Table 7. Conversion Factors for Units of Energy

T A B L E 8.-FORMER



22



ELECTRICAL EQUIVALENTS



*



Abbreviations : int., international ; emu, electromagnetic units ; esu, electrostatic units ;

cgs, centimeter-gram-second units.

RESISTANCE:

1 international ohm =

1.00051 absolute ohms

1.0001 int. ohms (France, before 1911)

1.00016 Board of Trade units (England,

1903)

1.01358 B. A. units

1.00283 “legal ohms” of 1884

1.06300 Siemens units

1 absolute ohm =

0.99949 int. ohms

1 “oractical” emu

io8c g S emu

1.11262 X lo-’’ cgs esu

CURRENT

:

1 international ampere =

0.99995 absolute ampere

1.00084 int. amperes (U. S. before 1911)

1.00130 int. amperes (England, before

1906)

1.00106 int. amperes (England, 190608 )

1,00010 int. amperes (England, 190910)

1.00032 int. amperes (Germany, before

1911)

1.W2 int. amperes (France, before

1911)

1 absolute ampere=

1.00005 int. amperes

1 “practical” emu

0.1 cgs emu

2.99776 x lo9 esu



ELECTROMOTIVE

FORCE :

1 international volt =



1.00046 absolute volts

1.00084 int. volts (U. S. before 1911)

1.00130 int. volts (England, before 1906)

1.00106 int. volts (England, 1906-08)

1.00010 int. volts (England, 1909-10)

1.00032 int. volts (Germany, before

1911)

1.00032 int. volts (France, before 1911)

1 absolute volt =

0.99954 int. volt

1 “practical” emu

lo8 cgs emu

0.00333560 cgs esu



OF ELECTRICITY :

QUANTITY

(Same as current equivalents.)

1 international coulomb =

1/3600 ampere-hour

1/96494 faraday



:

CAPACITY

1 international farad =

0.99949 absolute farad

1 absolute farad=

1.00051 int. farads

1 “practical” emu

10.” cgs emu

8.98776 X 10” cgs esu



1NDUCTANCE

1 international henry =

1.00051 absolute henries

1 absolute henry =

0.99949 int. henrv

1 “practical” emu

log emu

1.11262 X lo-’’ cgs esu

A N D POWER :

ENERGY

(standard gravity = 980.665 cm/sec-’) 1 international joule =

1.00041 absolute joules

1 absolute joule=

0.99959 int. joule

lo’ ergs

0.737560 standard foot-pound

0.101972 standard kilocram-meter

0.277778 X

kilowakhour



:

RESISTIVITY

1 ohm-cm = 0.393700 ohm-inch

= 10,000 ohm (meter, mmz)

= 12,732.4 ohm (meter, mm)

= 393,700 niicrohm-inch

= 1,000,000 microhm-cm

=6,015,290 ohm (mil, foot)

1 ohm (meter, gram) = 5710.0 ohm (mile,

Dound)

QCANTITIES :

MAGNETIC

1 int. gilbert

= 0.99995absolu.tegilbert

1 absolute gilbert = 1.00005 int. gilberts

1 int. maxwell

= 1.00046 absolute

maxwells

1 absolute maxwell = 0.99954 int. maxwell

1 gilbert

= 0.7958 ampere-turn

1 gilbert per cm = 0.7958 ampere-turn

per cm

= 2.021 ampere-turns

per inch

= 1 line

1 maxwell

= 10.’ volt-second

1 maxwellpercmZ= 6.452 maxwells per

in?



*This table is now superseded by the adoption of the new system of electrical units in January 1948



and



IS



given for reference only.



SMITHSONIAN PHYSICAL TABLES



T A B L E S 9-15.-SOME



MATHEMATICAL T A B L E S



T A B L E 9.-DERIVATIVES



d ax



=a d r



AND INTEGRALS



- X""

-,~

+

= logx



J.r"dr



J$



, unless



tc



=-I



Jc'dx



J r"'ds

J.r"'Pdx



d c=

d rn=



= r= d.r

= o ra'dx



Jli



d 1og.r



1

= 7d.r



J(a



.rr



= .r' ( 1



ti



Slog x d x

dv



+ Dx)"dx



+ log, x ) d.r



+ .r2)-'dx



= cos .r dx



J(n'



= - sin x d x



J(a' - x')-'dx



d tan .r



= secz x dx



J(a' - x')-+dx



d cot .r

d sec x

d csc x

d sin-' .r

d cos-' x

d tan-' x



= - csc'x d x

= tan x sec x d x



Jx(a' -C x')-fdx



d sin x

ti



cos .r



d cot-' x

d sec-' x

d csc-' x

d sinh x

d cosh x

d tanh x

d coth x

d sech .r

d csch .r



- - cot x ' csc x d x

= ( 1 - x p ) - *d x

-- (1-x2)-*dx

= (1 x*)-'dx

= - (1 + x l ) - ' d x

= c' (2- 1)-* dx

- - A = * (x z - 1) -4 dx

= cosh x dx

= sinh x d x

= sech' .r d r

- - csch'x dx

- - sech .r tanh x d.r

- - csch x.coth x d s



+



d sinh-lx = ( 2



+ 1 ) - * d.r



d cash-' x = ( x z - 1) -4 d.r

d t a n h - ' x = (1 - x Z ) - ' d x

d coth-'x = (1 - x 2 ) - ' d x

d sech-' x = - .r-' ( 1 - x ' ) -4 d.r

d csch-' .r = (x'

