728
•0008097166
-0008090615
•0008084074
•0008077544
•0008Q71026
•0008064516
H)008058018
-0008051530
-0008045052
•0008038585
-0008032129
*0008025682
*0008019246
*0008012821
*0008006405
-0008000000
*0007998605
*0007987220
-0007980846
-0007974482
*0007968127
*0007961788
-0007956449
-0007949126
*00079^812
*0007936508
•0007980214
*00079^930
-0007917656
•0007911392
•0007905188
-00078988^4
-0007892660
-0007886435
•0007880221
-0007874016
SQUABBS^ CUBES, BOOTS, AND Ei^IPROOALS.
No.
ia7t
1872
1275
1276
1377
1278
1279
1280
1281
1289
128d
1284
128^
1286
1287
1288
1280
1290
1281
1292
1298
1294
1296
1296
1297
1298
1299
1800
Id^l
1802
1808
1804
1805
1806
1807
1808
1809
1810
1811
1312
1818
1814
1815
1816
1817
1318
1819
Square
1
1
I
1
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
61 64 41
6179 84
62 06 29
62 80 76
62 66 25
628176
68 07 29
68 32 84
68 68 41
68 84 00
64 09 61
64 85 24
64 6089
64 86 66
6612 26
66 87 96
66 68 69
66 8944
66 15 21
66 4100
66 66 81
66 92 64
67 18 49
67 44 86
67 70 26
67 96 16
68 22 09
68 48 04
68 74 01
69 00 00
69 26 01
69 6204
69 78 09
70 0416
70 30 26
70 66 86
70 82 49
7108 64
71 84 81
716100
71 87 21
72 18 44
72 89 69
72 65 96
72 92 26
78 18 66
78 44 80
78 7124
73 97 61
0«be
2 068
2 068
2 062
2 067
2 072
2 077
2082
2 087
2092
2 097
2 102
2106
2111
2U6
2121
2126
2 181
2186
2141
2146
2161
2 166
2 161
2166
2171
2176
2181
2186
2191
2197
2 202
2 207
2 212
2 217
2 222
2 227
2 282
2^87
2 242
2 248
2 268
2 268
2 268
2 268
2 278
2 279
2 284
2 289
2294
^5611
076 648
983 417
798 824
671 876
662 676
440 988
386 962
24Q689
162000
071041
997 768
982187
874804
824 125
781 666
746903
719 872
700 669
689000
686 171
689 088
700 757
720 184
747 876
782 836
826 073
876 692
989 899
000 000
07^901
16»60&f
246127
3424!64
447 625
660616
681 443
810 112
940629
091000
243 231
403 328
671 297
747 144
980 876
122 496
822 Old
629 432
744 769
SquaseRoot
35*6510869
35-666 1090
86-6791255
35*6931366
36-7071421
36-7211422
35'7361367
35-7491258
35-7631095
36-777087Q
36-7910603
35-8050276
35-8189894
35-8329457
35-8468966
35-8608421
35-8747822
85-8887169
35-9626461
35-9166699
35-9304884
36-94440 J 5
35-9583092
35-9722115
35-9861084
36-0000000
36-0138862
36-0277671
36-0416426
36-0555128
36-0693776
36-0832371
36-09709J3
36-1109402
36-1247837
86-1886220
36-1524550
36-1662826
36-1801050
36-1939221
36-2077340
36-2215406
36-2863419
36-2491379
36-2629287
36-2767143
36-2904946
36-3042697
36-3180396
CnbeRoot
10-8821629
10-8350030
10-8378416
10-8406788
10-8436144
10-8463485
10-8491812
10-8520125
10-8548422
10'8576704
10-8604972
10-8633225
10-8661464
10-8689687
10-8717897
10-8746091
10-8774271
10-8802436
10-8830587
10-8858723
10-8886845
10'8914952
10-8943044
10-8971123
10-89991.86
10-9027236
10-9065269
10-9083290
10-9111296
10-91S9287
10-9167265
10-9195228
10-9223177
10-9261111
10-9279031
10-9306937
10-9334829
10-9362706
10-9390569
10-9418418
10-9446263
10-9474074
10-9501880
10-9529673
10-9557451
10-9585216
10-9612965
10-9640701
10-9668423
Eeciprocel
-0007867821
•0007861635
-0007855460
-0007849294
-0007843137
•0007836991
-0007830854
-0007824726
•0007818608,
•0007812500
•0007806401
-0007800312
•0007794232
-0007788162
-0007782101!
-0OO777G05O:
-0007770008;
•0007763975
•0007757952
-0007751938
-0007745933
-0007739938
-0007733953
-0007727975
-0007722008
-0007716049
-0007710100
-0007704160
-0007698229
-0007692308
-0007686395
-0007680492
-0007674597
-0007668712
-0007662835
•0007656968
-0007651109
-0007645260
-0007639419
-0007633588
-0007627765
•0007621951
•0007616146
•000761 035C
•0007604563
•0007598784
•0007593014
•0007587258
•0007581501
682 SQUARES, CUBES, ROOTS, ASB RECIPBOCA1&
No.
1820
1821
1822
1828
1824
1826
1826
1827
1828
1829
1880
1881
1882
1888
1884
1885
1886
1887
1888
1889
1840
1841
1842
1848
1844
1845
1846
1847
1848
1849
1860
1861
1862
1868
1854
1856
1856
1867
1868
1859
1860
1861
1862
1868
1864
1866
1866
1867
1868
Square
74 24 00
74 60 41
74 76 84
75 08 29
75 29 76
75 66 25
75 82 75
76 09 29
76 85 84
76 62 41
76 89 00
77 16 61
77 42 24
77 68 89
77 96 56
78 22 26
78 48 96
78 76 69
79 02 44
79 29 21
79 56 00
79 82 81
80 09 64
80 86 49
80 68 86
80 90 26
81 17 16
81 44 09
81 71 04
81 98 01
82 25 00
82 52 01
82 79 04
88 06 09
88 88 16
88 60 26
88 87 86
84 14 49
84 4164
84 68 81
84 96 00
85 28 21
85 50 44
86 77 69
86 04 96
86 82 26
86 59 66
86 86 89
87 14 24
Oabe
2 299
2805
2 810
2 816
2 820
2 826
2 881
2 886
2 842
2 847
2 852
2 857
2868
2 868
2 878
2 879
2884
2889
2 895
2 400
2406
2 411
2 416
2 422
2 427
2 488
2 488
2 444
2 449
2 464
2 460
2 465
2 471
2 476
2 482
2 487
2 498
2 498
2 604
2 509
2 516
2 621
2 626
2 682
2 687
2 643
2 648
2 564
2 660
968 000
199161
488 248
685 267
940 224
208125
478 976
752 788
089 662
884 289
687 000
947 691
266 868
698 087
927 704
270 875
621066
979 768
846 472
721 219
104 000
494 821
898 688
800 607
715 684
188 626
669 786
608 928
466192
911 649
876 000
846 661
826 208
818 977
809 864
818 875
826 016
846 298
874 712
911 279
456 000
008 881
669 928
189 147
716 544
802125
896 896
497 868
108 082
Square Root
Cube Root
Reciprocal
86*8318042
86-8455637
36-859dl79
86-8730670
86-3868108
36*4006494
36'4 142829
36-4280112
3C-4417843
36*4654523
36*4691650
36-4828727
36-4965752
36-5102725
36-5239647
36-5876518
36-5518338
36-5650106
36-5786823
36-5923489
36-6060104
36-6196668
36-6838181
36-6469644
36-6606056
36-6742416
36-6878726
36-7014986
36-7151196
36-7287358
36-7423461
36-7659519
36-7696526
36-7831483
86-7967390
36-8103246
36-8239053
36-8374809
36-8510515
36-8646172
36-8781778
36-8917335
36-9052842
36-9188299
36-9323706
36-9459064
36-9594372
36-9729631
36-9864840
10-9696181
10-9728825
10'9751606
10-9779171
10-9806828
10-9834462
10-9862086
10*9889696
10-9917298
10-9944876
10-9972445
11-0000000
11-0027541
11*0055069
11-0082583
11-0110082
11-0137569
11*0165041
11*0192500
n-0219945
11-0247377
11*0274796
11-0302199
11*0329590
11-0356967
110384330
11-0411680
11*0439017
11-0466339
11-0493649
11*0520946
11-0548227
11-0575497
11-0602762
11-0629994
11-0657222
11-0684487
11-0711689
11-0738828
11-0766003
11-0793165
11-0820814
11-0847449
11-0874571
11-0901679
11-0928775
11-0956857
11-0982926
11-1009982
•0007575758
•0007570023
•0007564297
•0007558679
•OOO7552870
•0007547170
•0007541478
•0007536795
•OOO7530120
•0007524464
•0007518797
•0007513148
•0007607508
•0007601875
•0007496252
•0007490637
•O0O7485030
•0007479432
•0007473842
•0007468260
•0C07462687
•0007457122
•0007451665
•0007446016
•0007440476
•0007434944
•0007429421
-0007423905
•0007418898
•0007412898
•0007407407
-0007401924
•0007396450
-0007390983
-0007386524
•0007380074
-0007374631
•0007369197
0007363770
•0007368862
0007362941
0007347639
0007342144
•0007336757
•0007331378
0007826007
0007320644
0007816289
0007309942
8<)UABEd, CUfifiS, ROOTS, AND BBCIPBOGAI^. gf)
No.
1869
1870
1871
1872
1878
1874
1875
1876
1877
1878
1879
1880
1881
1882
1888
1384
1885
1886
1887
1388
1889
1390
1391
1392
1893
1394
1895
1396
1897
1898
1899
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
Square
1 87 41 61
1 87 69 00
1 87 96 41
188 28 84
1 88 51 29
I 88 78 76
1 89 06 25
1 89 88 76
1 89 61 29
189 88 84
1 90 16 41
1 90 44 00
1 90 71 61
1 90 99 24
1 91 26 89
1 91 54 56
1918^25
1 92 09 96
1 92 87 69
192 65 44
1 92 98 21
1 98 21 00
198 4881
198 7964
1 94 04 49
194 8286
194 60 25
1 94 88 16
1 95 16 09
1 95 44 04
195 7201
196 00 00
196 28 01
1 96 56 04
1 96 84 09
1971216
197 40 25
1 97 68 86
1 97 96 49
198 24 64
1 98 62 81
1 98 81 00
1 99 09 21
1 99 87 44
1 99 65 69
1 99 98 96
2 00 22 25
2 00 50 56
2 00 78 89
Oobfl
2665
2 571
2 576
2 582
2 588
2 598
2 599
2 605
2 610
2 616
2 622
2 628
2 638
2 639
2 645
2 650
2 656
2 662
2668
2 674
2 679
2 685
2 691
2 697
2 708
2 708
2 714
2 720
2 726
2 782
2 738
2 744
2 749
2 755
2 761
2 767
2 778
2 779
2 785
2 791
2 797
2 803
2 809
2 815
2 821
5i827
2 888
2 889
2 845
726 409
858 000
987 811
630 848
282 117
941 624
609 875
285 876
969 683
662152
862 939
072 000
789 841
514 968
248 887
991 i04
741 625
500 456
267 603
048 072
826 869
619 000
419 471
228 288
045 457
870 984
704 875
547 136
897 778
256 792
124 199
000 000
884 201
776 808
677 827
587 264
505 125
481416
866143
809 812
260 929
221000
189 531
166 528
151 997
145 944
148 875
159 296
178 718
Square Boot
Cube Root
37-0000000
87-0136110
87-0270172
37-0406184
37-0540146
37-0675060
37-0809924
37-0944740
37-1079506
37-1214224
371348893
37a488512
37-1618084
371752606
37-1887079
37-2021505
37-2156881
87-2290209
37 2424489
87-2558720
37-2692903
37-2827037
37-2961124
37-3095162
37-3229152
37-3368094
87-3496988
37-3630834
37-3764632
37-3898382
37-4032084
37-4166738
37-4299345
37-4432904
37-4566416
37-4699880
37-4838296
37-4966665
37-5090987
37-5238261
87-5366487
37-5499667
37-5632799
37-6766885
87-5898922
37-6031913
37*6164857
87-6297754
37-6430604
11-1087026
11-1064054
11-1091070
11-1118073
11-1146064
11-1172041
11-1199004
11-1225955
11-1252893
11-1279817
11-1306729
11-1333628
11-1360514
11-1387386
11-1414246
11-1441093
11-1467926
11-1494747
11-1621655
11-1548350
11-1675133
11-1601903
11-1628669
U-1655403
11-1682134
11-1708852
11-1735658
11-1762250
11-1788930
11-1815598
11-1842252
11-1868894
1M896623
11-1922139
11-1948743
11-1 975334
11-2001913
11-2028479
11-2055032
11-2081573
11-2108101
11-2134617
11-2161120
11-2187611
n-2214089
11-2240564
11-2267007
11-2293448
11-2819^76
Reciprocal
•0007304602
•0007299270
0007293946
0007288630
0007283321
0007278020
0007272727
0007267442
0007262164
0007256894
0007251632
0007246377
0007241130
•0007236890
0007230658
0007226434
0007220217
0007216007
•0007209805
0007204611
0007199424
0007194245
0007189073
000718390g
0007178761
0007173601
000716845S
0007163324
0007158196
000715307€
0007147962
0007142857
000713776S
000713266^
0007127584
0007122507
0007117438
0007112376
0007107321
0007102273
0007097232
0007092199
0007087172
0007082153
0007077141
0007072136
0007067138
0007062147
0007057163
8Q1TA1ISB, CUBtty BOOTS, AlTD BBCIP80CAU3.
Ko.
Sqaan
1418
1419
1420
1421
1422
1428
1424
1426
1426
1427
1428
1429
1430,
1481
1482
1433
1484
1435
1430
1487
1488
1439
1440
1441
1442
1448
1444
1445
1446
1447
1443
1449
1460
1451
1452
1453
1454
1455
1456
1457
1469
1459
1460
1461
1462
1463
1464
1465
1466
20107 24
3 01 85 61
2 01 64 00
2 01 92 41
2 02 20 84
2 02 49 29
2 02 77 76
2 08 06 25
2 03 84 76
2 03 63 29
2 03 91 84
2 04 20 41
2 04 49 00
2 04 77 61
2 05 06 24
2 05 34 8d
2 05 68 66
2 06 92 25
2 06 20 96
2 06 49 69
2 06 78 44
2 07 07 2;
2 07 86 00
2 07 64 81
2 07 9? 64
2 08 29 41
2 08 51 86
2 08 80 25
2 09 OJ 16
2 09 88 09
2 09 67 04
2 09 96 01
2 10 25 00
2 10 54 01
2 10 83 04
2 11 12 09
2 11 41 16
2 11 70 25
2 11 99 86
2 12 28 49
2 12 57 64
2 12 86 81
2 18 16 00
2 13 45 21
2 18 74 44
2 14 03 69
21432 90^
2 14 62 25
2 14 91 56
OttlM
9861
2 867
2868
2 869
2 876
2 881
2 887
2 893
2899
2906
2 911
2 918
2 924
2080
2 986
2942
2 948
2954
2961
2067
2 973
2 979
2 985
2992
2 998
a 004
3 010
8 017
8 023
8 029
8 036
3 042
3 048
3 054
8 061
8 067
8 073
3 080
3 086
8 092
8 099
3105
3112
3118
8124
8181
8137
8144
8150
206 632
248 039
288 000
841461
403 448
478 967
658 024
640 625
786 776
841483
954 752
076 689
207 000
345 991
493 668
649 737
814 604
987 875
169 856
360 463
559 672
767 519
984 000
209 121
442 888
685 807
936 884
196125
464 536
741623
027 392
321849
626 000
986 851
257 408
586 677
924 664
271 875
626 816
990 993
368 912
745 679
136 000
635 181
943128
8^9 847
785844
219 625
662 696
Square Boot
87-6568407
87*6696164
87-6828874
87-6961536
87-7094153
37-7226722
37-7359246
87-7491722
87-7624152
87-7756635
87-7888873
87-8021163
87*81^408
87-8286606
87-8417769
37-8549864
87-8681924
87-8818938
87-8H5906
87-9077828
37-9209704
37-9841535
S7'9478319
37-9605058
87-973P751
37-9868898
38-0000000
380181556
38-0203067
88-0394532
38-06^952
88-0657326
380788665
38-0919939
38:1031178
88-1182371
3^-1318619
38-1444622
38a576G8l
38-1706C93
38-1837662
38-1968585
38-2090463
38-2230297
38-2361085
88-2491829
38-2622529
88-^758184
38-28aB794
CnbeRoot
11-2346292
11-2372696
11-2399087
11-2425465
11-2451831
11-2478185
ll-2504o27
11-2530866
11-2567173
11-2583478
11-2609770
11-2636050
11-2662318
11-2688573
11-2714816
11-2741047
11*2767266
11-2793472
112819666
11-2845849
11-2872019
11-2808177
11-2924323
11-2950457
11-2976579
11-3002688
113028786
11-3064871
11-3080945
11-8107006
113133066
11-8159094
11-3185119
11-3211132
11-3237134
11-3263124
11-8289102
11-8315067
11-8341022
11-8366964
11-3392894
11-3418818
11-3444719
11-3470614
11-3496497
11-3522368
11-3548227
n-3674075
11-35999U
Bedprocal
J
•OOO70521
•0007047^
•0007045
•0007037!
•OOO703'i
-0007027
•000702211
-00070171
•0007012^
•000700771
•0007002801
•0006997901
•0006993001
•000698812C
•000698324(
•0006978361
•0006973501
•0006968641
•000696378^
•000695894:
•000695410J
•0006949271
•000694444J
•0006939621
•000693481J
•000693000:
•000692520}
•000692041;
•000691562!
•0006910851
•OOO69O607J
•000690131:
•000689655:
•0006891791
•000688705
•000688231
•000687757
•000687285
•000686813
•000686341
•000685871
•000685401
•000684931
•000684462
•000683994
•000683527
•000683060
•000682593
•000682128
SQUARES, CUB£», .
ROOrs^ AND
RBCIPBOCALS. 685
Na
SqoaCB
Ooba
Square Boot
OabeRoot
Bedprocal
1467
215 20 89
8 167 114 668
38-3014360
11-3625735
-0006816688
1463
2 15 50 24
8 163 576 282
38-3144881
U-8661547
-0006811989
1469
2 15 79 61
8 170 044 709
88-3275368
11-3677847
•0006807362
1470
2 16 09 00
8 176 528 000
38-3405790
11-3703136
-0006802721
1471
2 16 88 41
8 188 010 111
38-3636178
11-3728914
-0006798097
1472
2 16 67 84
8 189 606 048
38-3666622
11-8764679
•0006793478
1478
2 16 97 29
8 169 010 817
38-3796821
11-8780433
•0006788866
1474
2 17 26 76
8 202 624 424
38-3927076
11-3806175
•0006784261
1475
2 17 66 25
8 209 046 875
38-4057287
11-3831906
•0006779661
1476
2 17 85 76
8 216 578 176
38-4187454
11*8857625
•0006VV6068
1477
2 18 15 29
8 222 118 888
88-4317677
11*8883332
-0006770481
1478
2 18 44 84
8 228 667 862
38-4447656
11*3909028
-0006765900
1479
2 18 74 41
8 235 226 289
38-4577691
11-3934712
•0006761825
1480
2 19 04 00
8 241 792 000
38-4707681
11-8960384
•0006756757
1481
2 19 83 61
8 248 867 641
38-4837627
11-8986045
-0006752194
1482
2 19 68 24
8 264 962 168
38-4967580
11-4011695
•0006747688
1488
2 19 92 89
8 261 645 687
38-5097390
11-4037832
♦0006743088
1484
2 20 22 56
8 268 147 904
38-6227206
11*4062959
•0006738544
1485
2 20 52 26
8 274 769 126
38-5366977
11-4088574
•0006784007
1486
2 20 81 96
8 281 879 256
38-5486706
11-4114177
•0006729476
1487
2 21 11 69
8 288 008 808
38-6616389
11-4139769
•0006724960
1488
2 21 41 44
8 294 646 272
38-6746030
11*4165849
•0006726430
1489
2 217121
8 801 298 1^9
38-6875627
U-4190918
•0006715917
1490
2 22 01 00
8 807 949 000
38-6005181
11-4216476
•0006711409
1491
2 22 80 81
8 814 618 7?!
38*6134691
11-4242022
•0006706908
1492
2 22 60 64
8 821 287 488
38-6264156
11*4267656
•0006702418
1498
2 22 90 49
8 827 070 167
38-6398582
U-4293079
•0006697924
1494
2 28 20 86
8 884 661 784
38-6622962
11-4818591
•0006698446
1495
2 28 60 25
8 841 862 876
88-6652299
11-4844092
-0006688968
1496
2 28 80 16
8 848 071 936
38-6781698
11-4869681
'0006684492
1497
2 24 10 09
8 864 790 478
88-6910848
11^395059
•0006680027
1498
2 24 40 04
8 861 517 992
88-7040050
11*4420525
-0006675567
1499
2 24 70 01
8 868 254 499
38-7169214
11*4445980
•0006671114
1500
226 00 00
8 876 000 000
88-7298385
11-4471424
-0006666667
1501
2 26 80 01
8 881 764 601
88-7427412
11-4496857
•0006662225
1502
2 26 60 04
8 888 518 008
88-7666447
11-4522278
•0006667790
1503
2 25 90 09
8 808 290 527
88-7685439
11-4547688
-0006648860
1504
2 26 20 16
8 402 072 064
88-7814389
11-4573087
-6006648986
1506
2 26 50 25
8 406 8B2 626
38-7948294
11-4598474
-0006644518
1506
2 26 80 86
8 415 662 216
38-8072158
U-4623850
-0006640106
1507
2 27 10 49
8 422 470 848
38-8200978
11-4649215
•6006685700
1508
2 27 40 64
8 429 288 512
38-8329757
11^74668
-0006681800
1509
2 27 70 81
8 486 115 229
88-8458491
U-46d99U
•0006626905
1510
228 0100
8 442 951 000
SS-85871^
11 •47^5^42
•0006622517
1511
2 28 81 21
8 440 796 881
38-8715884
11-4750562
•0006618134
1512
2 28 61 44
8 456 649 728
38-8844442
11-4775871
•0006618767
1518
2 28 91 69
8 468 512 697
38-8973006
11-4801169
•0006609886
1514
2 29 21 96
8 470 884 744
38 9101529
11-4826455
•0006606020
1615
229 62 26
8 477 266 876
38-9230009
11-4851781
•0006600660
ess
WtUABBS, OUBBS, IMOtB, AND KBCIPBOCAU.
Ko.
Sqiiart
Oabe
Square Boot
CabeBoot
Bedprocal
1516
2 29 82 66
8 484 156 096
38-9358447
11-4876995
•O0O6696306
1617
2 80 12 89
8 491 055 418
38-9486841
11-4902-249
-0006691958
1518
2 80 43 24
8 497 963 882
38-9616194
11-4927491
-0006587615
1619
2 80 78 61
8 604 881 859
38-9748505
11-4952722
•0006583278
1620
2 81 04 00
8 511 808 000
38-9871774
11-4977942
•0006578947
1621
2 81 84 41
8 618 743 761
39-0»00000
11-5003151
-0006574622
1522
2 81 64 84
8 520 688 648
39-0128184
11-5028348
•0006570302
1628
2 81 95 29
8 582 642 667
39-0256326
11-5063535
-0006665988
1624
2 82 26 76
8 589 605 824
39-0384426
11-5078711
-0006561680
1525
2 82 66 25
8 546 578 125
39-0512483
11-5103876
•0006557377
1626
2 82 86 76
8 558 659 576
39-0640499
11-5129030
•OOO6553080
1627
2 88 17 29
8 560 550 188
39-0768473
11-5154173
•0006548788
1628
2 88 47 84
8 567 549 952
39-0896406
11-6179306
-0006544503
1629
2 88 78 41
8 674 558 889
39-1024296
11-5204425
-0006640222
1530
2 84 09 00
8 581 577 000
39-1152144
11-5229585
•0006535948
1581
2 84 89 61
8 588 604 291
39-1279951
11-5254634
•0006531679
1682
2 84 70 24
8 595 640 768
39-1407716
11-5279722
-0006527415
1588
2 85 00 89
8 602 686 437
39-1535439
n-5304790
•0006523157
1684
2 86 81 66
3 609 741 804
89-1663120
11-5329865
-0006518906
1686
2 85 62 25
8 616 805 375
39-1790760
11-5354920
•0006514658
1686
2 85 92 96
8 623 878 656
39-1918359
11-5379965
-0006510417
1587
2 86 23 69
8 680 961 158
39-2045915
11-5404998
•0006606181
1588
2 86 54 44
8 638 062 872
39-2173431
11-5430021
-0006501951
1589
2 86 86 21
8 645 158 819
39-2300905
11-5455033
•0006497726
1540
2 37 16 00
8 652 264 000
39-2428337
11-5480034
■0006493506
1541
2 87 46 81
8 659 388 421
39-2555728
11-5505025
-0006489293
1542
2 87 77 64
8 666 512 088
39-2683078
11-5530004
-0006485084
1548
2 88 08 49
3 678 650 007
39-2810387
11-5554973
•0006480881
1544
2 88 89 86
8 680 797 184
39-2937654
11-5579931
-0006476684
1545
2 88 70 25
8 687 953 625
39-3064880
11-5604878
-000647-2492
1546
2 89 01 16
8 696 119 886
39-3192065
11-6629815
•0006468305
1547
2 89 82 09
8 702 294 828
39-3319208
11-5654740
•0006464124
1548
2 89 63 04
8 709 478 592
39-3446311
11-5679655
•0006459948
1549
2 89 94 01
8 716 672 149
39-3573373
11-5704559
•0006455778
1550
240 25 00
8 728 875 000
39-3700394
11-5729453
•0006451613
1561
2 40 56 01
8 781 087 151
39-3827873
11-5754336
•0006447453
1552
2 40 87 04
8788 808 608
39-3954312
11-5779208
•0006443299
1553
2 41 18 09
8 745 689 877
39-4081210
11-5804069
•0006439150
1664
2 41 49 16
8 752 779 464
39-4208067
11-5828919
0006435006
1555
2 4180 25
3 760 028 875
39-4334883
11-5853759
•0006430868
1556
2 42 11 36
8 767 397 616
39-4461658
11-5878588
•0006426735
1567
2 42 42 49
8 774 566 693
39-4588393
11-6903407
•0006422608
1568
2 42 78 64
8 781 888 112
39-4715087
11-6928215
•0006418485
1559
2 43 04 81
8 789119 879
39-4841740
11-5953013
•0006414368
1660
2 43 86 00
8 796 416 000
39-4968353
11-5977799
•0006410256
1661
2 43 67 21
8 803 721 431
39-5094925
11-6002576
-0006406150
1662
2 43 98 44
3 811 066 828
39-5221457
11-6027342
-0006402049
1568
2 44 29 69
3 818 860 547
39-5347948
11-6052097
•0006397953 ,
1664
2 44 60 96
8 825 694 144
39-5474399
11-6076841
-0006393862
SQUARBS, CUBES, BOOI8, AJSD BBCIPBOGALS. 68
No.
Square
2 44 92 25
Oobe
Square Root
CabeBoot
Beciprocal
1666
1 8 888 087 126
39-5600809
11-6101575
-0006d8977(
156612 45 28 56
8 840 889 496
39-5727179
11*6126299
-000638569(
1667
2 45 54 89
8 847 761 268
39*5853508
11-6151012
-000638162]
156B
2 45 86 24
8 865 122 482
39-5979797
11-6175716
-0006377551
1669
2 46 17 61
8 862 608 009
39-6106046
11-6200407
-0006373486
1670
2 46 49 00
8 869 898 000
39-6232255
11-6225088
-0006369427
1671
2 46 80 41
8 877 292 411
39-6358424
11-6249759
•000636537S
1672
2 47 11 84
8 884 701 248
39-6484552
11-6274420
-000636132S
1573
2 47 48 29
8 892 119 617
39-6610640
11-6299070
-000636727S
1574
2 47 74 76
8 899 647 224
39-6736688
11*6323710
-000635324C
1576
2 48 06 26
8 906 984 876
39*6862696
11*6348339
•0006349206
1676
2 48 87 76
8 914 480 976
39-6988665
11*6872957
-0006345178
1677
2 48 69 29
8 921 887 083
39-7114593
11-6397566
-0006341154
1578
2 49 00 84
8 929 852 662
39-7240481
11*6422164
•0006337186
1679
2 49 82 41
8 986 827 689
39-7366329
11*6446751
-0006333122
1580
2 49 64 00
8 944 812 000
397492138
11-6471329
•0006329114
1581 ; 2 49 96 61
8 951 605 941
39-7617907
11*6495895
•0006325111
1582 2 60 27 24
8 959 809 868
39-7743636
11*6520452
-0006321113
1688 2 60 58 89
8 966 822 287
39-7869325
11*6544998
-0006317119
1584 2 50 90 66
8 974 844 704
39*7994976
11-6569534
•0006313131
1585 1 2 51 22 26
8 981 876 625
39-8120585
11-6594059
•0006309148
1586 2 5158 96
8 989 418 066
39*8246155
11-6648574
*OOO63O5170
1687
2 51 85 69
8 996 969 008
39*8371686.
11*6643079
•0006301197
1688
2 52 17 44
4 004 629 472
39-8497177
11*6667574
•0006297229
1689 2 52 49 21
4 012 099 469
39*8622628
11*6692058
*0006293266
1590 , 2 52 81 00
4 019 679 000
39-8748040
11*6716532
*0006289308
1691 1 2 58 12 81
4 027 268 071
39-8873413
11*6740996
*0006285355
1692 2 58 44 64
4 084 866 688
39-8998747
11*6765449
*0006281407
1698 2 53 76 49
4 042 474 857
89-9124041
11-6789892
•0006277464
1594 2 54 08 86
4 050 092 584
39-9249295
11-6814325
•0006273526
1595 2 54 40 26
4 057 719 875
39*9374511
11*6838748
•0006269592
1596 2 64 7216
4 065 856 786
39*9499687
11-6863161
•0006265664
1697 2 56 04 09
4 078 008 178
89-96248-24
11-6887568
•0006261741
1598 2 55 86 04
4 080 659 192
89*9749922
11-6911955
•0006257822
1699 2 55 68 01
4 088 824 799
39*9874980
11-6936837
•0006253909
1600 2 56 00 00
4 096 000 000
40*0000000
11-6960709
•000625000C
1601 2 56 82 01
4 108 684 801
40*0124980
11-6986071
•0006246096
1602 2 56 64 04
4 111 879 208
40*0249922
11-7009422
•0006242197
1608 2 56 96 09
4 119 088 227
40*0374824
11-7033764
*000623830g
1604 2 57 2816
4 126 796 864
40*0499688
11-7058095
•0006234414
1605 2 57 60 25
4 184 520 125
40*0624512
11*7082417
*00062305SO
1606 2 67 92 86
4 142 263 016
40-0749298
11-7106728
*0006226650
1607 2 58 24 49
4 140 995 648
40*0874045
11-7131029
•0006222776
1608 2 58 66 64
4 167 747 712
40-0998758
11-7155320
•0006218905
1609 2 58 88 81
4 166 609 629
40*1123423
11-7179601
•0006215040
1610 . S 59 21 00
4 178 281 000
40*1248058
11-7203872
-0006211180
1611 2 59 58 21
4 181 062 181
40-1372646
11*7228133
-0006207325
1612 2 59 86 44
4 188 852 928
40*1497198
11-7252384
•0006203474
1618 2 60 17 69
4 196 668 897
40-1621718
11*7276626
-0006199628
B88 BQUARCaK, CtlB^, BOOTS, AKD EBdPAOCAlift.
He.
1614
Sqoate
Cube
BqaanRoot
Cube Root
Reciprocal
2 60 49 96
4 264 468 644
40-1746188
11-7800855
•0006195787
•OO06191950
1616 ' 2 60 82 25
4 212 288 876
40-1870626
11-7325076
1616 2 61 14 56
4 220 112 896
40-19950-26
11*7349286
•0006188119
1617 2 6146 89
4 227 952 118
40-2119886
11-7378487
•0006184292
1618 2 6179 24
4 285 601 082
40-2243707
11-7897677
'OOO618O470
1619 2 621161
4 243 669 659
40-2867990
11-7421868
•0006176652
1620 2 62 44 00
4 261 528 000
40*2492286
11-7446029
•00^6172840
1621 2 62 76 41
4 269 406 061
40-2616443
11-7470190
•0006169031
1622 2 68 08 84
4 267 298 848
40-2740611
11-7494341
•0006165228
1626 i 2 68 41 29
4 275 191 867
40-2864742
11-7618482
-0006161429
1624 1 2 68 78 76
4 288 098 624
40-2988834
11-7542613
•0006167636
1626 ' 2 64 06 25
4 291 015 626
40-3112888
11-7666734
•0006153846
1626 ; 2 64 88 76
4 298 942 876
40-3236908
11-7690846
•0006160062
1627
2 64 71 29 i 4 806 678 888
40-8360881
11-7614947
•0006146282
1628
2 65 08 84
4 814 626 162
40-3484820
li-7639039
•0006142606
1629
2 65 86 41
4 822 781 189
40-8608721
11-7668121
•0006138735
1680
2 66 69 00
4 880 747 000
40-8732585
11-7687193
•0006134969
1631
2 66 01 61
4 888 722 691
40-8856410
11-7711255
•0006131208
1682
2 66 84 24
4 846 707 968
40-3980198
11-7735306
•0006127451
1638
2 66 66 89
4 864 708 187
40-4103947
11-7759349
•0006128699
1684
2 66 99 66
1 862 708 104
40-4227668
11-7783881
•0006119951
1686
2 67 82 26
4 870 722 876
40-4351832
11-7807404
•0006116208
1686
2 67 64 96
4 878 747 466
40-4474968
11-7831417
•0006112469
1687
2 67 97 69 4 886 781858
40-4698666
11-7856420
•0006108736
1636
2 68 80 44
4 894 826 072
40-4722127
11-7879414
•0006105006
1639
2 68 68 21
4 402 880 119
40-4845649
11-7908397
-0006101281
1640
2 68 96 00
4 410 944 000
40-4969185
11-7927371
•0006097561
1641
2 69 28 81 4 419 017 721
40-5092682
11-7951335
•0006098845
1642
2 69 6164 4 427 101288
40-5215992
11-7975289
•0006090134
1648
2 69 94 49 4 486 194 707
40-6339864
11-7999234
•0006086427
1644
2 70 27 86 4 448 297 984
406462699
n-8038169
•0006082725
1645 2 70 60 25 4 451411125
40*5586996
11-8047094
•0006079027
1646 2 70 98 16 : 4 459 684 186
40*6709256
11-8071010
•0006075334
1647
2 71 26 09 4 467 667 028
40*6832477
11-8094916
•0006071645
1648
2 71 69 04
4 475 809 792
40-6956668
11-8118812
•O006067961
1649
2 71 92 01
4 488 962 449
40-6078810
11-8142698
•0006064281
1650
2 72 26 00 4 4^2125 000
40-6201920
11-8166576
•0006060606
1661
2 72 66 01
4 500 297 451
40*6824998
11-8190443
•0006066935
1652
2 72 91 04
4 508 479 808
40-6448029
11-8214301
-0006058269
1668
2 78 24 09 ; 4 516 672 077 1
40-6671027
11-8238149
•000604»607
1654
2 78 67 16
4 524 874 264
40-6693988
11-8261987
•0006046949
1665
2 78 90 26
4 683 086 875
40-6816912
11-8385816
•0006042296
1656
2 74 28 86
4 641 808 416
40*6939799
11*8809634
•O00603S647
1657
2 74 66 49
4 549 640 898
40*706264S
118138444
•0006035003
1658
2 74 89 64
4 657 782 812
40*7185461
11-8857244
•0006031863
166d.
2 76 22 81 ' 4 666 084 179 1
40-7808287
11-8881034
"0006027728
1660 2 76 56 00 ; 4 574 296 000
40*7480^76
11-8404815
'0006024096
1661 2 75 89 21 ' 4 682 567 781
40-7^.^3677
11-84285H6
•OOO6O2O470
1662
2 76 22 44 4 690 649 528
40-7676348
11*8452348
*OOOe016847
-
HqVARE&j OUDBS, ^OOSS, AK^ &£03pftOOAI«S.
689
^Ui,
1663
1664
1666
1666
1667
1668
1669
1670
1671
1672
1678
1674
1675
1676
1677
1678
1679
16Q0
1681
.1682
1688
1684
1685
1686
1687
1688
1689
1690
1691
1692
1698
1694
1695
1096
1697
1608
■]699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
Square
8^55 69
2 76 88 96
2 77 22 25
2 77 65 56
2 77 88 89
2 78:22 24
278 55 61
2 78 89 00
.2 79 22 41
2 79 55 84
2 79 89 29
2 80 22 76
2 aO 66 25
2 80 89 76
2 ai 28 29
2 81 56 84
2 81 80 41
2 82 24 00
2^82 57-61
^ 82 91 24
2 88 24 89
2 83 58 56
283 92 25
284,25 96
284 69 69
2 84 08 44
285:27 21
2 85 61 00
2 85 04 81
286 28 64
28662 49
2 86 96 86
287 80 25
Ji 87^4 16
.287 98:00
ii 68 82 04
288 66 01
a89 00 00
2 89 84 01
^89 68 04
2 90 02 09
200 36 16
2 00 70 25
2 9104 86
a 91 88 49
2 91 72 64
2 92 06 81
2 92 41 00
2 92 75 21
Cube
4 699 141 247
4 607 442 994
4 615 754 626
4 624 076 296
4 682 407 968
|4 640 749 632
4 649101809
4 657 463 000
^ 665 834 711
4 674 916 448
4 6^2 608 217
4 691 010 024
4699 421875
4 707 843 776
4 716 275 738
4 724 717 752.
4 788 169 889
4 741 632 000
4 750 104 241
4 758 586 568
4 767 078 987
4 775 581 504
4 784 094 126
4 792 616 856
4 801149 708
4809 692 672.
4818 245 769.
4 826 909.000
4 885 382371
4 848 965 688
4 852 559 567.
4 861163 884^
4 869 777 875
4 878 401586
4 887 035 873
4805 680 892:
4904 335 099
4918 000 000.
.4 921675 101
4 980 360 408;
4 989 055 027
4 947 761 064
4 956.477625
4 965203 816
4 978 940 248
4 982686 912
4 991 443 829
6 000 211000.
6 008..288 481
Square Root
40-7798970
40-7921561
40-8044115
40-8166633
40-8289113
40-8411657
40*8538964
40*8656335
40-8778669
40*8900966
40*9028227
40*9145451
40*9267688
40*9389790
40'95I1905
40*9633983
40-9756025
40*9878031
41*0000000
41-0121933
41*0243830
41-0365691
41*0487515
41*0609303
41-0731055
41*0852772
41*0974462
41*1096096
41*1217704
41*1339276
41*1460812
41-1582313
41*1703777
41*1825206
41*1946599
41*2067966
41*2189277
41-2310663
41-2431812
41-2563027
41-2674205
41-2796849
41-2916466
41-8037629
41*8168665
41-8279666
41-8400632
41*3521463
41*8642358
Cube Root ,
U«476100
11-8499843
11*8623676
11*8547299
11-8671014
11-8594719
11-8618414
11-86^2100
11*8665776
11-8689443
11-8713100
11-8736748
11-87603^7
11-8784016
11-8807636
11-8831246
11-8854847
11-8878439
11-8902022
11-8925696
11-8949159
11-8972713
11-8996268
11-9019793
11-9043319
11-9066836
11-9090344
11-9113843
11-9137332
11-9160812
Xl-9184283
11-9207744
11-9231196
11-9264639
11-9278073
11-9301497
11-9324913
11-9348319
11-9371716
11-9395104
11-9418482
11-9441862
11-9465213
11-9488564
11-9511906
11-9535239
11-9658563
11-9581878
11-9606184
Redpeocal
-0006013229
^006009615
-0006006006
•O006002401
^005998800
•0006995204
K)005991612
•0005988024
-0005984440
-0005980861
-0006977286
-0006973716
-0005970149
•0005966587
•0006963029
-0006959476
-0005956926
*0006952381
•0006948840
*0006945308
*0006941771
•0006938242
•0006984718
-0005981198
-0006927682
•000.6924171
•0005920663
•000^917160
•0006913661
•0006910165
-0006906675
-0006903188
-0006899705
-0005896226
•0006892762
•0005880282
-0005885815
-0005882363
•0006878895
-0006876441
-0006871991
-0006868545
•0005866103
-0006861665
-0005858231
-0005854801
•0005851375
-0005847963
-0005844535
Yy
90 CM^UAEBS, CUBES, BOOTS, AND BBOIPROOALS.
Bqnan
2 98
3 98
2 98
3 9i
3 94
3 94
3 96
3 95
3 95
3 96
3 96
3 96
8 97
3 97
3 97
3 98
3 98
3 98
3 99
3 99
2 99
8 00
800
8 01
8 01
8 01
8 03
8 03
803
8 08
8 08
8 08
8 04
8 04
8 04
8 05
8 06
8 05
8 06
8 06
8 06
8 07
8 07
8 08
8 08
8 08
8 09
8 09
8 09
09 44
48 69
77 96
13 86
46 66
8089
16 34
49 61
84 00
18 41
63 84
87 29
2176
66 36
90 76
26 29
69 84
94 41
29 00
68 61
98 24
82 89
67 66
02 25
86 96
7169
06 44
4121
76 00
10 81
46 64
80 49
16 86
60 26
8516
30 09
65 04
90 01
26 00
60 01
95 04
80 09
65 16
00 26
85 86
70 49
05 64
40 81
76 00
Oobt
6 017
5 026
6 086
5 044
5 068
5 061
5 070
5 079
6 088
5 097
6106
6116
6124
5183
6141
5160
6169
6168
5177
6186
6196
6 204
6 218
6 222
6 281
6 240
249
268
268
277
286
2 6
6 804
6 818
6 822
6 881
841
850
859
86S
877
886
896
405
414
428
488
442
461
6
6
6
6
6
6
776188
674 097
883 844
300 875
029 606
868 818
718 383
577 959
448000
838 861
319 048
130 067
081434
968136
886176
837 588
780 863
748 489
717 000
700891
696168
699 887
714 904
740 876
776 266
822 668
879 272
946 419
024 000
112 021
210 488
819 407
488 784
668 626
708 986
869 728
020 992
193 749
875 000
667 751
771008
984 777
309 064
448 876
689 216
945 098
211 612
488 479
776 000
SqnanBoofe
41-3763217
41*8884042
41*4004881
41-4125585
41*4246304
41-4366987
41*4487636
41*4608249
41*4728827
41*4849870
41-4969878
41-5090851
41*6210790
41*5831193
41-5461561
41*6571«95
41*5692194
41-6812467
41*5932686
41-6052881
41-6178041
41*6293166
41-6413266
41*6538312
41-6653333
41*6773319
41*6893271
41-7013189
41-7133072
41-7252921
41-7372735
41-7492515
41-7612260
41-7781971
41-7851648
41-7971291
41*8090899
41-8210473
41-8330013
41*8449519
41-8568991
41*8688428
41*8807832
41*8927201
41-9046587
41-9165838
41-9285106
41*9404339
41*9523589
CabeBoot
11-9628481
11-9651768
11-9675047
11*9698317
11*9721577
11*9744829
11-9768071
11-9791304
11*9814528
11-9887744
11*9860950
11-9884148
11*9907336
11*9930516
11*9958686
11-9976848
12-0000000
12-0023144
12-0046278
12-0069404
12-0092621
12-0115629
12-0138728
12*0161818
12*0184900
12*0207978
12-0281087
12-0254092
12^)277188
12-0800175
12*0323204
12-0846228
12*0369238
12-0392236
12r0415229
12-0488218
12*0461189
12*0484156
12*0607114
12O530063
12-0553003
12-0575986
12-0598859
12-0621773
12*0644679
12»0667576
12*0690464
12*0718344
12*0736215
Bedprocal
■0006841121
•0006837712
-0006834306
-0006830904
•0005827506
•0006824112
•0005820722
•0005817336
•0006818953
•0005810575
*0006807201
•0006803831
•0005800464
*0005797101
•0006798743
*0005790d88
•0006787087
-0005783690
•0006780347
-0006777008
-0006773672
•0005770340
•0005767013
•0005763689
0005760369
-0005767052
"0005763740
•0005750431
•0005747126
•0005748825
-0005740528
•0005787285
•0006783945
*0006780659
•0005727377
-0005724098
*0005720824
•0005717653
•0005714286
•0005711022
•0005707763
•0005704507
•0005701254
•0005698006
•0005694761
•0005691520
0005688282
-0005685048
'0005681818
SQUARES, CUBB8, ROOTS, AMD RBCIPBOCALS.
.691
Ka
1761
17d8
1768
1764
1766
1766
1767
1768
1760
1770
1771
1772
1778
1774
1776
1776
1777
1778
1770
1780
1781
1782
1788
1784
1785
1786
1787
1788
1780
1790
1791
1792
1798
1794
1796
1796
1797
1798
1799
1800
1801
1802
1808
1804
1806
1806
1807
1808
1809
Square
8 10 11 21
810 46 44
8 10 81 69
8 11 16 96
8 11 62 26
81187 66
812 22 89
812 68 24
812 98 61
818 29 00
818 64 41
8189984
S 14 86 29
8 14 70 76
816 06 26
8 15 41 76
8 15 77 29
8 16 12 84
8 16 48 41
816 84 00
81719 61
8 17 56 24
8 17 90 89
818 26 66
8 18 62 25
818 97 96
8 19 88 69
8 19 69 44
8 20 05 21
8 20 41 00
8 20 76 81
8 21 12 64
8 21 48 49
8 21 84 86
822 20 25
8 22 66 16
8 22 92 09
8 28 28 04
8 28 64 01
8 24 00 00
8 24 86 01
8 24 72 04
8 25 08 09
8 25 44 16
8 25 80 25
8 26 16 86
8 26 52 49
8 26 88 64
8 27 24 81
Oube
5 461 074 081
5 470 882 728
5 479 701 947
6 489081744
5 498872135
5507 728 096
5 617 084 668
5626 456882
5586 880 600
5645 288000
5 654 687011
5664 051648
6 678 476 917
5 682912 824
5692850876
6 601 816 576
6 611 284 488
6 020 762 952
6 680 252180
5 689 752 OOO
5 649 262541
5 658 788 768
5 668 815 687
5 677 858 804
5 687 411 626
5 698975 656
5 706 550 408
5 716 185 872
5725 782 069
5 785 889 000
5 744 966 671
6 754 586 088
5 764 224 257
5 778 874 184
5 788 584 875
5 798 206 886
5 802 888678
5 812 681 592
5 822 285 899
5882 000 000
6 841 725 401
5 851 461 608
5 861 208 627
6 870 966464
6 880 786 126
5 890 514 616
6 900 804 948
5 910 106 112
5 919 918 129
Square Root
41-9642706
41'9761837
41*9880935
42-0000000
42-0119031
42-0238028
42-0856991
42-0476921
42-0694817
42-0713679
42*0832508
42*0951304
42-1070066
42*1188794
42-1307488
42-1426150
42-1544778
421663373
42-1781934
42-1900462
42-2018967
42-2137418
42-2255846
42-2374242
42-2492603
42-2610932
42*27292271
42-2847490
42-2965719
42-3083916
42*8202079
42-3320210
42-3438807
42-3556371
42-3674403
42*3792402
42*3910368
42*4028301
42*4146201
42-4264069
42-4381903
42-4499705
42-4617476
42*4736212
42*4862916
42-4970587
42-5088226
42-5206833
42-5323406
Cube Boot
12*0769077
12-0781930
12*0804775
12-0827612
12*0850439
12*0873258
12*0896069
12-0918870
12*0941664
12*0964449
12*0987226
12*1009993
12*1032753
12*1056503
12*1078245
12*1100979
12*1123704
12*1146420
12*1169128
12-1191827
12*1214618
12-1237200
12-1259874
12*1282539
12*1305197
12*1827846
12-1860485
12*1873117
12*1398740
12-1418355
12-1440961
121463659
12*1486148
121608729
121631302
12*1663866
12-1676422
12-1598970
12*1621509
12*1644040
12*1666562
12*1689076
12*1711582
12*1734079
12-1756669
12-1779050
12-1801622
12*1823987
12*1846448
Reciprocal
*0006678592
*0005675869
•0005672150
•0005668934
•0006666722
•0005662614
*0< 106659310
•0006666109
•0005662911
•0006649718
•O005646627
•0005643341'
•0006640168
•0006636979
•0006633808
•0005630631
-0005627462
•0005624297
•0006621135
•0006617978
-0005614823
•00066U672
•0005608525
-0005606381
-0006602241
-0005699104
•0006696971
-0005692841
•0005589716
•0005686692
-0006688473
-0005680867
•0005577246
•0005574136
•0006671031
•0005567929
•0005564830
-0006661785
•0005658644
•0006555656
-0005552471
•0005649390
•0005646312
-0006548237
•0005540166
-0005537099
-0005534034
•0005630973
•0006527916
()92 SQUAftBS, CUBES, KOOTS, AND BBClPBOCAiaL
Ho.
Squaie
Oabe
1610
1811
181S
1818
1814
1815
1816
1817
1818
1819
isao
1881
1882
1888
1884
1886
1888
1887
1888
1889
1880
1881
1888
1888
1884
1886
1886
1887
1888
1889
1840
1841
1848
1848
1844
1845
1846
1647
1848
1849
1850
1851
1853
1658
1654
1865
1866
1667
1858
8 87 61 00
8 87 97 81
8 88 8144
8 86 69 69
8 89 05 96
8 89 48 86
8 89 78 66
8 8014 89
8 80 6184
880 87 61
8 81 84 00
8 61 60 41
88196 84
8888189
8 88 69 76
888 06 86
8 88 48 76
8 88 79 89
8 8415 84
8 84 68 41
8 84 89 00
8 66 85 61
8 86 68 84
8 86 98 89
8 86 85 66
8 86 78 85
8 87 08 96
8 87 45 69
8 87 68 44
8 88 19 81
8 88 66 00
8 88 98 81
8 69 89 64
8 89 66 49
840 08 86
8 40 40 85
840 7716
8 41 14 09
8 41 61 04
B 41 68 01
8 48 85 00
8 43 68 01
8 48 99 04
8 48 86 09
8 48 7816
8 44 10 85
8 44 47 86
8 44 84 49
6 46 8a 64
6S89741000
6989 674 781]
6 949 410^98
6 959 874 7S97
6 969141144
5 979€a8B75
6 988 906 496
6 996 606 6118
6 006716 488
6 018 686 859
6088 6680001
6 088 610 661
6048464 846
6 068 438767
6068 404 884
6 076 390686
6088867 976
6008 896 888
6 106 415 658
6118445 789
6188 467 000
6 188 689 191
6146*608 868
6 166:676 637
6106761704
6176 8K>7 896
6168 966.066
6 199 1063858
6 909iEL8 478
6 819 858 719
6 889 604.000
6 8391606 881
6 849 889666
6 860084 107
6 870 219 684
6 880 436 186
6890 648 786
6 800 678 488
6 811 113 198
6 8Sa 868 049
6 831636 000
6 841 898 061
6 353 188 808
6 868 477477
6 87iS788 664
6 868 101 875
6 398 430 016
6 408 769 793
64U130i718
8Q«wreBo9t
42-5l409l<8
43%')556456
425673988
42-5793377
42-5910769
42-61)28168
4d>01455a6
42-3262829
42-6880112
42-6ft97862
42*6614560
42'67817»6
42*0848919
42-6966046
42-7088180
42-7200187
42-7317212
42-7484206
42-766ai67
42'7€66096
42-7764992
42-7901658
42-8016691
42-<13d492
42-8252262
42-8366999
42-8485706
42-8602380
42-8719022
42-8885633
42-8952212
42-9068759
42-9186275
42-9301769
42-9416211
42-9584632
42-9661021
42-9767379
42-9883706
48-0000000
43-Oa 16263
43-0232496
43-0348696
43-0464865
43-0581003
480697109
43-0813185
430929228
43-1046241
Cube Boot
12'1666891
12-1691381
ie«a!9U3762
18*1986186
18*1966599
18*1981006
lCt2003404
ld*2025794
12*2048176
12'a070549
12-2002915
12-2115^72
12-2187621
12-2169962
12:2162295
12*2204620
12-2226986
12*2249244
12-2271544
12-2293636
12*2316120
13'8388886
12*2360663
12-2362923
12-2405174
12-2427418
12*2449653
12*24.71680
12-2494099
12*2516310
12*2538513
12*2580706
12*2682896
12*2605074
12-2627246
12*2649408
12*2671568
12-2693710
12*2715849
12*2737960
12*2760108
12-2782216
12-2804826
12r2826424
12-2648616
12-2870598
12-2682673
12-2914740
12-2986690
Bsciprookl
10005524862
•0005521611
^05518764
'0005516720
'OOOdSlSO?)
'0006509642
^0006606608
^05508677
*0005600560
"0005497526
'0006494605
0005491488
-0005468474
-0006466464
"0005462456
'0006479462
'0005476461
*0005478464
-00O647O460
«0005467469
•0005464481
'00064^496
;0005456615
•0005455587
*0005452663
•0005449691
•0005446623
*00ed448668
•0006440696
•0006437738
-0005434783
•0005431881
-0005428682
*0005426936
*0006422993
•0005420054
-0005417118
•0006414185
•0005411255
•0005406329
-0005405405
•0005402485
•0005399568
-0005396654
*000539874d
•0005390836
•0005887931
*000d385080
-0005882X31
SQ17ARES, CUBES, ROCKFS, AND HBCKPBOCAJLGU
696
Vo.
1889
I860
1861
1862
1063
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
187ft
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1898
1894
1896
1896
1897
.!&%
1899
1900
1901
1902
1903
1904
1906
1906
1907
Sqoan
8 45 58 81
8 45 96 00
3 46 83 21
3 46 70 44
8 47 Ot 69
347 44 96
8 47 8i 25
8 4819 56
8 48 56 89
8 48 94 24
8 49 81 61
8 49 69 00
850 06 41
350 43 84
3 50 81 29
35118 76
3 51 5^ 25
3 51 98 76
352 3129
3 52 68 84
353 06 41
353 44 00
3 53 8161
3 54 10 24
8 54 56 89
3 54 94 66
3 55 88 25
3 55 W 96
3 56 07 69
3 56 46 44
3 66 86 21
3 57 2100
3 57 58 81
3 57 96 64
8 58 34 49
3 58 72 86
3 59 10 25
3 59 48 16
8 59 86 09
3 60 'il4 04
8 60 62 01
8 61 00 00
8 61 38 01
3 61 76 04
3 62 14 09
8 62 52 16
8 62 90 25
8 63 28 86
8 68 08 49
Obbe
6 424 482 779
6 484 856 000
6 445 240 361
6 455 685 926
6 466 042^7
6 476 460 544
6 486 889 626
6 497829 696
6 607 781 868
6 618244 092
6 628 717 909
6 531>203 000
6 649699 811
6 560206 848
6 670 726 617
6 681 256 624
6 691 796 876
6 602 849 876
6 612 913 188
6 628 488 163
6 634 074 439
6 644 672 000
6 655 280 841
6 665 900 968
6 676 532 887
6 687175104
6 697 829 126
6 708 494 456
6 719171108
6 729 869 072
6 740 568 869
6 751 269 000
6 761 990 971
6 772 724 288
6 788,468 967
6 794 224 984
6 804 992 876
6 815 771 186
6 826 561 278
6 887 362 792
6 848 175 .699
6 859 000 000
6 869 835 701
6 880 682 808
6 891 541 827
6 902 411 264
6 918 292 626
6 924 185 416
6 936 089 648
SqaareBoot
43-1161223
431277173
43-1898092
431608980
43-1624837
43-1740663
43-1856468
48-19^2221
48-2087954
48-2203666
48-2319326
43-2434966
43-2850676
43-2666153
43-2781700
48-2897216
43-9012702
43-3128157
\ 43-3243680
48*3358973
48-3474336
43-3589668
43-3704969
43-3820289
43-3935479
43-4050688
43*4165867
43-4281016
43-4396132
434511220
43-4626276
43-4741302
43-48«6298
43-4971263
43^6086198
43-5201103
43-5815977
43-5430821 \
43-5545636
43-56«0418
43-577M71
43^6889094
43-6064687
43-6119B49
43-6233882
43-6348486
43-64$3057
43-6577599
43-6692111
Cube Boot
12-2968861
12-2980895
12-3002980
12-3024968
12-3046978
12-3068990
12-3090994
12-3112991
12-3134979
12-3156959
13-3178932
12-3200897
12*3222854
12^3244803
12-8266744
12-3288678
12*3310604
12-3332522
12-3354432
12-8376834
12-8898229
12-3420116
19-344199&
1SS463866
12-3485730
12-3507586
12-3629434
12-3561274
12-8673107
12-3694932
12-3616749
12-8638659
12-3660861
12-3682155
12-8703941
12-8725721
12-3747492
12-8769256
12-8791011
12-8812759
12*3834600
12-3856233
12-3877959
12-8899676
12-5921886
12-8943089
13-8964784
12-3986471
12-4008151
Bec^rooal
-000537923^
•0005376344
-0005873466
-0005870569
-0005367687
-0006864807
•0006861980
-0006359057
•6005856186
-0005358319
-0006830455
•0006847594
-0006844785
-0006841880
-0005839028
-0006836179
-0006838383
•0006330490
-0005327661
•0006324814
-00068219180
-0006319149
•00068163121
•0005813496
•0006310674
•0006307856
•0005806040
-0006802227
•0006299417
•0005296610
•0005293806
•0005291006
•0005288207
-0005286412
-0005282620
•0005279831
•0005277045
•0006274262
-0005271481
-0005268704
-0005265929
-0005263158
-0005260389
•0005257624
•0006254861
•0005252101
•00016249344
-0006246590
-0005248838
BQUABB8, CUBES, BOOTS, AND BBCIPAOCAL8.
Square
Onbe
SqnareEoot
Cube Boot
Eecipirocal
64 04 64 ' 6 946 006 81ft
43*6806598
12-4029823
0005241090
64 48 81 ; 6 966 988 489
43-6921045
12-4051488
0005288845
64 8100 6 967 871000
48-7035467
12-4073145
0005285602
65 19 81 1 6 978 881 081
43*7149860
12-4094794
•0005282862
1 66 67 44 6 989 788 688
43-7264222
12-4116436
-0005230126
(66 95 69 7 000 765 497
43-7378654
12-4138070
-0005227392
{66 88 96
7011789 944
43-7492857
12-4159697
•0005224660
) 66 78 85
7 088 786 876
43-7607129
12*4181316
*0005221982
1 67 10 66
7 088 748 896
43-7721373
12*4202928
•0005219207
3 67 48 89
7 044 768 818
43-7835585
12-4224533
*0005216484
B 67 87 84
7 055 798 688
48-7949768
1^-4246129
*0005213764
8 68 86 61
7 066 884 669
43-8063922
12*4267719
*0005211047
868 64 00
7 077 888 000
43-8178046
12*4289300
*0005208333
8 69 08 41
7 088 958 961
43-8292140
12-4810875
•0006205622
8 69 40 84
7100089 448
43-8406204
12*4332441
•0006202914
8 69 79 89
7 111 117 467
43-8520239
12*4354001
•0005200208
8 70 17 76
7 188 817 084
48*8684244
12*4375552
•0005197505
8 70 56 86
7 188 888 185
43-8748219
12*4397097
•0005194805
8 70 94 76
7 144 450 776
43-8862165
12-4418634
-0005192108
,8 7188 89
7 156 584 988
43-8976081
12*4440163
-0005189414
8 71 71 84
7 166 780 758
43*9089968
12-4461685
•0005186722
8 78 10 41
7 177 888 089
48-9203825
12*4483200
•0005184033
8 78 49 00
7 189 067 000
43*9317652
12*4504707
•0005181347
8 78 87 61
7 800 887 491
43-9431451
12-4526206
•0005178664
8 78 86 84
7 811 489 568
43-9545220
12-4547699
-0005175983
8 78 64 89
7 888 688 887
43-9658959
12-4569184
•0005173306
8 74 08 56
7 888 848 604
43-9772668
12-4590661
-0005170631
8 74 48 35
7 846 075 876
43-9886349
12-4612131
•0005167959
8 74 80 96
7 256 818 856
44-0000000
12-4633594
•0005165289
8 75 19 69
7 867 568 958
44-0113622
12*4655049
•0005162623
8 75 66 44
7 878 885 ^78
44-0227214
12*4676497
•0005159959
8 75 97 81
7 890 099 019
44-0340777
12-4697937
•0005157298
8 76 86 00
7 801 884 000
44-0454311
12-4719370
•0005154639
8 76 74 81
7 818 680 681
44-0567815
12-4740796
•0005151984
8 77 18 64
7 828 988 888
44-0681291
12-4762214
-0005149331
8 77 62 49
7 885 808 807
44*0794737
12*4783625
•0005146680
8 77 91 86
7 846 640 884
44-0908154
12-4805029
•0005144033
8 78 80 85
7 857 988 625
44*1021541
12-4826426
•0005141388
8 78 69 16
7 869 888 686
44-1184900
12-4847815
•0005138746
8 79 08 09
7 880 705 188
44-1248229
12*4869197
•0005136107
8 79 47 04
7 898 088 898
44*1361530
12-4890571
-0005133470
8 79 86 01
7 408 478 849
44-1474801
12-4911938
-0005130836
8 80 36 00
7 414 875 000
44-1588043
12-4933298
-0005128205
8 80 64 01
7 426 288 861
44*1701256
12-4954651
-0005125677
1 8 81 08 04
7 487 718 408
44-1814441
12*4975995
-0005122951
8 8143 09
7 449 150 177
44-1927596
12-4997333
-0005120328
8 81 81 16
7 460 698 664
44-2040722
12-5018664
-0005117707
8 88 80 25
7 472 058 876
44-2153819
12-5039988
•0005115090
8 88 69 66
7 488 680 816
44-2266888
12-5061304
•0005112474
SQUARES, OinSES, BOOTS, AND RECIPROCALS. ^95
No.
1967
1968
1959
1960
1961
196fl
1968
1964
1966
1966
1967
1968
1969
1970
1971
1972
1978
1974
1976
1976
1977
1978
1979
1980
1981
1982
1988
1984
1986
1986
1987
1988
1989
1990
1991
1998
1998
1994
1996
1996
1997
1998
1999
2000
2001
2002
2008
2004
2006
Sqiutre
Oob«
Square Boot
8 82 98 49
8 88 87 64
8 88 76 81
8 84 16 00
8 84 66 21
8 84 94 44
866 88 69
8 86 72 96
8 86 12 26
8 86 51 56
8 86 90 89
6 87 80 24
8 87 69 61
8 88 09 00
8 88 48 41
8 88 87 84
8 89 27 29
8 89 66 76
8 90 06 25
8 90 45 76
8 90 85 2d
8 91 24 84
8 91 64 41
8 92 04 00
8 92 48 61
8 92 88 24
8 98 22 89
8 98 62 56
8 94 02 25
8 94 41 96
3 94 81 69
8 96 21 44
8 95 61 21
8 96 01 00
8 96 40 81
8 96 80 64
8 97 20 49
8 97 60 86
8 98 00 25
8 98 40 16
8 98 8009
8 99 20 04
8 99 60 01
4 00 00 00
4 00 40 01
4 00 80 04
4 01 20 09
4 01 60 16
4 09 00 25
7 496
7 506
7 618
7 629
7 541
7 552
7 564
7 675
7 587
7 698
7 610
7 622
7 688
7 646
7 667
7 668
7 680
7 692
7 708
7 715
7 727
7 788
7 750
7 762
7 774
7 785
7 797
7 809
7 821
7 833
7 846
7 856
7 868
7 880
7 892
7 904
7 916
7 928
7 940
7 952
7 964
7 976
7 988
8 000
8 012
8 024
8 086
8 048
8060
014 498
609 912
017 079
636 000
066 681
609128
163 847
729 844
807 125
896 696
498 068
111 232
786 209
878 000
021 611
682 048
354 817
038 424
784 875
442 176
161 888
898 852
636 789
892 000
159 141
988 168
729 087
631904
846 625
178 256
Oil 808
862 272
724 669
599 000
486 271
888 488
298 657
215 784
149 875
095 986
058 978
028 992
005 999
000 000
006 001
024 008
064 027
096 064
160 126
mmmtmssmaa^mmm^^smFsms
44-2379927
44-2492938
44-2605919
44-2718872
44-2831797
44-2944692
44-3067568
44-3170396
44-8283205
44-3395985
44-3508737
44-3621460
44-3734165
44-3846820
44-3969467
44*4072066
44-4184646
44-4297198
44-4409720
44-4622215
44-4634681
44-4747119
44-4859528
44-4971909
44-5084262
44-5196586
44*5808881
44*5421149
44-5538388
44*6645699
44-576778L
44*5869936
44-5982062
44-6094160
44*6206230
44*6318272
44-6430286
44-6542271
44-6654-228
44-6766168
44-6878059
44-6989933
44-7101778
44-7218596
44-7326385
44-7437146
44-7548880
44*7660586
44-7772264
CabeBoot
12-5082612
12*5103914
12-5125208
12-5146495
12-6167776
12*5189047
12*5210813
12*6231571
12-6252822
12*5274065
12*6295802
12*5316631
12*5887763
12*5358968
12*6880176
12*6401877
12*5422570
12*6443757
12*5464936
12-5486107
12*5507272
12*6528430
12*5549580
12-6570723
12*5591860
12*5612989
12-5634111
12-5666226
12-5676384
12*6697435
12*6718529
12*5789615
12*5760695
12*5781767
12*5802832
12-5823891
12*5844942
12-5865987
12-5887024
12*5908064
12*5929078
12-5950094
12-5971108
12*6992105
12-6013101
12-6034089
12*6055070
12-6076044
12*6097011
Reciprocal
0005109862
0005107252
0005104645
•0005102041
•0005099439
•0005096840
*0005094244
•0005091660
*0005089069
-0005086470
-0005083884
*0005081301
•0005078720
•0006076142
*0005073567
-0005070994
•0006068424
-0005065866
*0005063291
•0006060729
•0005058169
-0006056612
-0006053067
•0006060505
•0005047956
•0005045409
-0005042864
*0006040828
*0006037783
*0005035247
*0005032713
-0006030181
•0005027662
-0006026126
•0005022602
*0006020080
•0005017561
•0005015045
•0005012631
*0006010020
-0005007511
•0006005005
-0005002601
•0005000000
•0004997501
•0004995005
•0004992511
•0004990020
•0004987531
SQUABSI, OUBSB) BOOTS, AUD BBCIPBOGAfiJU.
Sqnftn
109 40 86
4 02 80 49
4 08 20 64
4 08 60 81
404 01001
4 04 41 21:
404 8144
4 06 2169
4 05 61 96
4 06 02 25
4 06 42 66
4 06 82 89
4 07 28 24
4 07 68 61
4 08 04 00
4 08 44 41
4 08 84 84
4 09 25 29
4 09 65 76
OBbe
10 06 25
10 46 76
10 87 29
11 27 84
11 68 41
12 09 00
12 49 61
12 90 24
4 IS 80 89
4 18 71 66
4 14 12 26
4 14 52 96
14 98 69
15 84 44
16 76 21
16 16 00
16 66 61
16 97 64
17 88 49
17 79 86
18 20 26
18 61 16
19 02 09
19 43 04
19 84 01
20 26 00
20 06 01
2107 04
2148 09
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4 21 89 16
Mm
8072 216 210
8 064 294 848
8 006 884 512
8 108 466 729
8 120 001 000
8 182 727 881
8 1<44 866 728
8 167 016 107
8 169 178 744
8 181: 858 876
8 108 6140 096
8 205 788 918
8 217 949 8ft&
8 280 172 860
8 242 408 000
8 254 655 261
8 266 914 048
8 279 186 167
8 291 469 824
8 808 765 626
8 816 078 576
8 828 898 088
8 840 726 962
8 868 070 889
8 866 «27 000
8 877 796 791
8 890 176 708
8 402 069 987
8 414 975 804
8 427 892 876
8 489 822 666
8462 264 668
8 464 718 872
8 477 186 819^
8 480 664 000
8 602 164 921
8 614 658 088
8 627 178 507
8 639 701 184
8 652 241 126
8 664 798 886
8 577 867 828
8 680 934 692
8 602 628 640
8 616 126 000
8 627 788 661
8 1^0 864 608
8 668 002 877
8006 668 404
^
SqiMre'Root
44*788Sdl8>
44-7995586
44'8107130
44-8218697
44*8336285
44*8441746
44*8658230
44-8664685
44-8776113
44*8887514r
44*8998886
44*9110231
44-9221549
44-9332839
44:9444101
44*9555336
44*9666543
44*9777723
44-9888875
45-0000000
46*0111097
45*0222167
45-0338210
45-0444225
45*0555213
45-0666173
45*0777107
45-0888013
45-0998891
46-1109743
46*122e667
46-1831364
45*1442134
45*1562876
45-1663592
45*1774280
45-1884941
45-1995575
45-21061:82
45-2216762
45-2327315.
45-2437841
45-2648340
45-2658812
45-2769257
45-287»676
45-2990066
46-3100430
45-3210768
Cube Boot
12^117071
12'6188924
12-6159870
12-6180810
12-620] 743
12-6222669
12-6243587
12-6264499
12-6285404
12-6306801
12*682^192
12*6348076
12-6368053
12-6389823
12-6410687
12 6431543
12-6452893
12*647»235
12*6494071
12-6514000
12-6535722
12-6656638
12-6577346
12-6598148
12-6618043
12-6689731
12-6660612
12-6681286
12-6702053
12-6722814
12-6748667
12-6764814
.J^-6785054
12-6805788
12-6826514
12-6847234
12-6867947
12-6888654
12-6909354
12-6930047
12-6950783
12-6971412
12-6992084
12-7012750
12-7038409
12-7064061
12-7074707
12-7095346
12-7115978
Beeiprocal
0004985645
-0004982561
'000498008O
-0004977601
-0004976124
-0004072650
-0004970179
-00049^7710
-0004965243
-000496-2779
-0004960317
-0004957858
•0004955401
-0004952947
-0004950495
•0004948046
•0004945598
•000494S154
-0004940711
:-0004988272
-0004985834
I-000498S899
•0004980966
•000492S586
-0004926108
-0004923683
-0004921260
-0004918839
•0004916421
-0004014005
-0004011501
-0004909180
•0004906771
•0004904865
-0004901961
-0004899559
•0004897160
•0004894762
-0004892868
•0004889976
-0004887586
'0004885198
-0004882813
•0004880429
-0004878W9
•0004875670
•000487329*
-0004870921
'0004808|»49
«^»*!P
I I w
SQUARES, CUBBS, ROOTS, AND RECIPROCAI^^ 697
'
irb.
Square
2055
2056
2067
2058
2050
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
207S
2074
2075
2076
2077
2078
2079
2080
2081
2082
2088
2084
2085
2086
2087
2088
2089
2090
2091
2092
2098
2094
2095
2096
2097
2096
2099
2100
2101
2102
2108
4;
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
22 80 25
22 71 86
2ai2<49
23 53 64
23 9481
24 86 00
24 7r2l
251944
25 59^69
26 00^96
26 42925
26 8aJ56
27 24 89
27 6€t2l
28 07^61
28 4900
28 90141
29 0184
29 73 29
80 1^ 7B
80 5^25
86 9*; 76
81 89 29
818(184
82 2341
82 64 00
83 0^61
83 47 24
83 83 89
84 80 56
84 72i25
351SI96
85 5d 69
85 97 44
86 89 21
86 8100
87 22 81
87 64 64
88 0^49
88 49 36
88 9Q25
89 82 16
89 74 09
401004
40 58 01
410000
41 42 01
41 84 04
42 2&09
Otibe
8 6T8 816S75
8 690 ^t ^6
8 708 679 198
8 716 879 112
8 729 091 87&
8 741 816 000
8 754 552 981
8 767 802 328
8 780 064 047
8 792 888144
8 606 624 625
8 818 42S 496
8 881 284 763
8 844 058 482
8 856 894 509
8 869 748 000
8 882 60$ 911
8 895 477 248
8 908 868 017
8 921 261 224
8 984 171 875
8 947 094 976
8 960 080 538
8 972 978 652
8 985 989 039
8 998 912 000
9 Oil 897 441
9 024 896 868
9 037 906 787
9 060 028 704
9 063 964 125
9 077 012 0S6
9 090 072 503
9 103 146 472
9 116 280 969
9 129 829 000
9142489 671
9 165 562 688
9 168 698 857
9 181 846 584
9 195 007 875
9 208 186 736
9 221 366 673
9 234 565 192
9 247 776 299
9 261 000 000
9 274 236 801
9 287 485 208
9 800 746 727
Square Boot
Cube Hoot
45-3821078
45-3431362
45-3541619
46-3651849
46-3762(^2
45-38722^
45-3982378
45-4092501
45-4202698
45-4312668
45-4422711
45-4832727
45-4642717
45^4752680
45-4862616
45-4972526
455082410
45-5192267
45-5302097
45-6411901
45-5521679
45-56314'80^
45-5741165
45-685«853
45-5966525
45-6070170
46-6179789
46-6289382
46-6398948
45-6508468
46-6618002
45-6727490
45-6836961
46-694«386
45-7056795
45-7165178
45-7274584 •
45-7381865
45-749S169
45-760$447
46-77116&9
48-782#9^
45-793012I5
45-8039299
45-8148447
46-8257569
45*83666«6
46-8475786
45-8584779
12-7186603
12-7157222
12-7177835
12-7198441
12-7219040
12'7289632
12-7260218
12-7280797
127301-370
12-7321=985
12-7342494
12*736Sfi046
12-7383^92
12-7404131
12-7424664
12-7445189
12*7465709
1^*7486222
K*7506728
12*7527227
12-7547721
12-7568207
12-7588687
12-7609160
12-7629627
12-7660087
12-7670540
12-7690987
12*7711427
t2*7731861
12*7752288
12-7772709
12*7793128
12-7813581
12-7833932
12*7864326
127874714
12-7895096
12-7915471
12-7985840
12-7986202
12^7976568
12-7996907
12-8017250
12-8037686
12-8057916
12-80781239
12 809^56
12-8118866
ReeiproGtil
1.
I.,
'-0004866l«0
•0004863813
-0004861449
-0004859086^
•0004856727
•0004854369
•0004853014
-6004849(S6T
000484731Ct
000484496*^
000484%1&
r00O484(®71
0004837a^
-00014886600
-000483326a
F-00048309118.
e0(^828Sm-
0004826855
000482^27
k000482160I
•0004819277
-0004816966
-0004814636
?-00048l2320
•00048100(^
■600480769<2
•0004803382
•0004803074
f-0004 800768
0004798464
00047961^
0004793864
P -0004791567
-0004789272
-0004786979
•0004784689
•0004782401
•0004780115
0004777831
0004775549
•0004773270
•0004770992
•0004768717
•0004766444
-0004764178
-0004761905
•0004759638
i -0004757374
0004755112
t.i
SQUAEB8, GUBBS, BOOTS, AND BECIPR0CAL8.
STa
Sqnazt
Oube
Square Boot
CobeBoot
Bedprooal
104
4 42 68 16
9 814 020 864
45-8693798
12-8189170
•0004752852
105
4 4810 25
9 827 807 626
45-8802790
12-8159468
-0004750594
106
4 48 62 86
9 840 607 016
45-8911756
12-8179759
-0004748838
107
4 48 94 49
9 858 919 048
45-9020696
12-8200044
•0004746084
108
444 86 64
0 867 248 712
45*9129611
12-8220323
•0004743833
100
4 44 78 81
0 880 581029
45-9238500
12-8240595
-0004741684
1110
4 45 21 00
9 898 981000
46-9347363
12-8260861
•0004739386
an
4 45 68 21
9 407 298 681
45-9456200
12-8281120
•0004737091
Ilia
446 05 44
9 420 668 928
45-9565012
12-8301878
•0004734848
1118
4 46 47 69
9 484 066 897
45-9673798
12-8321620
•0004782608
1114
446 89 96
9 447 457 544
45-9782567
12-8341860
•0004730369
tllS
447 82 25
9 460 870 875
46-9891291
12-8362094
•0004728182
iiie
4 47 74 56
9 474 296 896
46-0000000
12-8882321
•0004725898
1117
4 48 16 89
9 487 785 618
460108688
12-8402542
•0004723666
0118
4 48 59 24
9 501 187 082
46-0217340
12-8422756
•0004721435
(119
4 49 01 61
9 514 651159
46-0325971
12-8442964
-0004719207
iiao
4 49 44 00
9 628 128 000
46-0434577
12*8463166
-0004716981
(121
4 49 86 41
9 541 617 561
46-0548158
12*8483361
•0004714757
(122
4 50 28 84
9 555 119 848
46-0651712
12*8503551
•0004712535
1128
1 4 50 71 29
9568 684 867
46-0760241
12*8523783
•0004710316
1124
1 4 51 18 76
9 582 162 624
460868745
12-8543910
•0004708098
1125
4 51 56 25
9 595 708 125
46-09VV223
12*8564080
•0004705882
tl26
4 51 98 76
9 609 256 876
46-1085675
12*8584243
•0004703669
^127
4 52 41 29
9 622 822 888
46-1194102
12-8604401
•0004701457
a28
4 62 88 84
9 686 401 152
46-1302504
12-8624552
•0004699248
a29
4 58 26 41
9649 992 689
46-1410880
12-8644697
•0004697041
aso
4 58 69 00
9668 697000
46-1519230
12-8664835
•0004694886
asi
4 54 11 61
9 677 214 091
46-1627655
12-8684967
•0004692633
a82
4 54 54 24
9 690 848 968
46-1735855
12-8705093
•0004690432
ass
4 54 96 89
9 704 486 687
46-1844130
12-8725213
•0004688233
U184
4 55 89 56
9 718 142 104
46-1952378
12-8745326
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1185
4 55 82 25
9 781 810 875
46-2060602
12-8765433
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U86
4 56 24 96
9 745 491 456
46-2168800
12-8785634
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1187
4 56 67 69
9 759 185 858
46-2276973
12-8805628
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1188
4 57 10 44
9 772 892 072
46-2385121
12-8825717
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1189
4 57 58 21
9 786 611 619
46-2493243
12-8845799
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S140
4 57 96 00
9 800 844 000
46-2601340
12-8865874
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tl41
4 58 88 81
9 814 089 221
46-2709412
12-8885944
•0004670715
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4 58 81 64
9 827 847 288
46-2817459
12-8906007
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(148
4 59 24 49
9 841 618 207
46*2925480
12-8926064
•0004666356
(144
4 59 67 86
9 855 401 984
46-3038476
12-8946115
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4 60 10 25
9 869 198 625
46-3141447
12-8966159
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1146
4 60 58 16
9 888 008 186
46-3249398
12-8986197
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4 60 96 09
9 896 880 528
46-3357314
12-9006229
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1148
4 61 89 04
9 910 665 792
46-3465209
12-9026255
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1149
4 61 82 01
9 924 618 949
46-3578079
12-9046275
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1150
4 62 25 00
9 988 875 000
46-3680924
12-9066288
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151
4 62 68 01
9 952 248 951
46-3788745
12-9086295
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1152
4 68 11 04
9 966 185 808
46-3896540
12-9106296
•0004646840
SQUARES, CUBES, BOOTS, AND BECIPBOCALS. 6^9
No.
12158
2164
2156
2150
2157
2158
2169
2160
2161
2162
2163
2164
2166
2166
2167
2168
2169
12170
2171
2172
2178
2174
2175
2176
2177
2178
2179
2180
2181
2182
2188
2184
2185
2186
2187
2188
2189
2190
2191
2192
2198
2194
2196
2196
2197
2196
2199
2200
3201
Sqnare
4 68 64
4 68 97
4 64 40
4 64 88
4 66 26
4 65 69
4 6612
Cube
66 56
66 99
67 42
67 86
63 28
68 72
4 69 16
69 68
70 02
70 46
70 89
7182
7175
7219
72 62
78 06
78 49
78 98
74 86
74 80
76 24
76 67
7611
76 64
76 98
77 42
77 86
78 29
78 78
7917
79 61
80 04
4 80 48
4 80 92
4 8186
4 8180
4 82 24
4 82 68
4 88 12
4 88 66
4 84 00
4 84 44
09
16
25
86
49
64
81
00
21
44
69
€6
26
66
89
24
61
00
41
84
29
76
26
76
29
84
41
00
61
24
89
61
26
96
69
44
21
00
81
64
49
86
26
16
09
04
01
00
01
i
9 980
9 998
10 007
10 021
10 086
10 049
10 063
10 077
10 091
10 106
10119
10188
10147
10161
10176
10190
10 204
10 218
10 2-^2
10 246
10 260
10 274
10 289
10 808
10 817
10 881
10 846
10 860
10 874
10 888
10 408
10 417
10 481
10 446
10 460
10 474
10 489
10 608
10 617
10 632
10 646
10 661
10 676
10 690
10 604
10 618
10 638
10 648
10 6^2
085 677
948 264
878 876
812 416
768 898
728 812
706 679
696 000
699 281
715 528
744 747
786 944
842 126
910 296
991463
085 682
192 809
818 000
446 211
5^2 448
761 717
924 024
109 875
807 776
619 288
748 752
981 889
282 000
495 741
772 568
062 487
865 604
6S1 625
010 866
868 203
708 672
077 269
469 000
863 871
261888
683 067
117 884
564 875
025 536
499 878
986 892
486 599
000 000
6166OI
Square Boot
46-4004810
46-4112055
46-4219775
46-4327471
46-4435141
46-4642786
46-4650406
46-4758002
46-4865572
46-4973118
46-5080638
46-5188134
46-5295606
46-5403051
46-6510472
46-5617869
46-5725241
46-5882588
46-5939910
46-6047208
46-6154481
46-6261729
46-6368953
46-6476152
46-6583326
46-6690476
46-6797601
46-6904701
46-7011777
46-7118829
46-7225855
46-7832858
46-7439836
46-7546789
46-7653718
46-7760623
46-7867508
46-7974358
46-8081189
46-8187996
46-8294779
46-8401537
46*8508271
46-8614981
46*8721666
46-8828327
46*8934963
46-9041676
46-9148164
Cube Boot
12-9126291
12*9146279
12*9166262
12*9186238
12-9206208
12-9226172
12-9246129
12-9266081
12-9286027
12-9305966
12-9325899
12-9345827
12-9365747
12-9385662
12-9405570
12-9426472
12-9445369
12-9465259
12-9485143
12*9505021
12-9524893
12-9544759
12-9564618
12-9684472
12-9604319
12-9624161
12-9643996
12-9663826
12-9683649
12-9703466
12-9723277
12-9743082
12-9762881
12-9782674
12-9802461
12-9822242
12*9842017
12-9861786
12-9881549
12-9901306
12-9921057
12-9940802
12-9960540
12-9980273
130000000
13-0019721
13-0039436
18-0059145
13-0078848
Bedprocal
-0004644682
-0004642526
-0004640371
-0004638219
•0004636069
-0004633920
•0004631774
•0004629630
-0004627487
-0004625347
•0004623209
•0004621072
-0004618938
•0004616805
•0004614675
•0004612546
•0004610420
•0004608295
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•0004601933
•0004699816
•0004597701
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•0004568296
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1*1829
6324
.2
ri275
*8869
10072
•1203
*62
1^6820
•6946
1*1833
6488
8
1-1838
*8781
1*0086
•1304
*63
1*6989
•5886
1*1438
6562
4
1-1603
•8694
1*0098
•1406
•61
1*7160
5828
1*1494
*6666
6
1*1618
•8607
1*0113
•1606
'56
1*7332
•6770
1*1651
5782
6
1*1736
•8621
1*0128
•1607
*6S
1*7607
•6712
1*1609
•6897
7
1*1853
•8437
10145
•1708
•67
1*7683
•5655
11669
*6014
8
1*1972
*8353
1*0162
*1810
•68
1*7860
•5699
1*1730
6131
9
1-2092
*8270
1*0181
•1912
*60
1*8040
•6643
1*1792
6248
0
1*2214
•8187
1*0201
•2013
•60
1*8221
•6488
1*1866
6366
11
1-2337
•8106
1*0221
•2116
•61
1*8404
•6434
1*1919
*6486
»
1-2461
•8026
10243
*2218
•62
1*8689
-6379
1*1984
'6606
3
1*2686
*7946
10266
2320
•68
1*8776
•6326
1*2061
6725
4
1*2712
*7866
1*0289
•2428
•64
1*8966
•6273
1*2119
6846
,6
1-2840
*7788
1*0314
•2626
•66
1*9165
'6220
1*2188
*6968
«
1*2969
*7710
I'OSU)
•2629
*66
1*9318
•6168
1*2268
*709O
17
1*3100
•7631
1-0837
•2733
•67
1*9642
•6117
1*2330
*7213
8
1*3231
•7558
1*0336
2837
*68
1*9789
•6066
1*2402
•7336
!9
1*3364
*7483
1*0424
•2941
*69
1*9937
*6016
1*2476
7461
10
1*3199
7408
1*0463
'3046
•70
20138
•4966
1*2662
-7686
;i
1*3334
•7334
1*0484
•3163
71
20340
*4916
1*2628
•7712
\2
1*3771
•7262
1*0616
•3255
•72
2*C614
*4868
12706
*7838
z
1-3310
*7189
1-0660
•S360
73
20761
•4819
1*27186
*7966
\i
1*4060
•7118
1*0684
'3466
*74
20969
•4771
1*2866
•80?4
16
1-4190
•7047
1*0619
*3672
'76
2*1170
•4724
1*2947
•8223.
(6
1*4833
*6977
10665
*3678
*76
21383
•4677
1-3:33
*8363
.7
1*4477
*6907
10692
•3786
•7T
2*1588
•4630
1*3114
-8484
8
1*4623
*6839
10731
3892
•78
21815
-4684
1*3199
•8616
9
1*4770
•6771
1*0770
*4000
•79
2*2334
-4638
1*3286
•8748
709
Exponential and Hyperbolic Functions— Cow^mued.
a.
««^-
«-*•
Codha;.
Sinh«.
X,
€"'
e^'
Cosh X.
Sinb X.
•80
2-2256
4493
13374
•8881
1*20
3*3201
•3012
1*8107
1^5096
•Bl
22479
4449
1-3464
•9015
1*21
33535
•2981i
1*8268
1-5276
'82
2'27C6
-4404
1-3555
•9150
1*2?^
3*3872
•2952
1-8412
1-5460
-83
2'2933
-4360
1-3647
•9286
1*23
3*4212
•2923
1*8568
1*5646
'84
2'3i61
-4317
1-3740
•9423
124
3*4556
•2894
1*8725
1'5831
'86
2*3396
•4274
1-3835
•9561
1-25
3*49C3
•2865
1-8884
1*6019
86
2*3632
-4232
1-3932
•9700
f26
36251
•2836
1*9045
1*6209
•87
2*3869
•4190
1^4029
•9840
1*27
3*5608
•2806
1*9208
1'6400
*88
2*4109
-4148
1^4128
'9981
1*28.
3'5366
•2780
1-9373
1*6593
•89
2*4361
-4107
14229
1*0122
1*29
36328
•2753
1-9540
1*6788
•90
246S6
-4066
14331
10266
130
3*6693
•2725
1-9709
1*6984
•91
2*4843
-4026
14434
1*0409
1*31
3*7C62
•2698
1-9880
1'7182
•92
2*5093
•3986
1-4539
l*066i
1*32
3*7434
•2671
20053
1*7381
-93
2*6346
•3946
1-4646
1*0700
1*33
3*7810
2345
20228
1*7583
•94
2*5600
•3906
1*4755
1*0847
1*34
3*8190
•261b
2*0404
1-7786
•96
2-6857
•3867
1^4862
1*C9.6
1*36
3*8574
•2592
2-0583
1-7991
•96
2-6117
•3829
14973
1*1144
1*36
3*8932
•2567
2*0764
1-8198
•97
2*6379.
•3791
15085
1*1294
1*37
3*9354
•2541
2-0947
1-8106
•98
i 26645
. 5-6912
•3753
1-5199
11446
1*38
3*9749
•2516
21132
1-8617
•99
•3716
15314
1*1598
1*39
4*0148
•2491
2^1320
1-8829
100
' 2*7183
•3679
1-5431
ri752
1*40
4*0552
•2466
2-1609
1*9043
101
2*7466
•3642
1-5549
1*1907
1*41
4*0960
•2441
2^1700
1*9259
102
2*7732
'3606
1-5669
1*2063
142
4*1371
•2417
2^1894
1*9477
1'0»
2*8011
•3570
1-5790
1*2220
1-43
4*1787
•2393
2-2090
1^9697
1*04
^8292
•3634
1-5913
1*2379
1-44
4*2207
•2369
2^2288
1*9919.
106
2*8676
•3499
1-6038
1*2539
146
4*2631
•2346
2-2488
20143
106
2*8864
•3465
1-6164
1*2700
1*46
4*3060
•2322
22891
2-0369
107
2-9161
•3430
1-6292
1*2862
1*47
4*3492
•2299
2^28i)6
20596
1'06
2'9447
•3896
1-6421
1*3025
1*48
4*3930
•2276
2^3103
2-0826
10»
2*9743
•3362
1-6C52
1*3190
1*49
4*4371
•2254
23312
2*1059
110
3*0042
•3829
1-6685
1*3356
1*60
4*4817
•2231
23524
2*1293
111
30S14
•3296
16820
1*3524
1*51
4*5267
•2209
23738
2*1629
112
30648
•326S
1-6958
1^3693
1*52
4*5722
•2187
2-3955
2*1768
113
3*0967
•3230
1-7093
1*8863
1*63
4*6182
•2165
2-4174
2*2008
114
31268
•3198
1-7233
1*4035
1*64
4*6646
•2144
2*4395
2*2251
116
3*1682
•3166
1-7374
1'4208
1-55
4'7115
•2122
2^46l»
2*2496
116
3*1B99
•3186
1-7617
1^4382
1*66
4*7588
•2101
2*4845
2*2743
1'17
3*2220
•3104
1-7662
1^4559
1*57
4*8066
•2080
2-5071
2*2993
118
3*2614
•3073
1-7808
14736
1*68
4*8550
•2o:k)
2-5305
2*3245
119
3*2871
•3042
1-7956
1*4914
1*69
4*9038
•2039
2-5538
2*3499
*:10
■■HHMMaHaM
EXPONENTTAT. AND HYPSBBOLIC FUNCTIONS^-Con^intf^d.
X,
«*•
e^'
Ooshx
Sinho;.
X.
««•
e-^'
Cosh X.
Sinha;.
•60
4*9530
*2019
2*5776
2*3756
2*00
7^3891
*1363
3*7622
3*6269
•61
60028
*1999
2*6014
2*4016
2*1
8*1662
*1226
4*1443
4*0219
•62
5*0531
•1979
2*6255
2*4276
2*2
90250
•1108
4*5679
4*4671
•63
51039
•1959
2*6499
2*4540
2*3
99742
•1003
5*0372
4*9370
6i
51552
•1940
2*6746
2*4806
2*4
11'0232
•0907
5*5570
6*4662
•65
5*2070
•1920
2*6995
2*5076
2-6
12*1826
•0821
6*1323
6*0aD2
•66
52593
•1901
2*7247
2*5346
2*6
13*4637
•0743
6*7690
6*6^7
•67
53122
•1882
2*7502
2*5620
2*7
14*8797
•0672
7*4736
7*4063
•68
63656
•1864
2*7760
2*5896
2*8
16*4446
•0608
82527
8*1919
•69
5*4195
•1845
2*8020
2^6175
2*9
18*1741.
•056Q
9*1146
9*0596
•70
6*4740
•1827
28283
2*6456
3^0
20*0856
•0498
10*0677
10*0179
•71
5*5290
•1809
2'8549
2*6740
31
22*1980
•Q450
11*12X5
11*0765
•72
5*5845
•1791
2^8818
2*7027
3*2
24'6325
•0408
12*2866
12*2469
•73
5*6406
1773
2^9090
2*7317
3*3
27*1126
•0369
13*5747
13*5379
•74
5*6973
•1756
2^9364
2*7609
3*4
29*9641
•0334
14*9987
14*9654
•76
5*7546
•1738
29642
2*7904
3*6
33*1155
•0302
16*5728
16*5126
•76
5*8124
•1720
2'9922
2*8202
3'6
36*5982
•0273
18*3128
18*2865
•71
5*8708
•1703
3*0206
2*8503
3*7
40*4473
•0247
20*2360
20*2113
•78
5*9299
•1686
30492
2*8806
3*8
44*70J2
•0224
22*3618
22*3394
•79
5*9894
•1670
3*0782
2*9112
3*9
49*4024
•0202
24*7113
24*6911
•80
6*0496
•1653
31075
2*9422
4-0
64*5982
•0183
27*8082
27*2899
•81
6-1104
•1635
3^1370
2*9734
4*1
60*3403
•0166
30*1784
30-1619
•82
61719
•1620
3^1669
3*0049
4-2
66*6863
•0150
33*3607
33*3357
•83
6*2339
•1604
3^1972
3*0367
4*3
736998
•0136
36*8567
36*8431
•84
6*2966
•1588
3^2277
3*0689
4*4
81*4509
•0123
40*7316
40*7193
•85
6*3598
•1572
3^2585
3*1013
45
90*0171
•0111
46*0141
46*0030
•86
6*4237
•1567
3^2897
3*1340
4*6
99*4843
•0100
49*7472
49*7371
•87
6*4883
•1541
3^3212
3*1671
4*7
109'947
•0091
64*9781
54*9690
•88
6*5535
•1526
3*3530
3*2036
4*8
121*510
•0082
60*7693
60*7511
•89
6*6194
1511
3*3852
3*2342
4*9
134*290
•0074
67*1486
67-1412
•90
6*6859
•1496
3*4177
3*2682
5*0
148*413
•0067
74*2099
74-2032
•91
6*7531
•1481
3^4506
3*3025
5*1
164*022
•0061
82-0140
82'0079
•92
6*8210
•1466
3*4838
3*3372
5^2
181*272
•0056
90*6388
90*6333
•93
6*8895
•1452
35173
3*3722
5-3
200*337
•0050
100*171
100*167
•94
6*9668
1437
3*5512
3*4075
5-4
221*406
•0046
110*706
11O-701
•96
7*0281
1423
3*5865
3*4432
5*6
244*693
•0041
122*348
122*344
•96
7*0993
•1409
3*6201
3*4792
5-6
270*426
•0037
136*215 .
136*211
•97
71707
1396
3*6551
3*5156
5-?
298*867
•0034
149*436
149*432
•9&
72427
1381
3*6904
3'6523
5-S
330*300
•0030
165-161
165*148 1
•99
7'3155
1367
3*7261
3*5894
5'9
366037
•0027
182*520
182*517 1
_
-
6-0
403*429
•0026
201*716
201*713 1
HYPERBOLIC LOGARITHMS.
711
Table of Hyperbolic Logarithms.
To find the kyperbolic logarithm of a number multiply
the oommon logarithm of the number by the figures
2*302585052994, and the product is the hyperbolic loga-
rithm of that number.
Example. — The common logarithm of 3*75 is -5740313 ;
the hyperbolic logarithm is then found by multiplying
2-302585 by -5740313 = 1-3217659, the hyperbolic loga-
rithm.
Mo.
1-01
1*02
1-08
1-04
105
1-06
1-07
1-08
1-09
10
11
12
13
14
15
16
17
18
19
1-201
1-21
1-22
1*23
1-24
1-25
1-36
1-27
1*28
1>29
1-30
1-31
1-32
1-38
1*34
LogKrithm
No.
1-35
Logarithm
No.
169
Logarithm
•5247284
No.
203
Logarithm
-0099503
•3001046
•7080357
•0198026
1-36
•3074847
1^70
•5306282
2-04
•7129497
-0295588
1-37
•3148108
1^71
5364933
206
•7178399
•0392207
1-38
•3220833
172
•5423241
206
•7227058
•0487902
1-39
•3293037
1^73
•5481212
2-07
•7275486
•0582690
1-40
•3364721
1-74
•5538850
2-08
•7323678
-0676686
1-41
•3435895
1'75
•6596156
2^09
•7371640
•0769610
1-42
•3506568
1*76
•6653138
2-10
•7419373
•0861777
143
•3676744
1-77
•6709795
211
•7466880
•0953102
1^44
•3646431
1-78
•5766133
2-12
•7514160
•1043600
1^45
-3715635
1^79
•6822156
2^13
•7561219
•1133285
1-46
-3784365
1-80
•5877866
2-14
•7608058
•1222174
1-47
•3852623
1-81
•6933268
216
•7654680
•1310284
1-48
•3920420
1-82
•5988365
2-16
•7701082
•1397614
149
•3987762
1^83
•6043159
2^17
•7747271
•1484199
1-50
•4054652
1^84
•6097653
2^18
•7793248
•1570038
1-51
•4121094
1-85
•6151855
2-19
•7839014
•1655144
1-52
•4187103
1-86
6205763
2-20
•7884573
'1739534
1^53
•4252675
1-87
•6259384
2-21
•7929925
•1823215
1-54
•4317823
1-88
•6312717
222
•7976071
•1906204
1-55
•4382550
1-89
•6365768
2-23
•8020015
•1988607
1^56
•4446868
1-90
•6418538
2-24
•8064758
•2070140
1-67
•4610766
1-91
•6471033
2-25
•8109303
•2151113
1-58"
•4574247
1-92
•6523251
2-26
•8153647
•2231435
1-59
•4637339
1-98
•6575200
2-27
•8197798
•2311161
1-60
•4700036
1-94
•6626879
2-28
-8241754
•2390167
1-61
•4762341
195
•6678294
2-29
•8286618
•2468601
1-62
•4824260
1-96
•6729445
2-30
•8329089
•2546422
1^63
•4885801
1^97
•6780335
2-31
•8372474
•2623643
1-64
•4946959
1-98
•6830968
2-32
-8415671
•2700271
1-66
•5007752
1-99
•6881346
2-33
•8458682
•2776316
1-66
•5068176
2-00
•6931472
2-34
•8501509
•2851787
1-67
-6128237
201
•6981347
2-35
•8544164
•2926696
168
•5187938
202
•7030974
2-36
•8686616
VA
HTPEBDOLTG
LOGARITHMS.
T^o.-
2-85
LoiMthm
333
liOffuntiim
To?"
3-81
LoMSm"
•37
•8628899
1^0473189
1-2029722
1-3376291
•38
•8671004
286
10608216
334
12059707
382
1-3402604
•39
•8712933
2-^7
10643120
3-36
12089603
3-83
1-3428648
•40
•8754686
2^88
10577902
336
1-2119409
384
1-3454723
•41
•8796266
2^89
10612564
3-37
1-2149127
3-86
1-3480731
•42
•8837676
2-90
10647107
338
12178767
3-86
1-3506671
•43
•8878912
291
10681629
339
1-2208299
3-87
1-3632644
•44
•8919980
2-92
1-0716836
3-40
1-2237764
3-88
1-3558361
•45
•8960879
2-93
r0760024
341
12267122
3^89
1^3684091
•46
•9001613
294
10784096
3-42
1-2296405
3-90
13609765
•47
•9042181
2-95
10818051
343
1 •2325606
3-91
1-3635373
•48
•9082585
2^96
10861892
344
12364714
3-92
1-3660916
•49
•9122826
2^97
1-0886619
345
1 2383742
3-93
13686396
•60
•9162907
298
10919233
346
1 •2412686
394
1-3711807
•51
•9202825
2^99
1-0962783
3-47
1-2441646
3-9^
1-3737166
•62
•9242689
3^00
10986124
3-48
l«2470322
3-96
1*3762440
•63
•9282193
301
1-1019400
3-49
1-2499017
3-97
1-3787661
54
•9321640
302
1-1062668
3-60
1-2627629
3-98
1-3812818
•65
•9360934
303
11086626
3*61
1*2666160
3*99
1-3837911
•66
•9400072
304
1-1118676
3-62
1^2684609
4-00
1-3862943
!-67
•9439068
306
1-1151416
3-63
1-2612978
401
1*3887912
!-58
•9477893
3-06
1-1184147
3-64
1*2641266
402
1-^912818
!-69
•9616578
307
1-1216775
3*66
1-2669475
403
1-3937763
1-60
•9666112
3-08
11249296
3-66
1-2697606
404
1-9962446
1-61
•9693602
309
1-1281710
3-67
1*2726666
4-06
1-3987168
!'62
•9631743
310
1-1314021
3^68
1-2753627
4-06
1-4011829
1-63
•9669838
3-11
11346227
369
1-2781621
407
1-4036429
1-64
•9707789
312
1^1378330
3-60
1-2809338
4K)8
1-4060969
1-66
•9745696
313
11410330
3-61
12837077
409
1-4086449
1-66
•9788259
314
1-1442227
3-62
1-2864740
410
1-4109869
{•67
•9820784
3-15
1-1474024
3-63
1-2892326
4-11
1-4134230
J-68
•9868167
316
1-1506718
3-64
1-2919836
4-12
1-4158631
5-69
•9896411
317
11537315
3-65
1^2947271
4-13
1-4182774
}-70
•9932618
318
M5688U
366
1-2974631
414
1-4206957
1-71
•9969486
3^19
1-1600209
367
1-8001916
4-15
1*4231083
5-72
I'OOOeSlS
320
1^1631608
3-68
18029127
4-16
1-4266160
173
10043016
3^21
ri662708
3-69
1-3056264
4*17
1*4279161
^•74
10079679
322
1^1693813
3-70
1-3083328
4-18
1-4303112
l^75
10116009
3-23
1-1724821
3-71
1-8110318
4-19
1-4327007
1.76
1-01&2306
324
1-1756733
3-72
1'3137236
4-20
1-4350644
1.77
1^0188473
3-25
M786649
3-73
1-3164082
4-21
1-4374626
1.78
1^0224609
3-26
1-1817271
3^74
1-3190856
4-22
1-4398361
1.79
1^0260416
3-27
1-1847899
3-75
1'3217669
4-23
1-4422020
{•80
10296193
3-28
1-1878434
3-76
1-3244189
424
1-4446632
1-81
10331843
829
M908875
3-77
1-3270749
4-26
1-4469189
1.82
1 '0367368
5-30
M939224
378
1-3297240
4-26
1-4492691
1.83
1'0402766
3-31
1-1969481
379
1-3323660
4-27
1-4516138
(.84
l^O438O40
332)l.M999647 1
3-80
13360010
428
1*4639630
HYPERBOLIC LOGARITHMS.
7ia
4-29
iiOgwritlun
ToT"
ixNianuuik
TloT
5-26
LiOfraritbin
^oT
jjosantbrn
1'4662867
4-77
1*5623462
1-6582280
5-73
1-7467165
4*80 1-4586149
4-78
1*5644406
5-26
1*6601310
5-74
1-7474591
4*31
1-4609379
4-79
1-5665304
5-27
1*6620308
6-76
1*7491998
4S2
1-4632553
4-80
1-5686169
5-28
1-6689260
6-76
1-7609874
433
1*4655675
4-81
1-5706971
5-29
1*6658182
6*77
1*7626720
4<34
1-4678743
4'«2
1-6727739
5-80
1-6677068
6-78
1-7544086
4*36
1-4701758
4-83
1-6748464
5-31
1-6695918
6-79
1-7661323
4-36
1*4724720
4-84
1-5769147
6-82
1-6714733
5-80
1-7678579
4-87
1-4747630
4-85
1-6789787
6*33
1-6738512
5-81
1-7695805
4*38
1-4770487
4-86
1*5810384
5-34
1*6762256
5-82
1-7618002
4-39
1*4793292
4-87
1-5880939
5-86
1*6770965
5-83
1-7630170
4-40
1-4816045
4-88
1-5861452
6*36
1-6789639
5-84
1-7647808
4*41
1*4838746
4-89
1-5871923
5-37
1*6808278
5-85
1-7664416
4*42
P4861396
4*90
1-5892352
5-38
1*6826882
6-86
1*7681496
4'43
1*4883994
4-91
1-5912739
6-39
1-6846453
6-87
r7698546
4*44
1*4906543
4-92
1-6933086
6-40
1-6863989
5-88
1*7715567
4-45
1*4929040
4'93
1'6953389
6*41
1-6882491
5-89
1-7732659
4-46
1*4951487
4*94
1-6973658
5-42
1*6900968
6-90
1*7749623
4-47
1*4973883
4-95
1-5993876
5*43
1-6919391
6-91
1-7768458
4*48
1*4996230
4-96
1-6014067
5*44
1-6937790
5*92
l-77«3364
4*4^
1*6018527
4-97
1-6034198
5-45
1:6956156
6*93
1*7800242
4-60
1*6040773
4-98
1-6054298
6*46
1-6974487
6-94
1-7817091
4-51
I'5062971
4-99
1'6074368
5*47
1-6992786
6-96
1-7833912
4*52
1-5085119
500
1-6094377
6-48
1-7011061
6-96
1-7850704
4*53
1*6107219
501
1-6114359
5-49
1-7029282
6*97
1-7867469
4*64
1*5129269
502
1-6134300
6-50
1-7047481
5-98
1-7884205
4*56
1-5151272
603
1-6154200
5-51
1-7066646
5-99
1*7900914
4*66
1-5173226
5*04
1-6174060
6*52
1*7083778
6-00
1-7917596
4*5:7
I-5196132
5-06
1-^193882
5-63
1-7101878
601
1-7934247
4-68
1*6216990
506
1-6213664
5-64
1*7119944
6-02
1*7950872
4-69
1*6238800
507
1*6233408
5*66
1-7137979
6-08
1-7967470
4*60
1*5260563
5-08
1-62531 12
5-66
.1*7165981
604
1-7984040
4-61
1*6282278
5*09
1-62727T8
5-57
1*7173950
606
1-8000582
4*62
1*5303947
510
1-6292405
5-58
1-7191887
606
1-8017098
4*63
1-6325568
5-11
1*6311994
6-59
1*7209792
607
1*8038586
4'64
1*6347143
512
1-6331544
5-60
1-7227660
608
1*8050047
4-65
1-5368672
513
1-6351057
5-61
1-7245607
6-09
1-8066481
4*66
1-5390154
514
1-6370680
6*62
1*7268316
6-10
1-8082887
4*67
1*5411590
5*15
1-6389967
6-68
1-7281094
6-11
1-8099267
4*68
1<5432981
616
1-6409365
5-64
1-7298840
612
1-8115621
4*69
1-5454325
517
1-6428726
5-65
1*7316665
6-13
1*8131947!
4-70
1*5476626
sas
1-6448050
5-66
1-7334238
614
1-8148247
4-71
1-5496879
5*19
1*6467836
5-67
1-7361891
615
1-8164520|
4*72
1*6618087
5-20
1-6486586
5-68
1-7369512
616
1-8180767
4-73
1<5539252
6-21
1-6505798
6*69
1-7387102
6-17
1*81969881
4*74
1-5560371
5-22
1-6524974
5-70
1-7404661
6-18
1*821818^
4-75
1*5581446
5-23
1*6644112
5-71
1-7422189
6-19
1-8229861!
4*76
1*5602476
5-24
1*6563214
5-72
1-7439687
6-20
1-824649^
714
HYPERBOLIC LOOABtTHMS.
%
TjogSSST
7-65 203470561
7*66 2-0360119'
7-67 20373166
7-68 20386196
7*69 2-0399207
7-70,2-0412203
7-71 i0426181
7*72^ 2-0438143
7-78 2*0461088
7-74 20464016
7*76 2-0476928
776 2*0489823
7-77 2-0502701
7-78 2-0615563
7*79 2-0628408
7-80 2-0641237
7-81 2*0554049
7*82 2*0566845
7*83 2*0579624
7*84 2-0592388
7-86 2O605135
7*86 2-0617866
7-87 2*0630580
7:88 2*0643278
7-89 2-0655961
7-90 2-0668627
7*91 2-0681277
7-92 2-0693911
7*93 2-0706630
7-94 2-0719132
7*96 2-0731719
7-96 2-0744290
7*97 2-0766845
7-98 2-0769384
7-99 2-0781907
800 2-0794414
8-01 2-0806907
8*02 2-0819384
8*03 2-0831846
8-04 2-0844290
8-06 2*0856720
8-06 2-0869135
8-07 2-0881534
8*08 2*0893918
8*09 20906287
8*10 2-O918640
8-11 2-0930984
8-12 2-0943306
no.
6*21
6-22
6-28
6*24
6*25
6*26
6*27
6-28
6*29
6-30
631
6-32]
6-33
6-34
6*36
6-36
6*37
6*38
6*39
6*40
6-41
6-42
6*43
6*44
6*46
6*46
6*47
6*48
6*49
6-60
6-61
6-52
6*68
6*64
6*56
6-66
6-67
6*58
6-59
6-60
6*61
663
6*63
6-64
6-66
6*66
6-67
6-68
JJoffftrttlun
1-8261608
1-8277699
1-8293763
1-8309801
1-8325814
1*8341801
1*8367763
1*8373699
1*8389610
1*8405496
1*8421366
1*8437191
1*8463002
1*8468787
1*8484547
1*8500283
1-8515994
1-8531680
1-8547342
1-8562979
1-8678592
1-8594181
1-8609745
1*8625285
1*8640801
1-8656293
1*8671761
1*8687206
1-8702626
1-8718021
1-8733394
1-8748748
1-8764069
1-8779371
1-8794660
1-8809906
1-8826138
1*8840347
1-8855533
1-8870697
1*8886837
1*8900964
1*8916048
1-8931119
1*8946168
1-8961194
1*8976198
1*8991179
6-69
6-70
6*71
6-72
6*73
6*74
6-76
6*76
6-77
6*78
6-79
6*80
6*81
6*82
6*83
6-84
6-85
6-86
6*87
6-88
6-89
6-90
6*91
6*92
6*93
6-94
6-95
6-96
6-97
6-98
6-99
7-01
702
703
7*04
705
7-06
7-07
7-08
7'09
7*10
7*11
712
713
7*14
715
7*16
1*9006138
1*9021075
1-9035989
1*9060881
1*9065751
1*9080600
1*9095426
1-9110228
1-9125011
1-9139771
1*9154509
1-9169226
1*9183921
1-9198594
1*9213247
1*9227877
1*9242486
1*9257074
1*9271641
4*9286186
1*9300710
1*9315214
1*9329696
1*9344157
1*9358598
1*9373017
1*9387416
1*9401794
1-9416162
1*9430489
1*9444805
7*00 1-9459099
1*9473376
1-9487632
1*9501866
1*9516080
1*9630275
1*9544449
1*9568604
1*9572739
1*9586853
1-9600947
1-9615022
1-9629077
1*9643112
1*9657127
1*9671123
1-96860991
H
o.
Xio0u1thm
7-17 1-9699056
7*18 1-9712993
7*19 1*9726911
7-20, 1-9740810
7*211 1*9764689
7*22! Ir9768549
7*23 1<9782390
7*24, 1-9796212
7*26 1*9810014
7*26 1*9823798
7*27 1*9837562
7-28 1*9861308
7-29 1-9865035
7-30 1*9878743
7*31 1*9892432
7-32 1*9906103
7*33 1*9919764
7-34 1*9933387
7-35 1*9947002
7-36 1*9960599
7*37 1*9974177
7*38 1-9987736
7-39 20001278
7*40 2*0014800
7-41 20028306
7-42 20041790
7*43 20065258
7-44 2-0068708
7*45 2^82140
7*46 2-0095563
7*47 2*0108949
7*48 2^122327
7*49 20135687
7*50 20149a30
7*51 2-0162354
7*62 20176661
7-63 20188960
7*54 2*0202221
7*55 2-0215476
7*56 20228711
7*57 2*0241929
7*58 20255131
7*69 2-0268315
7*60 20281482
7-61 2 0294631
7r62 20307763
7*63 20320878
7*64 2*0333976
HYPERBOLIC LOGAUITHMP.
715
No.
Iiocariihin
^foT
LogaritQin
woT
909
Iiocuithm
soT^
LiORftrithih
813
2-0955613
8*61
2-1629243
2-2071748
9*67
^2586332
B-14
2-0967906
8-62
2-1640851
9-10
2-2082744
9-68
2-2696776
8-16
2-0980182
8-63
2-1552445
911
2-2093727
9-59
2*2607209
8-16
2*0992444
8-64
21664026
9-12
2-2104697
9-60
2-2617631
8-17
21004691
8*66
21575693
9-13
2*2115666
9-61
2-2628042
8-18
21016923
8-66
2-1587147
9-14
2-2126603
9-62
2-2638442
8-19
21029140
8-67
21698687
9*16
2-2137538
9-63
2-2648832
8-20
21041341
8*68
21610216
9*16
2-2148462
9-64
2-2659211
8-21
21053629
8*69
21621729
9*17
2-2159372
9-66
2-2669679
8-22
21066702
8-70
21638230
9-18
2-2170272
9-66
2-2679936
8-23
2-1077861
8-71
21644718
919
2-2181160
9*67
2-2690282
8*24
2-1089998
8-72
2-1656192
9-20
2*2192034
9*68
2-2700618
8-25
2*1102128
8-73
21667653
9*21
2*2202898
9-69
2-2710944
8-26
21114243
8*74
21679101
9*22
2-2213750
9-70
2*2721258
8-27
21126343
8-75
21690536
9*23
2-2224590
9*71
2*2731662
8-28
21138428
8*76
2-1701959
9-24
2-2235418
9-72
2-2741866
8-29
2*1160499
8*77
21713367
9-25
2-2246236
9^73
2-2752138
8-30
21162665
8-78
2-1724763
9*26
2-2257040
9-74
2-2762411
8-31
21174696
8-79
2*1736146
9-27
2-2267833
9*76
2-2772673
8-32
2*1186622
8*80
21747517
9-28
2-2278616
9-76
2*2782924
8*33
2*1198634
8-81
2-1768874
9*29
2-2289386
9-77
2-2793165
8-34
2*1210632
8-82
2-1770218
9-30
2-2300144
9*78
2-2803396
8-35
2*1222616
8*83
2-1781550
9-81
2-2310890
9*79
2-2813614
8-36
21234684
8-84
21792868
9*32
2*2321626
9*80
2-2823823
8-37
21246639
8-85
2-1804174
9-33
2-2332350
9-81
2-2834022
8-38
21258479
8-86
2-1815467
9-34
22343062
9*82
2-2844211
8-39
21270406
8-87
2-1826747
9-35
2*2353763
9-83
2-2854389
8-40
21282317
8-88
2-1838015
9-36
2-2364462
9-84
2*^3864666
8-41
2*1294214
8-89
2-1849270
9-37
2-2375130
9-86
2*2874714
8-42
21306098
8-90
21860512
9-38
2*2385786
9-86
2*2884861
8*43
21317967
8-91
21871742
9-39
2*2396452
9*87
2*2894998
8-44
2-1329822
8-92
2-1882959
9-40
2*2407096
9-88
2-2905124
8-45
2*1341664
8-93
2*1894163
9-41
2-2417729
9-89
2-2915241
8'46
2*1353491
8-94
2-1905355
9-42
2-2428350
990
2*2926347
8-47
2-1366304
8-95
21916535
9-43
2-2438960
9-91
2-2935443
8-48
21377104
8-96
21927702
9-44
2-2449559
9*92
2*2945529
8-49
2-1388889
8-97
21938856
9-46
2*2460147
9-93
2*2956604
8-50
2*1400661
8-98
2*1949998
9-46
2*2470723
9*94
2-2965670
8-51
2-1412419
8-99
2*1961128
9-47
2-2481288
9*95
2-2975725
8-52
2-1424163
900
2-1972245
9-48
2*2491843
9-96
2-2985770
8-53
2-1436893
901
2-1983350
9-49
2-2502386
9-97
2-2995806
8-64
2-1447609
9-02
2-1994443
9-60
2-2512917
9-98
2-3005831
8'55
2-1459312
903
2-2005523
9-51
2-2523438
9-99
2-3015846
8-56
2-1471001
904
2-2016591
9-52
2-2533948
10-00
2-3025861
8-57
2-1482676
906
2-2027647
9-53
2-2544446
11-00
2-3978952
8-6a
21494339
9-06
2-2038691
9-64
2-2564934
12*00
2-4849066
8-69
21506987
9-07
2-2049722
9-55
2-2565411
15-00
2-7080502
8-60
21617632
908
2*2060741
9*66
2*2676877
2000 2*99673221
716 NATITRAL SINES, TANGENTS, SECANTS, ETC.
Tablb of Natural Sinss, Tangents, Secants, &c.
Sine
4
2
i
I
1
6
8
1
1
1
1
i
4
10
t
Deg.
•OOOOOO
•004363
•008727
013090
•017452
•021815
•02«177
•030539
•094900
•089260
•043619
•047978
•062336
•056693
•061049
•065403
•069757
•074109
•078459
•082808
•087156
•091502
•095846
•100168
•104629
•108867
•113208
•117637
•121869
•126199
•130526
184851
•139173
•143493
•147809
162128
•156435
•160748
•166048
•169350
•173648
•177944
•182236
Cosocant
Tangent
Cotangent
Infinite
229-1839
1 14*5930
76*89655
5729869
4584026
38-20155
32-74554
28-65371
2647134
22-92559
20-84283
1910732
17*63893
16-38041
1628979
1433559
18-49373
12*74650
12*07610
11-47871
10-92877
10-43343
9-981229
9-566772
9-186531
8-883672
8-607930
8-205509
7-923995
7-661298
7-415596
7185297
6*968999
6-765469
6-578611
6-892453
6-221128
6058858
5-904948
6'758771
5-619760
6-487404
Cosine
•000000
•004363
•008727
•013091
•017455
•021820
•026186
•030553
•034921
O39290
•043661
•048033
•052408
•056784
•061163
-065544
•069927
•074318
-078702
-083094
•087489
•091887
•096289
•100695
-105104
-109518
•113936
•118358
•122785
-127216
•131653
-136094
•140541
•144993
•149461
•153915
•168384
•162860
•167343
•171831
-176327
-180830
•185389
Secant
Infinite 1
I229-18I71
114^58871
76-39001 jl
67^289961
45-82935 1
3818846 1
32-78026 1
28-63625 1
25-451701
22-90377 1
20-81883 1
1908114 1
17*61056 1
16*84986 1
15-25706 1
14-30067 I
13-45663 I
12-70621 I
1203462 1
11-48005 1
10-88292 1
1^-38540 1
9-931009 1
9-514365 1
9180985 1
8-776887 1
8-448957 1
8144346 1
7860642 1
7-695754 1
7-347861 1
7116370 1
6-896880 1
6-691156 1
6-497104 1
6*313752 1
6-140230 I
5-975764 I
5-819657 1
5-671282 1
5-630072 I
5-395517 I
Secant I Cotangent
Tangent
•000000
•000010
•000038
•000086
•000162
•000238
•000343
•000467
•000610
-000772
•000953
•001153
•001372
•001611
•001869
-002146
•002442]
-002757
-003092
•003446
•003820
•004213
•004625
•005067
-005508
•005979
•006470
•006980
•007510
•008060
•008629
•009218
•009828
-010467
•011106
-011776
•012465
-013175
-013905
-014656
*015427
-016218
-017080
Cosecant
Cosine Deg.
1-00000
•999991
•999962
•999914
•999848
•999762
•999657
•999534
r999391
•999229
•999048
•998848
•998630
•998392
•998135
•997859
•997564
•997250
•996917
•996666
•996195
•995805
•995396
•994969
•994522
•994056
•993572
•993069
-992546
•992005
•991445
•990866
-990268
•989651
•989016
•988362
•987688
•986996
•986286
•985556
•984808
'984041
983255
90
89
88
87
86
85
84
83
82
81
Sine
80
79I
Deg.
NATURAL SINES, TANGENTS, SECANTS, ETC. 717
Pag.
10 J
11
i
12
13
14
15
16
17
18
19
20
21
22
Sine
•186524
•190809
•195090
•199368
•203642
•207912
•212178
•216440
•220697
•224951
•229200
•233445
•237686
•241922
•246153
•250380
•254602
•258819
•263031
267238
•271440
•275637
•279829
•284016
•288196
•292372
•296542
•300706
•304864
•309017
•313164
•317306
•321440
•325568
•829691
•333807
•337917
•342020
•346117
•350207
•354291
•358368
•862438
•366501
•370557
•374607
Beg, Cotine
CkisecaBt
5*361239
5-240843
5-125831
5015852
4-910584
4-809734
4-713031
4^620226
4-531090
4-446412
4*362994
4*283658
4*207233
4*133566
4-062509
3-993929
3^927700
3-863703
3-801830
3-741978
3-684049
3-627955
3-673611
3-520937
3-469858
3-420304
3-372208
3-325510
3*280148
3*236068
3-193217
3-151545
3111006
3^071554
3033146
2995744
2*959309
2*923804
2*869196
2*855451
2*822538
2-790428
2*769092
2*728504
2*698637
2*669467
Tangent
•189856
•194380
•198912
•203452
•208000
•212557
•217121
•221695
•226277
*230868
•236469
•240079
•244698
•249328
•253968
•258618
•263278
•267949
*272631
*277325
*282029
*286745
•291473
•296214
•300966
•305731
•310608
315299
•320103
•324920
•329761
•334596
•339454
•344328
349216
354119
■369037
•363970
*368920
•373886
•378866
•383864
•388879
•393911
•398960
•404026
Secant Cotangent
Cotangent! Secant
6267152
5144654
602734011
4-916167
4^807686
4-70463011
4-605721
4^610709
4-419364
4331476
4*246848
4*165300
4086663
4010781
3937509
3*866713
3*798266
3*732051
3*667958
3*606884
3*645733
3*487414
3-430845
3*375943
3*322636
3*270863
3-220526
3*171595
3128999
3*077684
3032695
2*988685
2*946905
2*904211
2*863660
2*823913
2*765281
2747477
2*710619
2*674622
2*639465
2*60500)
2*671496
2*538648
2*506620
2-475087
Tangent
1'017863
1*018717
019591
1*020487
1*021403
022841
1*023299
1*024280
1*025281
1026304
1-027349
1*028416
1029503J
1030614
1-031746
1-032900
1034077
1-035276
1036498
1037742
1-039009
1-040299
1041613
1*042949
1*044309
1*045692
1047099
l^048529
1*049984
1*051462
1*052965
1054492
1*056044
1057621
1^059222
1-060849
1*062501
1*064178
1*065881
1*067609
1*069364
1071145
l^072952
1074786
1*076647
1*078535
Cosecant
Coilne
-982450
•981627
•980785
•979925
•979046
-978148
-977231
-976296
-975342
-974370
-973379
•972370
•971342
•970296
•969231
•968148
•967046
•965926
•964787
•963631
•962455
•961262
•960050
•958820
•967571
•956305
•955020
•953717
•952396
•951057
•949699
•948324
946930
•946619
•944089
•942642
•941176
•939693
•938191
•936672
•935135
•983580
•93^008
•930418
•928810
•927184
Sine
NATURAL SINES, TANGENTS, SEOANTS, ETC.
Bine
•378649
•382683
•386711
•390731
•394744
•398749
•402747
•406737
•410719
•414693
•418660
•422618
•426569
•430511
•434445
•438371
•442289
•446198
•460098
•453991
•457874
•461749
•465615
•469472
•473320
•477159
•480989
•484810
•488621
•492424
•496217
•500000
•603774
•507538
•511293
•515038
•518773
•522499
•526214
•529919
•533615
•537300
•540976
•544639
•548293
•561937
CodnA
Gotecsnt
2-640971
2*613126
2-68591 1
2-559305
2533288
2-507843
2-482&50
2458593
2-434756
2-411421
2-388576
2-366202
2-344288
2*322821
2-301786
2-281172
2-260967
2-241159
2-221736
2*202689
2-184007
2-165681
2147699
2-130055
2-1 12737
2096739
2079051
2-062666
2^046576
2030772
2015249
2^000000
1^986017
1-970294
1^955826
1-941604
r927624
1-913881
1-900368
1-887080
1-874012
1-861159
1-848516
1-836079
1-823842
1-811801
Secant
Tangent
409111
414214
419335
424475
429634
434812
440011
445229
460467
455726
461006
466308
471631
476976
482343
487733
493146
498682
604042
609625
515034
520667
526126
63U09
537319
642966
548619
554309
560dl$7
565773
57TB*7
577350
583183
589045
594938
600861
606815
612801
618819
624869
630953
687070
643222
649408
666629
661886
Cota pei t
Cotangent Secant
2-444326
2-414214
2-384729
2-366862
2-327563
2^299843
2272673
2246037
2-219918
2-194300
2*169168
2144507
2120303
2096544
2^073216
2*060304
2-027799
2-006690
•983964
•962611
•941620
-920982
-900687
-880727
•861091
•841771
•822769
-804048
^786629
•767494
-749637
•732051
-714728
'697663
-680849
-664280
-647949
-631852
•615982
•600335
-584904
•669686
-554674
-639865
-526264
-510835
1-0804501
1-082392
1-084362
1086360
1-088387
1-090441
1-092624
1094636
1-096777
-098948
•101148
-103378
•106638
-107929
-110260
-112602
•114986
-117400
•119847
•122326
•124838
•127382
•129969
•132570
•135216
-137893
•140606
-143354
•146137
-148966
•161810
•154701
•167628
•160692
•163594
•166633
-169711
-172828
•175983
•179178
•182414
•185689
•189006
•192363
•195763
•1 99206
Tangent | Cosecant
Corine
-926641
•923880
•922201
•920505
•918791
•917060
-915312
•913546
•911762
•909961
-908143
-906308
•904455
•902585
-900698
•898794
•896873
•894934
-892979
-891007
•889017
•887011
-884988
-882948
•880891
•878817
•876727
•874620
•872496
•870366
•868199
•866025
•863836
•861629
•869406
•857167
-854912
•852640
•860352
•848048
•845728
•843391.
-841039
-888671
•836286
•83^886
bine
I>eg.
67
66
65
64
63
62
61
60
Deg.
NATURAL SINES, TANGENTS, SECANTS, ETC. . 71
Beg:
Blrie
•656670
•669193
•662805
•666406
•669997
•673676
•677145
•680703
•684250
•687785
•691310
•694823
•698326
•601816
•606294
•608761
•612217
•616662
•619094
•622616
•625924
•629320
•632706
•636078
•639439
•642788
•646124
•649448
•652760
•656069
•659346
'662620
•666882
-669131
•672367
•676690
•678801
•681998
•686183
•688366
•691613
•694668
•697791
•700909
•704016
•707107
Peg.
Gosiiie
Ooeecant
•799952
•788292
•776815
•765517
•764396
•743447
•732666
•722061
•711697
•701302
•691161
•681173
1-671334
•661640
•662090
•642680
•633407
•624269
•616264
•606388
•597639
•589016
•680516
•672134
•663871
•655724
•647691
•639769
•631957
•624263
•616665
•609161
•601768
•494477
•487283
•480187
•473186
•466279
•469464
•462740
•446104
•439567
•433096
•426718
•420425
•414214
Secant
Tangent
•668179
•674609
•680876
■687281
•693725
•700208
•706730
•713293
•719897
•726543
•733230
•739961
•746735
•763554
•760418
•767327
•774283
•781286
•788336
•796436
•802585
•809784
•817034
•824336
•831691
•839100
•846663
•854081
•861666
•869287
•876977
•884726
•892634
•900404
•908336
•916331
•924391
•932616
•940706
•948966
•967292
•966689
•974167
•982697
•991311
1^00000
Cotangent
Cotangent Secant
496606
482661
468697
465009
441494
428148
414967
401948
389088
376382
363828
351422
339162
327045
316067
303225
291518
279942
268494
267172
245974
234897
223939
213097
202369
191754
181248
170850
160567
160368
140282
130294
120405
110613
009142
091309
1-081794
072369
1-063031
053780
1^044614
036630
026529
017607
008766
000000
1
1
1
1
1
1-
1-
1-
1'
I
1
1
1
1-
1^
1
1
1
1
1
1
1
1
1
1
1
1
1
I
1
1
1
1
1
1
1'202690
Tangent-.
•206218
•209790
•213406
•217068
•220775
•224627
•228327
•232174
•236068
•240011
•244003
•248044
•252136
•256278
•260472
•264719
•269018
•273371
•277779
•282241
•286760
•291335
•295967
•300658
•306407
•310217
•316087
•320019
•326013
•330071
•385192
•340380
•345633
•350953
•366342
•361800
•367328
•372927
•378699
-384344
•390164
•396059
•402032
•408083
•414214
Cosine
•831470
•829038
•826590
■824126
•821647
•819152
•816642
•814116
•811674
'809017
•806445
•803857
•801264
•798636
•796002
•793353
•790690
•788011
•785317
•782608
•779885
•777146
•774393
•771626
•768842
•766044
•763233
•760406
•767666
•754710
•761840
•748966
-746057
•743146
•740218
•737277
•734323
•731364
•728371
•725374
•722364
•719340
•716302
•713260
•710185
•707107
Coee^'ant Sine
i
r>e
pgQ LOGARITHMIC SINES, TANGENTS, SECAmPfi, ETC.
Table 07 LoeABiTHXic SnTEs, Tangents, Secants^ &c.
Deg.
1
4
f
JL
1
8
1
f
a
>eg.
Bine
— 00
7-63982
7-94084
8-11693
8-24186
8-33876
8-41792
8-48485
8-54282
8-59395
8-63968
8-68104
8-71880
8-75363
8-78568
8-81560
8-84358
8-86987
8-89464
8-91807
8*94030
8*96143
8-98157
9-00082
9-01923
9-03690
9*05386
9-07018
9*08589
910106
9-11570
9-12985
914356
9-15683
9-16970
9-18220
9*19433
9-20613
9*21761
9-2d878
9*23967
9-25028
9-26063
Cosiiie
Coeecaot
+ 00
12-36018
1205916
11-88307
11-75814
11-66125
11-58208
11-51516
11-45718
11-40605
11-36032
11-31896
11-28120
11-24647
11-21432
11-18440
11-15642
11-13013
11-10536
11-08193
11-05970
11-03857
11-01843
10-99918
10-98077
10-96310
10-94614
10-92982
10-91411
10-89894
10-88430
10-87015
10-85644
10-84317
10-88030
10-81780
10-80567
10 79887
10-78289
10-77122
10-76033
10-749721
10-78937
Secant
Tangent
7*63982
7-94086
8-11696
8*24192
8-33886
8-41807
8-48505
8-54308
8*59428
8*64009
8*68154
8-71940
8-75423
8-78649
8*81653
8-84464
8-87106
8*89598
8*91957
8-94195
8*96325
8*98358
9-00301
9-02162
9*03948
9-05666
9-07320
9-08914
9-10454
911943
9*13384
9-14780
9-16135
9-17450
9-18728
9-19971
9-21182
9-22361
9-23510
9-246S2
9-25727
9-26797
CotangKii
CatKOffffDt
+
12-36018
12-059U
11-88304
11*75808
11-66114
11-58193
11-51495
11-45692
11-40572
11*35991
11-31846
11-28060
11-24577,
11-21351
1118347
11-15536
11-12894
11-10402
11-08043
11-05805
11-03675
11-01642
10-99699
10-97838
10-96062
10-94334
10-92680
10-91086
10-89546
10-88057
10-86616
10-85220
10-83865
10-82550
110-81372
i 10-80029
10-78818
10-77639
110-76490
10-75368
10-74273
10-73203
Tftngent
Secant I Cosine
10-00000
1000000
1000002
10-00004
10-00007
10-00010
10K)0015
10-00020
10-00026
10-00034
1000041
10-00050
10-00060
10-00070J
10-000811
10-00093
10-00106
10-00120
10-00134
1000149
1000166
10-00183
10-00200
10-00219
10-00239
1000269
10-00280
10-00302
10-00325
1000349
10-00373
1000399
1000425
10-00452
10-00480
10-00608
10-00588
10*00568
10*00600
10-00632
10-00665
10*00699
10-00733
I>eg
Cosecant
10-00000
9*99999
9-99998
9-99996
9-999^3
9-99990
9-99985
9-99980
9-99974
9-99967
9-99959
9-99960
9-99940
9*99930
9-99919
9-99907
9-99894
9-99880
9-99866
9-99861
9-99834
9-99817
9-99800
9-99781
9-99761
9-99741
9-99720
9-99698
9*99676
9-99661
9-99627
9-99601
9-99576
9-99548
9-99620
9-99492
9-99462
9-99432
«-99400
9-99368
9-99335
9-99301
9-99267
Sine
wmmmt
90
89
3.
1
1
4
88
1
1
4
87
86
1
1
1
4
85
84
83
1
1
4
1
I
4
82
81
80
79i
Deg.
LOGABITHMIO SIN£S, TANGENTS) S£GANTS» ETC.
7!
Deg.
%
\
'?5
8^
Ii
t!
^1
82
i
4
>
i
4
i
11
i
\
12
13
U
15
16
t
i 17
18
1
1
1
4
!
19
1
4
20
1.
1
i
21
Sine
-22
>efir.
9-27073
9-28060
9-29024
9-29966
9-30887
9-31788
9-32670
9 33634
9-34380
9-35209
9-36022
9-36819
9-37600
9-38368
9-39121
9-39860
9-40586
9-41300
9-42001
9-42690
9-43367
9-44034
9-44689
9-45334
9-45969
9-46594
9-47209
9-47814
9-48411
9-48998
9-49577
9-50148
9-50710
9-51264
9-51811
9-52350
9-52881
9-53405
9-53922
9-54433
9-54936
9*65433
9-55923
9-56408
9-56886
9-57358
Goeecant
Cosine
10-72927
10-71940
10-70976
10-70034
10-69113
10-68212
1067330
10*66466
10-66620
10-64791
10-63978
10-63181
10-62400
10-61632
10-60879
1060140
10-59414
10-58700
10-57999
10-57310
10-56633
10-55966
10-56311
10-54666
10-54031
10-53406
10-52791
10-52186
10-51589
10-51002
10-60423
10-49862
10-49290
10-48736
10-48189
10-47650
10-47119
10-46596
10-46078
10-45567
10-45064
10-44567
10-44077
10-43592
10*43114
10-42642
Secant
Tangent
9-27842
9-28865
9-29866
9-30846
9-31806
9-32747
9-33670
9-34676
9-35464
9-36336
9-37193
9-38035
9-38863
9-39677
9-40478
9-41266
9-42041
9-42805
9-43658
9-44299
9-45029
9-45760
9-46460
9-47160
9-47852
9-48634
9-49207
9-49872
9-50629
9-51178
9-61819
9-52462
9-63078
9-63697
9-54309
9-54916
9-65614
9-56107
9-66693
9-67274
9-67849
9-58418
9-58981
9-59540
9-60093
9-60641
Ciotangent
10-72168
10-71135
10-70134
10-69154
10-68194
] 0*67253
10-66330
10-65424
10-64536
10-63664
10-62807
10-61966
10-61137
10-60323
10-59522
10-58734
10-57969
10-57196
10-56442
10-65701
10-64971
10-54260
10'53540
10-62840
10-62148
10-51466
10-60793
10-50128
10-49471
10-48822
10-48181
10-47548
10-46922;
10-46303
10-45691
10-46085
10*44486
10-43893
10-43307
10-42726
10-42151
10-41582
10-41019
10-40460
10-39907
10-39359
Secant
Cosine
Cotangrat Tangent
A ati
10-00769
10-00806
10-00843
10-00881
10-00920
10-00960
10*01000
10-01042
10-01084
1001128
10-01172
1001217
10-01263
1001310
10-01357
10-01406
10-01456
10-01506
10*01557
10-01609
1001662
10-01716
10*01771
10-01826
1001883
10-01940
10-01999
1002068
10-02118
1002179
10-02241
10-02304
10-02368
10-02433
10-02499
10-02666
10-02633
10-02701
10-02771
10-02841
10-02913
10-02986
10 03058
10-03132
10-03207
10-03283
Cosecant
9-99231
9-99195
9-99157
9-99119
9-99080
9-99040
9-99000
9-98968
9-98916
9-98872
9-98828
9-98783
9-98737
9-98690
9-98643
9-98594
9-98646
9-98494
9-98443
9-98391
9-98338
9-98284
9-98229
9-98174
9-98117
9-98060
9-98001
9-97942
9-97882
9-97821
9-97769
9-97696
9-97632
9-97667
9-97601
9-97436
9-97367
9-97299
9-97229
9-97169
9-97087
9-97016
9-96942
9-96868
9-96793
9-96717
Sine
J 2 LOGAEITHMIC SINES, TANGENTS, SECANTS, ETC.
Bine
9-57824
9-58284
9-58789
9-59188
9-59632
9-60070
9-60603
9-60931
9-61354
9-61773
9-62186
9-62595
9-62999
9-63398
9-63794
9-64184
9-64571
9-64953
9-65331
9-65705
9-66075
9-66441
9-66803
9-67161
9-67515
9-67866
9-68213
9-68557
9-68897
9-69234
9-69567
9-69897
9-70224
9-70547
9-70867
9-71184
9-71498
9-71809
9-72116
9-72421
9-72723
9-73022
9-73318
9-73611
9-73901
974189
Cosine
Cosecant
10-42176
10-41716
1041261
10-40812
10-40368
10-39930
10-39497
10-39069
10*38646
10-38227
10-37814
10-37405
10-37001
10-36602
10-36206
10'35816
10-35429
10-35047
10-34669
10-34295
10-33925
10-33559
10-33197
10-32839
10-32485
10-32134
10-31787
10-31443
10-31103
10-30766
10-30433
10-30103
10-29776
10-29453
10-29133
10-28816
10-28502
10-28191
10-27884
10-27579
10-27277
10-26978
10-26682
10-26389
10-26099
10-25811
Secant
Ttuigent
9-61184
9-61722
9-62256
9-62786
9-63310
9-63830
9-64346
9-64858
9-65366
9-65870
9-66371
9*66867
9-67360
9-67850
9-68336
9-68818
9-69298
9-69774
9-70247
9 70717
9-71184
9-71648
9-72109
9-72567
9-73023
9-73476
9-73927
9-74375
9-74821
9-76264
9-75705
9-76144
9-76580
9-77015
9-77447
9-77877
9-78306
9-78732
9-79156
9-79679
9-80000
9-80419
9-80836
9-81252
9-81666
9-82078
Cotangent
Cotangent
10-38816
10-38278
10-37744
10-87215
10*36690
10-36170
10-35654
10-35142
10-34634
10-34130
10-33629
10-33133
10-32640
10-32150
10-31664
10-31182
10-30703
10-30226
10-29763
10-29283
10-28816
10-28352
10-27891
10-27433
10-26977
10-26524
10-26073
10-25625
10-25179
10-24736
10-24295
10-23856
10-23420
10-22985
10-22553
10-22123
10-21694
10-21268
10-20844
10-20421
10*20000
10-19581
10-19164
10-18748
10-18334
10-17922
Tangent
Secant
10-03360
10-03438
10-03517
10-03597
10-03678
10K)3760
10-03843
10-03927
10-04012
10-04098
10-04185
10-04272
10-04361
10-04451
10-04642
10-04634
10-04727
10-04821
10-04916
10-05012
1005109
10-05207
10-05306
1005407
10-05508
10-05610
10-05714
10-06818
1005924
10-06030
10-06138
10-06247
10-06367
10-06468
10-06580
10-06693
10-06808
10-06923
10-07040
10-07168
10-07277
10-07397
10-07618
10-07641
10-On65
10-07889
Coeeoant
Cosine I>eg.
9-96640
9-96562
9-96483
9-96403
9-96322
9-96240
9-96157
9 96073
9-95988
9-95902
9-95816
9-95728
9-95639
9-95549
9-95468
9-95366
9-96273
9-96179
9-95084
9-94988
9-94891
9-94793
9-94694
9-94693
9-94492
9-94390
9-94286
9-94182
9-94076
9-93970
9-93862
9-93763
9-93643
9-93532
9-93420
9-93307
9-93192
9-93077
9-92960
[ 9-92842
9-92723
9-926J3
9-92482
9-92369
9-92235
9-92111
Sine
67
66
65
64
63
62
61
I
60
59
LOaAtlltHMIC SINES, TANGENTS, SECANTS, ETC. 729
Peg.
33|
34
35
36
37
38
39
40
41
42
43
^4
45
D0g.
Sine
9-74474
9-74756
9-76036
9-75313
9-75687
9-75869
9-76129
9-76395
9-76660
9-76922
9-77182
9-77439
9-77694
9-77946
9-78197
9-78445
9-78691
9-78934
9-79176
9-79415
9-79652
9-79887
9-80120
9-80351
9-80580
9-80807
9-81032
9-81254
9-81475
9-81694
9-81911
9*82126
9-82340
9-82551
9-82761
9-82968
9-83174
9-83378
9-83581
9-83781
9-83980
9-84177
9-84373
9*84566
9-84768
9-84949
Goeecant
10-25526
1025244
10-24964
10-24687
10-24413
10-24141
10-23871
10-23605
10-23340
10-23078
10-22819
10-22561
10-22306
10-22054
10*21803
10*21555
10*21309
10-21066
10-20824
10*20585
10*20348
10*20113
10*19880
10-19649
10-19420
10-19193
10-18968
10-18746
10-18525
10-18306
10-18089
10-17874
1017660
10-17449
10-17239
1017032
10-16826
10-16622
10-16419
10-16219
10-16020
10-15823
10-15628
10-15434
10-15242
10-15062
Ckmae
Tangent
Secant
9-82489
9-82899
9-83307
9-83713
9-84119
9-84623
9-84926
9-85327
9-85727
9-86126
9-86524
9-86921
9-87317
9-87711
9*88105
9-88498
9*88890
9*89281
9*89671
9*90061
9*90449
9*90837
9-91224
9-91610
9-91996
9*92381
9-92766
9-93160
9-93633
9-93916
9-94299
9-94681
9-96062
9*95444
9-96826
9-96206
9*96586
9-96966
9-97345
9-97725
9-98104
9-98484
9-98863
9-99242
9-99621
10-00000
Cotangent
10-17611
10-17101
10-16693
10-16287
10*16881
10*15477
10-15075
10-14673
10-14273
10*13874
10*13476
10-13079
10*12683
10*12289
10*11895
10*11502
10*11110
10*10719
10*10329
10*09939
10*09551
10*09163
10*08776
10-08390
10-08003
10-07619
10-07234
10-06860
1006467
10-06084
10-06701
10-06319
10-04938
10-04566
10-04175
10-03796
10-03414
10-03034
10-02665
10-02276
10-01896
10*01616
10-01137
1000758
10-00379
10-00000
Secant
Cotangent
10-08016
10-08143
10-08271
10-08401
10-08631
10-08664
10-08797
10-08931
10-09067
10-09204
1009343
10-09482
10-09623
10-09765
10-09909
10*10063
10*10199
10-10347
10-10496
10-10646
10-10797
10*10950
10-11104
10-11259
10*11416
10-11675
10-11734
10-11895
10-12068
10-12222
10-12387
10*12554
10*12723
10-12893
10-13064
10-13237
1013411
10*13687
10*13766
10-13944
10-14124
10*14307
10*14490
10*14676
10*14863
10*16062
Tangent
Cosine
Coeecant
9-91985
9-91867
9-91729
9-91699
9*91469
9*91336
9-91203
9-91069
9-90933
9-90796
9-90657
9-90518
9-90377
9-90235
9-90091
9-89947
9-89801
9*89663
9-89605
9*89354
9-89203
9-89050
9*88896
9-88741
9-88584
9-88426
9-88266
9-88106
9-87942
9*87778
9*87613
9-87446
9-87277
9-87107
9-86936
9-86763
9-86589
9*86413
9-86236
9-86056
9-86876
9*86693
9*85510
9*85324
9*86137
9*84949
Sine
INDEX.
**A" BRACKETS, 3*4- -
. xX. Abeolato tempena^tare, 406
Acceleratioa. M ^
diagrram, 83 •
Accommpdation, pa«seng«r, 479
Aoooraoy of oalotilati«us, 6
Acid, dippinjr, 601
Aome screw threads, 358
Admiralty coelflciexit,' 180-4 ;
" inatmotiona te ' iniw
steel, 264
" — limiting aize of plates,
237
^- 1 rivetfng regulations.
289-92 .
' tests. C«.. also Tetts)
*"' for cable, ^25
. —chain, 629
• ■■ materials, 233
— . -s*w. rope, 678,
679
Advance Cateeringr), 3B5.
Aerodynamics, 406
■ *— biplane effects, 424
■ • calcalations for
lift, and drift, 427
— centre of pressure,
414
pressure, 422
plates, 416
409
mal, 406
■distribution of
force on curvBd
— - other bodies,
plates, nor-
tanoe, 410
Bienfis, 428
' ' ■ inclined, 411
friotiohal resis-
model ezperi-
■ shape of edgrea, 421
Aeronautics, 431
• descripfcibn of air-
ahipa, 4SI-8 :
"" • envelope tnaterial, 446
■ tensions in,
438-444
'form of airships, 444
*~Iiftiab8 power, 436
model fficp^ments.
Aeronautics, resistance, 488
stability, 435
Aeroplane, v. Aeroaynamica
Aground, stability- of ship, 142
Air and i^ater^ results in, 431
— circuits, power in, 898
— craft, wire tope for, 681
— efflux of, 460
-^ gap, eifect of, ^408
— movement of, 396-401
— — quantity required, 399, Ml ^
— resistance of, 397
'■—' pressure on bodies, 409
plates^ curved^ 416-
>■' ■ JIM. *>*p|gna, iOB" \
418
ship, V. AeronautioB
-^- weight of, 396
Ale measure, 608
Algebraic symbols^ 1
Alloys, table o^ 269
Altitude of airships, 487
Aluminium bronze, 258
Amsler-Laffon integrator, 126
Anchor, cable attachment, 546
Anchors, 'tests -and number of, -
483, 626. 627, 630, 632
Aiigl» bulb, weight, etc., 250
' iron weight, 223
— • measurement of, 8
steel, weight, etc., 226,246
Angular motion, 64
velocity, 82
iljinealing, 267, 264, 294
Antilogarithm, 5
■■ table, 704 •
Apothecaries' weight, 606
Apparent wave peciod, 149
Appendage, 96
<airship«), 442
' u » ■ tesistanoe, 162
Arc, circular, 22, 2S, 60 ■
Ardendy, 208 ' .
Area of circles (table), 647
fig ares, general rules, 42-8 -
— — geometrical figures, 36-41
^ midship section, 91 "
sections, 91
segments of circle (table).
662
surfaces, V. Surface
Armament particulars, 1C8, 380-8
Armour, 108, 378, 888
726 INDEX.
Annonx bars* air resistance, 896
bolte, 888
Armstrong VBlom. 880
Aspeot ratio. 411 ff.
Atwood's formula, 115
AToirdnpois weight, 606
Axis of rolling, 161
B;ABBIT'S white metal, 258
> Backstay, screw for, M5
Baggage, <10d
Balanoe of rudders, 866
Ball-bearing friction, 86$
Ballast for tnolUUng ezpenmenty
188
Balllstios» 880-6
Ballonet, 482
Balloon, V. A^onautiof
Bttns« weight of oast iron, 28A
Barges, towing data, 196
Barnes* method for stability,
118-124
Bars, round, elliptio, square, •to.r
weight, 222
Bath, size of, 666
Beam, determinatian of, 891
effect on speed 172
— -stabiUty. 129
Beani, BJC. and S.F., general
800^ .. ,
I I .L . ■ I ■ « ■ ■ grapnioai,
803-6
.. ■ B.H. and deflection, 301-8,
822, 823
oontinuous, distribution of
load, 820
■supports to, 822
of equal strength, 820, 821
stress due to bending, 812
•shetir, 832
-— strongest out from log, 29
uAsymmetrioal, 826
Beams, scantUags of, 608, 617,
622. 624
Bearing pressure on screw, 356
values of rivets, 286-8- -
Bearings, fri^ion of, 863
shaft, distaace betmeenr
836
■ • '■ ■ working pressure on,
885, 870. 3TI , «,^
Beaufort soale for winds^ 210
Beer measure, 608
Bell months to tranking, 399
Belt gearing, 367-9
~- length of, 867
— itrength on slip of, 867-9
Bend in pipe, 404
Bend in trunk, SOT .
mnti^ng moment, airship, 44b
•beam, 800-8,882,
>(BqiiiTalent on
•launching, 876
• rudder head,870
ship. 846-4B
sbafte, 838
Berths. 478, 608
BiUpe Iwel. IW
Bilging, effect of, 140-2*
Biplane, 424
Bituminous paint, 695
BladC) area of screw. 191
BlakB s stopper, 640
Blechynden^s formula, 167
Block eoiefficisnt, 93
Blocks, cathead^ 662
■ engineers tackle, eSS
galTsaized^ 669
leading. 660
— Uf t and snatok, 542, 670
— — — miscellaneous, 671
snatch, 642. 670 ^ ^^^
--**-» Weston, differential, 355,
567
Board of Trade regulations--^
Machinery, boilers, 450-58
distillers. 466
- eTaporator8,v468
'• furnaces, m7
— — — — materials, 448
I mountings, 468
refrigerators, 466.
— ^— safety valves, 469
shafts, 464
spare gear, 463
steam pipes, 469
superheaters, 468
Hotor taunohes, 468
— — ^ vessels, 466
Ships, anchors 627
cables, 628
— ^^"oertiftoates Cvarious), 469
■ ■ compasses, 477
' . distress signals, 477
ismigrant, 400. 608
fire appliances, 476
hawsi&s, 578, 677
life-saving appUaofles,
495
master's ^and
sp&oes, 4?7
passenger
ecew's
aMommoda-
tlon, 479
-^-—plying limits, 469
— pumps, 476
feteerisg gedr, 476
-rantilatiOtt, 400
Boats, number and complement,
496
INDEX.
727
BoAi0, pasMngors allowed in, 483
leaatlings and weijrhU.
664, 666
slings for, 399
Bobstay, 686-9
Boiler monntings, 462
Boiler -rooms, size of, 389
'• air required, 899
Boilers, iron, rales for, 468
steel, rules for, 460.
tests for. 448
Bolt, armour, 106, 888
deok, 656
— screwed, strenerth of, 839
— steel, tests for, 270
Boltlueads, weight of, 238
Bolt rope cordage, 574
Bon jean ourres. 346, 873
Bow lights, 487
Bowsprits, 686, 690, 692
Box couplings, 836
Braced structures, 294-9
Brus, 267, 279
(Narval), r. Jfaval brass
plate, weight of, 230
Breadth (Lloyd's), 607, 6S0
Bricklaying, 661
Bridges, scantlings of, 610, 514
Bristol OhanneL distances down,
.216
British Corporation rule for
rudders, 871
Standard keys and keyways.
839,840
633-7
gles, 260
screws and pipes,
sections, 241-64
bulb an-
263
vOGo^
244
gles, 246
242
angles, 248
channels,
equal an-
I-beams,
tees, 262
unequal
thermal unit, 406
tonnage, 490
seds, 264.
Bronze, 257
Builders' O.IT. tonnage, 494
Bnlb angles, 230
— tees and plates, 253
Bulk modulus, 810
BulkheadSf Ck>mmittee*s report,
483
Lloyd's rules, 608, 516
- — ! — number ret^uired, 478
Bulkheads, strength and defleo-'
tiob, &&, 824. 488
weight and O.G., 104
Bullivant's B.W. rope, 680-2
• winch (crab), 683
Bulwarks, height of, 469, 479, 481
Bunks and berths, 478, 603
Buoyancy, 89
airships, 437
centre of, ▼, Centre of
buoyanoff
— ' curve of, 112, 118, 316
lifeboats. 496
Butts and straps, ^9, 290, 621
C&.BLB, attachment to anchor,
646
iron tests, 276
link, proportions of,
638, 666
provision of and tests
for, 626, 628, 632, 634
stowage, 644
Oaloulations, numerieal, 6
Calculus, differential and in-
tegral, 19-21
Camber of launching ways, 878
Canvas, 282, 283
Cargo, allowable, 606
Carlisle Bridge, distances from,
214
Carrigaloe Ferry, distances from,
21S»
Cask gauging, 604, 618
Cast iron, 266, 276, 458
baUs, weight, 231
pipes, weight, 2^
teeth, 3i2
' steel, 256, 272, 285, 450
Castings, shrinkage of, 235
Cat chain, 629, 538
Catenary, 16-18, 27
Cathead block, 562
Cattle, 482, 605
Caulking, wood, 697
Cavitation, 192
Cement, 602
Centignde, 406
Centre of buoyancy, 89, 91. 112-14
longitudinal,
182
ehift due to
trim, 188
effort, 208
flotation, 96, 131, 132
gravity, rules for, 62-9
geometrical
flgurea, 69-62
hull, lor
masses, 79
728
INDEX.
Centre of gravllv. ships. 108-9, 118
■ litterai resistance* 208
■ peroossion, 87
^— ~— pressure (taydrostatioal).
CmoviBg in air),
414, 418. 422
« Cmoving in
•water)» 369
Centrifagal force, 87, 358
Gertifioate, Boanl oi Trade* 469
Chain (catenary), 15-18
— — for bloolES, 667
■ ■ ' proportions of link, 628"
weight and tests, 529
Channels (British Standard), 244
Characteristic (logarithm), 4
Chargre, weight, etc., 380-7
Chrome yanidium steel, 258
Circle, Area and circumference,
rules, 87, 38
' ' « table,
642, 647
properties, 39
Circular aro, 22, 23, 60
bars, ▼. Ro^nd
- ' measure, 8
■ ' — table, 639
ring, flat, area, 40
sector, area, 40
C.Q., 61
■segment, area, 40
Ctable), 652
CO., 61
■zone, area, 40
Circumference of -circle (table),
642
Clear hawse slip,- 551
Cleat, belaying, 612, 554
Closet, ▼. Water closet
Clyde, distance down, 213 .
Coal, effect on bilging, 140
endurance, 189
measure, 608
— per H.P. per hour, 390
stowage and weight, 103
Coamings, 609
Coefficient of air resistance, 406
■ fineness, 93
friction, 358, 363,
376, 377
■^ta^iB^^
propulsion, 161, 1G3
rigidity, 310
Coins, English, 633
Coir rope, 672
Coke measure, 609
Collisions, prevention of, 487 .
Colours. for drawings, 565
■ harmony oft 696
Columns, strength of, 328
Committee, Bulkhead, recommen-
dations, 483
Comparison, Law of, 169, 184,
428
Compartment, timet to empty,
400, 403
Compass, points of, 636
provision of, 477, 482
Complement of an angle, 7
Compound pendulum, 87
Compressive strength and stress,
328-31
Conduction of heat, 405
Cone, air resistance, 409
CO., 61
development, 30
— ^— volnme, 49
■ of fruatrum, 60
Conic seciions, 13, 27
Conservation of energy, 85
Constant system of notation, 176
Consumption of fuel, 390
Convection of heat, 406
Copper, 267, 279
pipes, weight, 236, 459
plates, weight, 230
— ^— rods, weight, 234
Cordage, hemp, 672
'" manilla, 283, 673
■ various, 674
Corn measure, 608
Coznish, *Dr. Vaughan, wave
dimensions, 148, 149
Corresponding speeds, 169, 428
Cosecant, cosine, 7
• (table). 718
Cotangent, 7
(table). 716
Cotton fabrio for airships, 447
Countersinking, 265, 291
Couple, mechanical, 77
Couplings, 835
Course, indication of, 490
of ship turning, 364
Coventry Ordnance Co., guns,
386
Coversed- sine, 7
Cowl, 400
Grab winch, 583
Cranes, hand and steam, 362
Creosotirig timber, 658
Gr«w space, 477
Critical angle, 412, 418
Cross curves of stability, 117 .
Crosshead for davit, 516
Cube, oube root (table), 655
Cubic measure. 607, 614
Curvature, radius of, 20
Curve of buoyancy, 112, 113
displacement, 92
metacentres, 112
' ■ midship section areas, 91
sectional areas, 91, 173, 393
''— ' — tons per inch, 92
INDEX.
729
Carved plates, pressure on, 41&
Cwt. and kilogrammes (table),
626
iona (table), 629 .
-*-^
Cycloid, etc., 18. 33
area, 41
Cylinder, aJT resistance, 409
O.a., 61
^— — — — development, 30
stability, 128
strength, 330, 331, 457-9
SQriace and volame, 49
Cylisidri«al ring, 61
D
AVIDSON, Messrs., fans, 395
Davit, Admiralty tests, 2o3
diameter required, 326
fittings. 546
number and positions.
495-502
Deadweight, 494
Deadwood, effect on steering, 3S6
Decimals of a foot and yard, 630
• ' — an inch, 631
Deck, Woyd's rules for, 509, 511,
-617
bolt, 656
bulkhead, 486
coaming, 609
pipe, 638-
planking, 478, 503, 609,
619, 622, 524
weight of, 106
Decks, height between, 478, 481,.
503
Declivity, building and launch-
ing, 378
Definite integral, 21
Deflection of beams, .322, 327 .
bulkheads, 32 1
plating under lateraJ
pressure, 325
Degree (angular), 8
Denny^s formula for wetted sur-
face, 167
Density (v. also Weight), $S
Depth (Lloyd's), 507
Derrick strength and stability,
296
Design notes> 391-4
Detrimental surface, 427
Devonport, distances around, 221
Diameter of screw propeller, 190-5
Diaphragm in pipe, 404 .
Differential block, 667
calculus, 19 ,
Dimensions, determination of
(design), 391
■ effect of addition of
weight, 392
Dimensions, increased speed,
392
— ■■ li change on
stability, 129
Dipping acids, 601
Direct method for stability. 125
Disc area ratio, 191, J.93
Discharge through an orifice (air),
400
■ — (water),"
403 .
Discount table, 634
Dispensary, 604
Displacement, 90
(appendages), 96
curve, 92
effect on resistance,
169
launching, 374
out of trim,. 132
sheet, 94-102
and tonnage 498
Distances, Bristol Channel, 215
'— Carlisle Bridge to
Wicklow Head, 214 "
Oarrigaloe Ferry to
Boche Ft., etc., 219
.Clyde, 213 .
Devonport to Portland
Bill, 221
216
201
foreign porb, 694
Humber. 218
Liverpool to Holyheaid,
measurement of,. 23, 29
measured (principal),"
•
Southampton' and Isle
of Wight, 220
'Thame?, 212
Tyne, 217
Distemper, 596
Distiller, regulations, 465.
Distress signals, 469, 477, 490
Division of a line, 22
Dock gates, 339
Dockingr, effect on stability, 143...
Donkey-engine, 463
Door (v. also W.T. Door), 474,
603
— screw for, 356
Double bottom, 486, 487, 607
Draught at perpendiculars, etc.,
131-3
'• — change In fresh water,
132
effect of addition of
weight or increase of speed in
design, 392
— ^ effect on stability, 129
steering, 366'
to read accurately, t38
730
INDEX.
Drawing pftpen, aixes of, 613
Drawings, oolonn for, 565
Drift. 413, 429
Drillinsr, effect on ttreiigth, t94
Driving power, 209
Drum for i.w. rope, 575, 580
Dry meaaare, 606
Dryen, 696
Donmore, dietanoes from, 219
Daodeolmals, 659
Dnralnmin, 268, 434
Dynamical etability, 115, 116,
124
Dynamic!, 84
# CbtiMt of natuiml logarithms), 8
Sarthe, rook9» etc., 636
Bcoentriaally loaded pillars, 829
Scoentriotty (ellipse, eto.)» 14
Lconomioal speed, 190
Edge strips and laps, 289, 290
Sffeotive horse-power, 161
SIBoiency, aerpdynamloal, 412
hull. 164
propeller, 191, 192
• propalsire, 162
——' screw, 855
Bfflnx of air, 400
Bffort, centre of, 208*
Elastic coefficients, 809
limit, 809
Elasticity, modulus of, 260, 310, 811
Elbow in pipe, effect of, 404
Ellipse, 14, 23-5
■' area, 41
— momental, 73
■ perimeter, 49
Ellipsoid, volume, 61, 52
— ^— volume of frostrum, 52
Elliptical spring, 836
Elswick Ordnance Oo., guns, 880
Emigrant ships. 502
Enamels, 695, 603
Energy, 85
Engineers' tackle blocks, 668
Engine-room, air for, 400
' size, 389
English weight, eto., 606-12
■■ ■ comparison
with metric, 615-38
Entablature plate, 82
Envelope, arirship, 437, 445
Epicycloid, 18, 34
Equipment, 108, 469, 476, 482,
630, 582
" of boats, 800
"; yachts, 526
Equivalent bending and twisting
moments, 332
BvaDorators, rules concerning,
Bvolute of curve, 18
Excursion limits, 469
Expansion due to heat, 405
Experiment for finding itability,
135-6
Exponential fnnotions (table), 708
Bxtbiotion of rolling; 152-4
Eyebolt, eyepiate, 654
FABRIO, airship, 447
Factor of safety, 286, 809
811, 829, 460
Fahrenheit thermometer, 406
Fans, ventilating, 895
Fastenings for wood vachts, 621
Fastnet, distances, 219
Feathering paddles, 197
Feed heaters, 458
Feet to inches and yards (table),
680
— metres and millimetres, 624
Fenton's white metal, 268
Figure of merit, 878
Fine screw threads (B.S.), 536
Fineness, coefficient of, 93
Fire, prevention of, 468, 469, 476
Fittlnn. ship, 538-56
Five-eight rule, 46
Flanges of couplings, 835
Sipe (B.S.), 634
, weight, 224
— surfaces, pressure on, 325,
454. 458
Flexible steel wire rope, v. Bojm
Float area for paddles, 196
Floats, number of (paddles), 196
Flotation, centre, 96, 131, 132
Flow of water through pipes, 403
Flywheel, stress In, 828
Focus oi: oonfc, 23-5
Fog signals, 489
Force and motion, 84
— — resolution of, 76-9
Forced and free rolling, 156
Foreign measures, weights, etc.,
609-88
Forged steel, tests, 870-2 285, 460
Forgings, wrought iron, tests, 274
Form, determination of ships, 393
of airships, 444
Four-way piece, volume, 66
Frames, 507, 622
plate, design of, 838
weight and CO., 104
Framework, stresses in, 295
Framing of airships, 434
Freeboard, 469, 520
of lifeboats, 498
Freeing ports, 479, 481
INDBX.
7B1
FreqiM&oy of ▼ibrailon, 88, 837
Fresh-water allowanoe, 1(X3
— ^— effect on draught,
183
Friotion, air In tranki, 897
belting, 868
— ooeffloients of, 868, 868,
876, 877
Joamale and pivots, 868
water In pipes, 404
Friotioaal resistanoe in air, 410,
489
— — — — ^— of ships,
165-8
Froade^.B., propeller data, 190
W.,. law of comparison, 169
Fuel, oil, density, 108 i
--^ per H*P. per hoar, 890
Funioolar polygon, 808
Fomaces, 467
a, 8, 84-6
Ghslvanizing, 606
Ganges, plate and wire, 838, 239
Oanxe, air resistance, 409
Geometry, practical, 22
Girders for flat surfaces, '456
plate web, 833
Girth, mean, 167
of yachts, 520
Grftphioal, JB.F, and B.M., 808-8
Gravity, motion under, 85
Glue, 602
Grease for launching, 876, 378
Greek alphabet, 8
GroM tonAs^ge, 490-93
Grounding, • effect on stability,
143
Gudgeons, 871
Gunfire, heel due to, 165
Gfmmetal, 267, 278. 468
Guns, particulars of, 879-88
Gyration, radius of, 69 (v.
Momfini of inertid)»
HAMKOOK cloths, 696
Harmony of colours, 596
Hatches. W.T., 474
Hawse pipe, 538
Hawsers, manilia and hemp, 673,
682
provision of, 626, 631
S.W., ddta concerning, 676,
577
Head (of air), 897
Heat (conduction, etc), 406, 406
i ' generated In Journals, 863
specific, 406
Heel due to added weight, 134
gunfire, 155
Height between deoks, 478, 481,
608
Heights of bridges, useful, 579
Helical spring, 837
Helm, T. Steering
akigle, 866
Hemisphere, etc., O.G., 61
Hemp rope, 572, 582
High tensile steel, tests, 266, 269
Hollow and full lines, 174-6
-^— ~ pillars, strength of, 828
Holyhead to Llveipooli distances,
216
Hook, strength and form of, 838
Hooke's Joint, .861, 549
Hoop-iron, weight, 286
Hbr&on, dip and distance of, 36
Horse, work done by, 860
Horse-power, 85
— — of air, 898
■'■ of ships, V. Speed
—— transmitted by teeth,
843
Hospital, ship's, 401, 604
Hull emoiemcy, 162-4
survey of, 478
— — weight and 0;G., 103-7
Humber, distances down, 218
Hundredweights and kilogrammes
(table), 826
' ' ■ tons, 629
Hydraulic test of boilers, 448
Hydrogen, properties of, 436
Hydrostatics, 88
Hyperbola, 14, 25, 26
— ^— rectangular, 15, 27
■ ■ ■■ area of, 41
Hyperbolic functions, 13
(table), 708
> logarithms, 4
- (table), 711
Hypooycloid, 18
————— to draw, 35
I BARS, Weight and properties,
242
Ignition, 467
Impact, 88
Impregnation of timber, 558
Inaccessible objects, distances,
28, 29
inches, decimals and fractions
(table), 631
, ' to feet and yards (table);
J 680
782
INDEX.
Inohee to meires (table). v<~. ^ .
mllIlm«trM (table), 616,
623.
Inpliaed plane, 65, 35i
—— — plate, air pressnre oa,
411
Inolimag.ezperimenfc, 135-8
Index ef* speed at which resis-
tance varies, 166, 169
Indiarubber, 281
Indicated horse-power, 161
Inertia, moment of, v. Moment
of inertia •
Ingot steel fors^ings, 271, 286
Ink, 604
Inlets, air, 896-401
Inner bottom, 486
Insulation, beat, 405
Integral oaloalns, 20
definite, 21
Integrator, 126
International rules for prevention
of collisions at sea, 487
Inverse functions, 19
Involute curves, 18
' ef a circle, 86
Iron, r. Metalt Caet iron. Pig
iron^ eto,
boilers, 468
cable, tests, 276
hoop, weight, 280
in place of steel, 286
ore, 266
pipes, weight, 232
Irreversible screws, 366
Isle of Wight, distances, 220
JOHNS, A. W., aerodynamics,
406
— '■ ■■ aeronautics, 431
— curves of resis-
tance, 188
>-* dimensions ef
A brackets, 344
Joint, universal (Hooke's), 361,
649
Joints in a perfect frame, 29i
riveted, 286
Joule's equivalent, 406
Journal, friction of, 963
Judgment items, 103, 107 : .'.
KEEL, 607, 522, 524
Keys and keyways (British
Standard), 339
Eilbgrammes and os., lb.; qrs.,
cwt., and tons (table), 625, 626
Kilometres and littutical miles,
200
Kinetio energy, 85
Kingston valves, 462
Knots and kilometres (table), 200
- statute inches (table).
199
202
time (speed tablss),
LACQUER, 600
Ladders, oompe4;ilott* 604
Lagging (heat), 406
Land measure, 606, 607
Lapped butts and edges* dimen-
sions, 289, 280
Lateral resistance, centre of, 208
i Launch, motor, 468
passengers allowed in, 483
Launching calculations, 372-6
- declivities and friction,
377
general, 372-8
of Ltuitania and
Mauretania, 376
» stresses, 376
weight of ships, 374
Law of comparison, 169, 184
Lead, 280
line, 476
pipes, weight, 234
plates, weight, 230
Leading block (anchor gear)* 650
Leakage of air, 400
Leather belting, 866 '
Length, cable, 638
circalar arc, 22. 37
— : curves (general), 48, 49
determination of ship*8,
391
effect on speed, 170-2
entrance and run, 172^
' e volutes, 49
favourable, 171
Lloyd's, 607, 620
r- of rivets for ordering, 292
of waves, 146, 148
Lever, 353
Lifeboats, 495-602
Lifebuoys and jackets, 497
Life-rafts, 600
life-saving applianoos, 469, 49o-'
602
Lift (aerodynamical), 412, 429
— position of, when launching.
876
Lift-drift ratio, 417 £E.
Lifting power of airships, 436
Lighting, electric, eto,, 479, 604
INDEX.
738
Lighta for liMwoys, 477
— required at Ma, 487
Liille fa formula for pillars, 329'
I^imlto for plying, 469
liine, ooTTed, CO., 60, 62
Liineal measure, 606, 613
Liinea, preparation of, 393
Liink, proportioa of, 638
liinkfl, end and enlarged, for
oable, 656
Liquids, density of, 262
Liverpool to Holyhead, distaacqp,
216
Lizard Point, distances, 221
Lloyd's rules for ^anchors and
oableai 690-2
676. 677
ging, 664-93
hawaersCS.WOf
masts and rig*
riyetiiig.239'^2
' ' ■' ■■ shafts, 606
ships, 607-19
■ vaohts, 620-6
tests for snip materials,
284
Load of timber, 661, 636
liOads, curve of ressels, 317
of beams, 901
Logarithmic series, etc., 10
(table),720
Logarithms, 4
(table), 700
* — ^hyperbolic(table>,711
London, dietanoes near, 212, 694
Long measure, 606, 613
Longitudinal number (Lloyd's),
607
stability, 181-4
Lubrication, effect on friction,
863
liUsitania, particulars of launch,
876
MAOHINERY, Board of Trade
roles for, 448
-** spaces; «ir for.
899-401
tonnage, 498
deduction ftom
weights, spnoe.
etc., 108, 988-90
Magazines, air for, 400
Malleable cast iron, 266, 273, 276,
277
Manchester Ship Oanal, 679
Manganese bronse, 268
Ma&lUa eordage, 288-
' hawser, 678
Mantissa, 4
Marking of ship, 494
Master's and crew spaces, 477
Masts, Lloyd's rules for, 690, 692
Materials, notes on, 256 ff. -
• * ' weight and dimensioAA,
222 ff.
262
strength.
Mauretania,. launehittg data^ 376
Mean girth, Bieohynden's, 167'
Mean of means, 197
Measure, timber, -668
Measured miles, principal, 201
— speed tables, 202
trials on, W
Ujeasntes, eompswis^m Snglish
and metric, 616-38
■1 BngUsh, 606
foreign and English, 632
- ■■■ '■ metric, 613
miscellaneous, 609-12
Mechanioal powers, 353-63
Melting-points of metals, 405
Men, space occupied by, r.
strength of, 860
weight of, 108, 109
Messing space, air required for,
399-401
Metaeentre, 95, 111.14
longitudinal, 96, 133
" transverse. 96. 101
Metacentres, curve of, 118
Metacentric diagram, 112, 113
height. 112, 114
Metal sheets, weight, 230
Metals, conducting power of, 405
^■^— expansion of, 405
weight and strength, 260
Metres to feet, 62 1
ludhes, 622
Metric system, 613, 614
comparison with
English, 616-38
Midship section area, 91
■ coefficient, 93
Mild steel, Admiralty instruo-
tloBs, etc., 264
— r- tests, 966. 968. 284
449
Miles, tables of measured, 47
nautical, and kilometres
(table), 200
— — — ' statutr
(table). 199
Millimetres to feet (table), 621
inches (table),
616, 623
Miscellaneous constants, 223
' substances, weight,
and strength, 262"
784
IHDSlt.
Mixing paini, 696
Mcdel ezperimentt, aeroplanes
4aB
■ airships, 438
■' ships, 1T6
ICodolns of elaslioiiy, 360, 261,
810, 811
" ' seoQOtt, 818
solid and
hoUow shafts, 818, 819
Moments of figure, r, Ctntr* of
gravity,
— — — force or eoaple, 77-9
inertia, 89-78, 89
aiest 10, TS-i
841-84
about
rarions
seofcions,
oiroalar seofcions
(table), 819-62
-geometrical figures.
76, 818-17
868
•polar, 78
ships. 160-9, 849-
> waterplane, 94, 99,
112, 133
— — resistance, 818 fl.
sfcablUty, 116
weights, 80
to ohMige trim one inch,
188, 184
Momentai ellipse, 78
Hommtom, 84
— — — angular, 84, 87
Money, foreign and English, 632
table. 688
Mooring ohain, formolii for,
16-18
Moseley's formula, 116
Motion, 81
Motor-boats, 498, 666
— and engines, rating,
launches, 468, 666
— ^— ▼essels,'*466
Mountings and gunr, 879
Mnnts metal, 267
Muzzle Telocity of projeetilee,
879-88
W
-AVAL brass, 268, 277, 278
Ket, suspended in stream, 16
Ket tonnage, 492, 493
Keutral axis, 813
Kewall's steel wire rope, 676-7
Kewcastle-on-Tyne, distances, 213
' high-lerel
bridge, 678
Nickel steel, tMfcs, 867^ 2T0
Normal to ourve, 19, », 24
Ndcmand's formula, 114, 134
NumbezB, miscellaneons, 618
'"■ useful, 687
Nuts, sites, 636
weight, 886
OAKUM, 697
Observations of rolling, 160
Oil fuel, effect on stability, 140
per H.P. per hour, 8iX)
weight, 108, 140
Oil tanks, 466
Open link ehain, 638
Openings in boilers, 464
Ordnance, particuUwB of, 380-7
Ore, iron, 266
Orifice, flow of water through,
402
Ounces and kllosrammes (tabl^,
pounds (table), 629
P ADDLES, 196, 197
Paint mixing, 696
quantity required, 636, 697
Fainting, etc, 696-606
Panama Oanal tonnage, 498
Paper, size of, 613
tracing, 604
Parabola, 18, 26
— — area, 41
0.0., 61
■ of Tarions orders, area,
48, 48
Pkrabolio cylinder, stability, 128
■ half segment, O.G., 61
Paraboloid, O.G., 61
— — — derelopment, 88
— — and frustrum, rolume,
68, 68
Paraffin, consumption of, 890
PtoaUel foroes, 77, 78 .
plates, air pressure, 484
Parallelograai. area, 86
Parallelepiped, O.Q., 61
. surface. 48
Passenger accommodation, 479
Passengers, number and space,
468, 479-83, 499, 602, 604^
— — weight and O.Q., 109
Pence and shillings (table), 638
Pendulum, compound, 87
errors when measuxiof
rolling, 160
tttDEl.
786
Pendnlom period over large arcs,
86
iimple, 88
Peroussion, centre of, 87
Perforation of armour, 378, 388
Perimeter of ellipse, i9
Period of dipping, 161
— pitching, 169
rolling, 160-8
' vibration; 337
Permeability of airship enve-
lopes,. 4^
Perpendicular, to draw. 22
Petrol consumption, 390
Phosphor bronze, 258, 279
Pi C»), 3, 39, 637
Pig iron, 256
Pillar, strength of, 328
Pillars, Lloyd's roles for, 609,
616, 621.4
Pintle, rudder, 370, 371
Pipe flanges, 634
threads, 637
Pipe*, bilge suction, 487
— — — deck and hawse, 628
- flow of water through, 403
— — oil or petrol, 467
weight, 223, 232-6
Piping fourway, volume, 61
Pitch and pitch ratio, 190-6, 356
for caulking, 697
of rivets, 289
Pitched chain, 629
Pitching, 169
Pivot, friction o^ 363
Pivoting point, 366
Planking, outside, 624
weight, 222
Plastering, 661
Plate gauges, 238
Umiting sizes (Admiralty),
237
■pressure on, in air, 406 ff.
water, 368
rudder, thickness of, 871
web girder, 8M
Plating, compressive stress
allowable, 330
Lloyd's rules, 609, 618,
621-4
■ stress due to bending, 812
lateral pressure,
826, 464
weight, 222, 230
and CO., 104
Plying limits, 469
Plymouth, distances near, 221
Points of the compass, 636
Poisson's ratio, 310
Polar co-ordinates, 46, 66
moment of inertia, 72
Polygon, area, 36, 37
Polyhedron, volume and surfaoe,
61
Pontoon lifeboats, 498
Portland Bill, distances near, 221
Ports, distances from London^ 694
Potential energy, 85
Poundal, 84
Pounds and kilogrammes (table),
625, 626
r- ounces (table), 628
tons (table), 628
Power, 86
air circuits, 398
■ alteration due to change
of speed, S^2
increase due to additional
resistance^ 201
to carry sul, 209
transmitted by belting.
358
Powering ships, t. Speed
Powers, table of two-thirds, 184
Preservation of timber, 668
Pressure, air, 396
- airship (interior), 432,
448
98
atmospheric. 89
centre of, 89
closed surface, 89
distribution of, 416, 422
immersed plane surface.
liauid, 89
of water at various
heads, 402
Prices, table of, 686
Prism, volume and surface, 49
Prismatic coefficient, 93
effect fm
speed, 171, 176
Prismoid, volume, 60
Product of inertia, 72, 89
Progressive trials, 184
^ojeotiles, motion and range.
879, 888
— : ^partionlarsiof various,
380-7
Promenade space, 487
Proof spirit, 263
——' stress and load, 809
Propeller, screw, 190-6
—————— clearance, 894
- effect of propul-
sion, 164
fUiA
effect on turning,
efficiency, 191, 192
■ shafts. Board of Trade
rules, 464
— -' ■ Lloyd's rules,
606
struts. 844
786
.IMI>B}(.
frppprtioDfl, offeot on speed, nZ'$
ropalBive ooeffioient, 161
— coefficient (fttnl)ip«).
438
yalues oi,
163
Proyifliona, 505
^— in Navy, 663
weight, 108
PttUey, strees in» 828
Pal^jB, sjatem of, 354, 862
Pamps, provision of, 475
■ eiae and duty, 402
Punching, weakness due to, 293
Pyramid, CO., 61
TOlome and surface, 49
— ^— — ■^— —— — of
frastmm, 49
QaADRILATSRAL, area, 37
CO.. 69. 60
yoartera and kilosrrammes, 625
tons, 629
Qnenchinff steel, 256
BA.DIAL integration, 46, 66
Radiation of heat, 406
Radius of action, 189
■ curvature, 20
gyration, 69 (t>. Mo-
mtmt of i»0rtia)
to draw, 22
Rafts, 500
Itail, height of. 469, 479. 481, 503
Range of stability. 116
Rankine (r» pillars), 329
Rating of mdtor-boate and en-
gines, 888
Reanmur temperature scale, 406
Reciprocal diagram, 295
Reciprocals (table), 655
Rectangular bars, Weight, 222,
224
■ plates, compressive
stress, 330
springs, 836
Refrigerators, 465
Register tonnage, 496
Resistance (aeroplaaesV, 427
.(airships), 438
coefficient, effect of. 201
' friotional, 166, 410
in air, v. Air preature
of air in trunks, ffi)7
^ of water in pipes, 404
rolling, 162-5
(ahips), y. Speed
Restitution, coefficient ol, 88
Resultant acceleration, 82
force, 76-9
' stress, 832
velocity. 81
Reverse frames, 507, 612
Reversibility of screws, 35$
Revolution, solid of, surface, 66
volume, 63
Rigging ciiain, 629, 538
slip and screw, 639, 612
standing, 586-®
Rigidity, coefficient of, 310
Ring, cylindrical, volume and
surface, 61
plane, 0.0. of sector, 60
strength of, 338
Ripples, 148
Rivet-holes, correction for, 352
spacing, 289, 462, 486, 621
Riveted jointe. design and
strength, 285-94, 452
size of butt
etc., 289, 290.
Rivets, countersink of, 291
effect of punching, 294
in tension, strengtU of,
288, 289
-lengths and preparations
e/.^ 291, 292
shearing
and bearing
values of, 286-8
tests of, 268-70, 285
Roche Point, distances from, 219
Rod, weight of copper, 234
Rolling of ships, 150-60
among waves, 156-7
axis of, 151
effect of bilge keels,
154, 156
observations of, 160
' period, 160, 161
resistance and extinc-
tion, 152-5
with water chambers.
157-9
Rope, hemp and wire, 672
manilla, 283, 673
steel wire, 575-82
Admiralty, 578,
679
680-2
sheaves, 675
Bullivant's,
drums and
various, 674
Lloyd's, 686-9
Newairs, 676-T
notes on, 632
Rota, Colonel (on ship resis-
tance), 172
IHD>BZ.
787
BooBd boia, weiffkl, 322, S$l,
Bndder <al»lhip), 486
flhipa, area, 866
' ■ ■■ head, eize of, 869,
871, 682
■ ■ jpregsare, 867
Raddert, twin, 162, 866
Rale Of the road at eea, 409
V* (Siaif AJ, 8
^ Safetr, factor of, 809, 8tl,
828, 460, 461
— ■■' ' " 'valTO, 469
Sailing and sail area, 208, 496,
620. 664
mlest 489^
ehip. rlfif,' 686, 698
Screen for bow liehta, 487
Screw- efficieac/, 864, 866
— — propellers, t. Propeller
ritrfiring, 642, 646, 647
' ateerinff gear, 81
threads, B.S. flae, 686
B.S. (pit}e8), 687
■ B.S. (Whit worth),
683-6
■ ' form' of, 336, 683-6
Screwed bolte, ateength, 839
Soappers, 462
Sea wares, 146, 148, 149
Seacocks, 46S
Seasonintr timber, 667 '
Secants, 7 (table), 716
Sectional areas, cunre of, 91,393
Sections, method of, 298
Segment of circle, area, 40
(table), 662
Seller's screw thread, 866
Semicircle or semi-ellipse, O.G.,
6t
Set, 809
Shackle, anchor, 646
bow and straight, 648
■ of cable, 638
Shaft brackets, 844
horse-power, 161
strengrth, 334
Shafts, Board of Trade rales, 464-
Lloyd's rules, 606
Shallow water, effect on speed,
189
waves, 14T
Shear of rivets, 286-8
and resultant stress, 838
— ^— stress in beams, 332
ships, 862
B
Shear stress working Talnes, 884
Shearing force curve in ships,
847. 868
' in beams, 800-8
■ when launching,
876
Sheathing bolt, 666
copper, 280
Sheaves for S.W. rope, 676, 680
Sheer legs, 298
Sheet metals, weight, 280
Shells, strength, 830
Shift of flgore, effect on C.G.,
67. 80
Shillings ' and fjeriqe (table), 638
Ship fittings, 688-66
Ships, longitudinal stress in, 316
Shrinkage of castings, 285
Shrouds, Lloyd's rules for, 686-9
screw for, 546
Side pressure in air. 897
scuttles, 474, 486, 604
Signals, distress, 469, 477, 490
fog, 489
Signs and symbols, 1.
Simpson's roles, 42-6, 64, 94
Sine, etc., 7 (table), 716
Sirocco fans, 895
Size, effect on ship's resistance,
109
Skin friction, v. Friational re-
Slackness, 208
Sleeping spaces, air for, 899
Slide rule, 7
Slings for boats, 899
Slip, building, declivity of, 878
— clear hawse, 664
— of propeller, 190, X93, 196
—r rigging, 639, 646, 647
Sluice, discharge through, 402
Smooth water limits, 469
Snatch block, 642
Solders, table of, 259
Solid measure, 607, 614
Sounding line, 476
Southampton, distances from, 220
Space necessary for crew, 478, 499*
fassen^era.
Spacing of rivets. 289
Spare gear, 468, 466, 476
Specific gravity, 88
heat, 405
Speed and horse-'power, 160-'90
effect of dimensions on,
892
tions, 178
form and propor-
lenjrth, 170
size, 169
bb
on turning, 866
788
INI>SX.:
Speed. firiotiooaJ cesiitanoe, 166
— — — loM of, doe tQ fo«l bottom,
etc., 201
— ^-* lovr» and eoonomicaJU 190
minim uia , of aeroplane,428 .
powering bj Admiralty
eoefficient» 180-4
data, 106.9
perimeato, 179
approximate
methodioal ex-
modela» 176
— ^^— — progreaaiTe
triala, 184
propulaive ooefBoient».161
radloa of action, 189, 190
'■ realduary resiitanoe, 168
ahallow-water eileot, 189
taUea, 202
trials. 197
wake and thmat deduet
tion, 163
Speeda, oorreaponding, 169. 428
of belting, 858
Sphere, air pretsore on, 409 .
' development of, 81
— — — Tolome and anrfaoe, 60
— ^— — — of aegmfkt, 60
' of zone, 61
Spherical shells, strength, 830
Spirit, proof, 263
Springe, helical, streee, etc, 836,
461
Spunyam for oaaUdng. 698
Square baxB, weight, 222, 231
meaenre, 607, 613
■ seotiona under tension, 836
Sqnares, square roots (table),
656-99
Stability, 110-43
Barnes' method, 118-24
— — — change with change of
dimensions, 129^1, 891
— — — condition for, 110
— — — cross curves, 117
curves of statical and
dynamical, 116
— — — direct method, 125
— — dynamical, 116
effect of adding weight,
bilging, 138-42
134, 136
experimental determina-
tion, 136-8
— — geometrical bodies, 128
' integrator method, 126
large angles of heel.
114-28
light draught, 125
longitudinal, 131-4
Ul. 114
- metacentre above O.B.,
StaMUty, malaeeatria : diagsa&
.... ■ , height-, 112,
U4
- of aeroplanes, 418
— — — — of ship aground, 142
— ^— — io ensure, in design, 397
Stabilising plane, 436
Staining wood, 603
Stairways, 603, 604
Stanchions, 469, 479, 6USL
Stays for boilers 454
Steam pipes, 449, 459
Steel, T. Mild ateel, east steel,
etc.; for weights, ▼. Angle, ate.
— — — notes on, 256
— r-»- fcitbes, tests, -274
•^-— wire rope, r- Rope
wire, -weight and etreng^h,
240
yaohts, 621
Steering (airship). 436
-. chain, 609
gear Cecrew), 81, 866
indicator, 864
■ ' ■ ■■ influence ■ of features
on, 366
path when turning, 364
■' rudder pressure, 867-9
— — rules, 489
— — — strength of rudder
head, pintles, etc., 369
' tevms used, 364
— — — — turning trials, 366
Stem, 607, 621
Stempost, 607, 621
Stiffeners for bulkheads, 322,
324, 483
■ Webs and plate frames,
Stocks for anchors, 683
Stopper, Blake's, 640
Stoves, 463, 466
Stowage of chain cable, 644
■ provisions, 663
sulwtances (various),
662
Strain, longitudinal and shear,
309
volumetric, 310
Strength, v. also Stress and Teste
—— — of dumlumin, 259
metals, 860
— misceUaneoua sub-
stances, 262
ships under longi-
tudinal bending, 346-62
' steel wire, 240
structures and fit-
tings (general), 809-12
beams,312-23,
832
INDEX.
789
Strensth of s|imotare ^arinfti, 8B6
' ' '1 ' ■ > ' ' bulkheadfl,
322, 483
335
eottpfisjt,
328
•««•
keywaya, 339, ^0
338
-<^linden,330
-davits, 326
' derricks, 296
'hooka, 838
keys* and
pillaf8,e28-30
plate webs,
compression, 839
-platiB^T QAder
lateral pressure
r^
lati&g nnder
369
ribffs, 338
-rudder beads.
839
ets, 344
891a
boats, 299
337
ing, 341^
■^screwed bolts,
— shaft brack-
-shafts, 833,
— slinjs '• for
—springs, 838,
■» toothed gear-
oal beams, 326
timber, 261
— — — tmsymmetri-
Stress, 300
allowable, 852
bending, 312
-^— • in beams under yarioui
loadings, 323
bulkheads, 322-5
• cylinders and shells,
330
pression, 330
flywheels, 328
flat plating under com'
lateral
pressure, 8S0
unsymmetriesil beams,
326
62
33b
lon«ritadinal, in ships, 345-
shear and resultant, 332
' in beams, 382
»— shafts, allowable,
->— B^i— «MMa
Springs, 336
339
working, for screwed bolts,
Stress woAing in various materials.
312
Stmts, air resistance. 409, 410
strength of, M8
Stud-linked cable, 628, 538
Submarine, stability, 128
Suction, bilge, 487
pump, 476
Sues Panal dimensions, 578
' tonnage, 493
Superheaters, 458
Sui>erstructnre, strength of, 882
Supplement of angle, 7
Supports to beams, best posi-
tion, 322
Surface, hydrostatic pressure, 89
of geometrical figures, 49-
51
general solids, 57-9
solids of revolution, 56
stability. 115
wetted, 167
Swivel for cable, 555
Symbols, I
Synchronous rolling^ 156
TAOKLB blocks, 668
Tactical diameter, 865
Tangent (of an angle), 7; (table),
716
C*o curve), 19, 21
Tank, oil, 466
Taylor, D. W., approximate
curves of resistance, 186, 187
TchebychefTs rule, 46, 100, 128
Tee steel, weight and propor-
tions, 226-9, 262
Teeth, shape and strength of, 842
Temperature scales, 406
Tempering steel, 256
Tensile strength of rivets, 288
Tension, airship envelope, 447
Test pieces, 266, 448
Tests, Admiralty, 263
brass and copper.
279
263
265
cable iron, 275
canvas, 282
cast iron, 276
davits and derricks,
form of test pieces,
guHmetal, 278
indiarubber, 281 *
lead, 280
manilla edfdage, 283
Naval brass, 277
phosphor bronze, 279
740
IKDEX.
Tests, ttoel oMtingt. 919 ^^ ^
totzingat 970-2
. . ■ plAtM, twurs,
and rivaU, 966-70
InbM, 974
ings, 274
wood, 281
wroosht iroa forg-
lino. 981
Board of Trade^ 4a
hjdiaoUo,
lor boUeci, 447
riTOts, 449
iagi and oaatinga. 460
- aaohon, 697
eable, 628
ehain, 629
platoi and
ioboa, forg-
(Lloyd's). 284. 286 ^ ^
Thames, distances down, 212
Thermal properties of materials,
406
Thickness of propeller blades,
194
Thimbles, 688
Thomson's mlo, 46
Thread, sorew, 866, 633-7
Three-ten mlo. 68
Thrust dednotioa, 163-^
horse-power, 164, 191
Tides, 6(56, 604
Timber loads, 636
measure and eonireraion,
669
presenringr, 668
sea;8oning, 667
to cut beam from, 29
Time and knot table, 202
- to empty compartment (air),
400
(water),
408
Tin, notes on, 257
Tonnage, 490
yacht, 496, 623
Tons and kilogrammes (table),
625, 626
lb., qr., owt. (table),
628, 629
per inch, 92, 93
ourTO, 92
Toothed gearing, 341
Topmasts, 686
Torque on rodder head, 370
Torsion, 833, 334
Towing trials, 196
Tow line, provision of, 531
Towne's hook, 338
Traoing-pfkper, 604
Transverse number (Lloyd's), 607
Trapezoid, area, 36, 145
TfapMoid, CG., 60
— — momeak of inertia, 76
Trapesoidal role, 48
Tisy for motor, 468
Tresidder's fonanla, 878'
Triaaglo, area, 11, 86
O.O^ 68
■ momani of inertia, 76
■ properties of, 10
solation of, 11, 12
Trisonometrioal ratios (oartain
angles), 19
— — — - CB«aeral),
716
Tiigoaometry, 7-13
— ^— general fonnalss, 9
Trim, 18141
— due to ffyroaeopio notion, 87
— o effeok on speed, 170
Trochoid, 18, 887446
' lo draw, 84, 345
Itoohoidal wave theory, 144
Troy weight, 606
Tkonkinff, air, 896, 401
Tal>e plates, 467
Tnbea steel, tests lor streagth.
274, 460, 464
Togs, propalsive daia, 196
Turbine machinery, partioalan
of, 889, 890
shafts, 465
Taming trials, 866
Twist of shafts. 833
Twiskiag meakeat, equivalent,
888
— — — ia shaft*, 833.
834
Two-thirdt powers (table), 184
Tyne, dista.noes down, 217
ULTIMAT8 strength, 309
Uneasy rolling, 161
Uniform force, 86
Univwsal joint, 861, 649
Unsymmetrioal beams, 826, 327
Unwin, faotor of eafety, 811
Useful numbers, 687-
VALVB, safety, 469
Varnish, 696, 699
Velocity, angular, 82
— — — oompositioa of, 81
diagram, 81, 82
of air, 890
— — projectiles, 379-88
Ventilating iuis, 895-
Ventilation, 896-401
— — - of crew space, 477
' ' motor compart-
ment, 467
741
VMUUmton* 401
. r, Vibeation, fraquenoy of, 66, 88T
Vicken' fftAt,lpftrU«ilMs oi; 888
Volume of fonrway piping,- 66
gvomotrioal soUda, 49-63
. I toUd* in gonenil, 68-6
— of revolation, 63
3 • vrtdgm, 65
I Volametrio itniii mad modvlvm,
\ 9iQ
WAOBS table. 638
Wake, 168-6, 191
Wall-eided ehip, stebUity, 128
Wanhips, anchors and eables,
Waate in converting timber, 561
Water, iMlmiaaion to flhip, 138-42
and air, compariaon of re-
•ulta in, 431
ohambera
167-9
Canti-roUing),
402
oloaeta, 462, 478, 480, 6M
diaoharge frMA orilloe, 408
flow through pipee, 408
free aurfaoe effect, 189
freah, allowance, 108, 606
jpreaaare at varioaa heada,
pampa, pipea, etc., 402
weight of freah and aalt,
22b
Water-line ooefBcient, 98
— — — effect on atability, 391
Watertight aabdiviaion, 473, 478,
486
Watt, 86
Watta, Sir Philip Cra water-
chambera), 167
Wave aasamed in atrength cal-
oolationa, 346
• alope, effective, 156
Wavea, 143, 144
• accompanying ahip, 168 ff.
effect on rolling, 166-9
energy of, 146
momentum of. 146
period, apparent and real.
149
periods and lengths
Ctable), 146
preaaure within, 146
ripplea, 148
aea, dimenaiona of, 148,
149
ahallow water, 147
aubanrfacea of, 144
trochoidal, atandard, 846
Waya, laonohlng, 376.-8
Wedge, O.a., 67
Wedge, effect of, 333
volume, 66
Wilight cnrre for ahipa, 846-8
■ effect of addition to ahip,
181, 392
of ai», 896
■ alrahip, 437
armament, 108, 880-7
armour, 106
ban, round Mid ellip-
tical, 222
iaqaava and rect-
aagnlar,. 222
— boata, 666
cable, 688
' <diain, GUB
t caat-iron balla, 234
-pipea, 233
231, 236
223
copper roda and pipes,
i
eattha, ete., 636
equipment, 108
freah water (conatauts),
hoop iron, 236
^* hull, 103-7
lead pipe, 234
machinery and fuel,
106, 890
malleable flat ateel,
224, 225
iron pipea, 232
metala, 260
miaoellaneona
aub-
atancea, 262
nuta and boltheada, 236
pipea and anglea, rules
for, 223
— plating and planking.
222
223
231
aalt water (constants),
aheet metala, 230
ahip, 102-9
ahip when launched, 374
ateel angle and T, 226-9
round and aquare,
wire, 240
rope, 676-81
236
timber, 261
wire of variona metala.
Weighta and propertiea of B.S.
bulb anglea, 250
bulb tees.
253
244
gles, 246
■ ohannela,
■•qoal an-
teea, 252
742
Weights Mid prtperties, vaeqiial
nngltB, M ^^ «
" Zada, SH
,coinp>rigcm, Bngiish ftnd
metric, 616 88
Englith, 686
- fdfeign and Saglieh,
683
BMtrio, 614
miaeeHaneoni, 60^12
Weatoa pulley blocks^ fl67
Wetted enrfnoe, 167
White, Sir Williaro-H., rm waves,
149
White metal, 868
Whitworth thtfiad, 583
Widdow, distsnoee from, 214
Winch, BoUivaot's orab, 683
Wind, force of» and Beaufort
scale, 210
— ^— real 'and apparent motion
of, 2(»
Wine meacure, 607
Wire, air resistance of, 409
gauge, legal etandard, 239
weight aad atrez^tht 236,
2i0
Wood cleat, 6*2 „ ^
fuel per H.P. per hour.
390
staining, 603
weight aad etrength, 261
Wood yachts. seaxKllng*, ete., SOI
W^l ■mtfBsnre, 609
IPooHey's neasave, M
Work, 64
> done in oKtinetioo, 163
-— *- — • ■ by m«&. Bad aaimals^
860
Working load and stcentpth, SOS
■■ ahaar stress, 834
stress (direot), 318
Wrenoh, 79
Wrought iron, 256
— — — — ^— — - ohesi, 664
Y^CHX tonnage, 495, 580
Yachts, international racing
620-6
rardi. 584, 690 ^^
feet and inches, 630
Young's modnlas, 260, 810, 311
ZAHH'S formula for air fric'
Z-bacs, B.S., weights and pro^
pertiea, 864
Zigsajr riretioff, 898
Zinc 267, 281
for galvanismg, 605
plates, weight, 230
Stephen Austin and Sons, Limited, Printers, Hertford,
tc,i
.13
i
311
lae
31!
fi5
THIS BOOK 18 DTJS OK THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 26 CENTS
WILL BK ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO SI. GO ON THE SEVENTH DAY
OVERDUE.
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^fttf
APf?lC*68-i?w
DgO 11 1941
JUH 27*65-4 PNl
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