I ) - * d.r



+



SM!THSON!AN



PHYSICAL. TABLES



1

n + r

-_ log __

2a

a-x



= & (a' & x z ) *

- - $ cos x sin x

Jsin' x d x

1.= sin x cos x fx

Jcos' x d x

= 1 sin'x

.[sin x cos x d x

J(sin x cos x)-'dx = log tan x

Itan x d x

- -logcosx



+



Jtan' x d x

Jcot x d x

.fcot' x d x

Jcsc x d x

Jx sin x d x

Jx cos x d x

Jtanh x d x

Jcoth x d.r

Jsech x d x



+



= tan .r - x

= !og sin x

= - cot x - x

= log tan i x

= sin -p - x cos x



= cos .r + .r sin x



= log cosh I

= log sinh x

= 2 tan-'cz = ,9d 11



.fcsch x dx



= log tanh 3

2



.[.r sinh x d x

Jscosh x dx

Jsinh' x dx

.fcoshax dx



= r cosh x - sinh x

= x sinh x - cosh x

= $ (sinh x cosh x - x )



Jsinh



.T



= (sinh x cosh r

cosh x d x = icosh ( 2 x )



+x)



23



24



T A B L E 10.-MATHEMATICAL

(x



+y)"=



+ fx"-'y



x"



$.



-



~



n ( n 1)

2!



xn-syp



+.. .



n ( n - 1 ) . .. ( n - m

m!



( 1 f2)" = 1 2 nx



+ n ( n -2l!) x l



SERIES



+ 12_ X"-



n(n-1)(n-2)x2

3!



ym



+ .. .



+...+

(r1 ) % ! x k

(n- k ) ! k !



+...



n ( n + l ) xzT n ( n + l ) ( n + Z ) ~ "

(l+z)"=lTwX+---2!

3!

( n k - 1)x'"

(%-1)!k!



+



+2 T X J +

+,. .

+ 32x 1 ~ 4 +~ '5 2 , 6r6+.. .

h2

h"

f ( x + I c ) = f ( x ) + h f ' ( x ) + 5 f " ( X ) +.. .+ 2 f'"'(X)



(1 f x ) - ' = 1 T

(1 k x)" = 1



X



X'TXJ



~



X2



f(x)=f(o) +fP(o) +fl."(o)



x2



x'



x3



l+x+a+j-j+a



a'=



l+xloga+----



log (1 f. x )



( x log u ) 2 +



( x log a)8



3!



2!



(x-



= x - f r 2 + 4 x8 - a .z'



f...



+...



e'=



= ( x - 1) -4



X"



+... ,t?f'"'(o)



+. . .



+....



+ 3 ( x - 1)s -.



..



+.. ..



x3

x6

x'

1

)=x--+---+

...

sin x = 2i (e'= - e""

3!

5 ! 7!

1

x2 x'

xe

cos x = - (e'" e-'=) = 1- - - - ..= 1 - versin x

2

2 ! 4! 6 !

x3

2x6 1 7 2

62

tanx= x

-xD . .

3

15

315

2835

7r

x3

1 3 x 6 1 3 5 x T

- - cos-'x = x + - + -. - . -+-. - .2

6

2 4 5

246'+..

7r

1

1

1

tan-lx = - - cot-'x = x - - x8 -2 --x'

.

2

3

5

7

1

1

1

- 2r ;+3x'52



+



+



+ +-



+-+



+.



+.



+. .



+



-+...



-_-



1

x3

x6

XT

sinhx=-(e'-e-")

=x+-+--f-+

...

2

3 ! 5! 7 !

1

xz

x'

xe

coshx= - ( e a e-)) = 1 - 4-- i- ..

2

2!

4!

6!

(contiwued)



+



SMITHSONIAN PHYSICAL TABLES



+



+.



(Y'






T A B L E 10.-MATHEMATICAL



1

tanh .r = z - - r3

3 .



25



17

+ --125 .r6- r7 + . . .

315.



-_. . .



1 1

1

2 2.f'

2

1 1

1

cosl1-'.r= IO~~.I'------------2 2.r' 2

1

1

tanlir'x = .r - ra - r5

5.

3 .

1

1

gd.r=@=.r-.r 3 + -21



3 1

1 3 5 1

7' --4 4.r'

2 4 6 6.r'

3 1

1 3 5 1

4 4 2 2 4 6 6.P

+ -71 -r7 + . . .

61

.r5

5040 .z'+.. .



= IOg2.r+------



+



S E R I E S (concluded)



+



'' '



( x small)



-__.



1 sechS.r 1 3 sech'~

- - - secli. .I' - - ____ - - - _ _ -



2 4 5

...

1

61

- & __ @'+.

..

24

5040



2



3



( x large)



+

+

1

77r.r

f ( x ) = b" + bl cos - + b* cos

f.. .

2

7rn

27rn

+ sin +- azcos +. . . (-c < x < c)

lr,r

a, = +Jk ~ ( ssin

) -dx

x



1

= gd-'@ = @ + @3

6

7r.r



-



a1



111



b,=



+J'f- f ( x ) cos-d.z



iii7r.r



T A B L E 11.-MATHEMATICAL



CONSTANTS

Numbers



e = 2.71828 18285

c? = 0.36787 94412

111= logloe= 0.43429 44819



(Af)-'=



Logarithms



= 3.14159 26536

7r2 = 9.86960 44011

1

7r = 0.31830 98862



0.49714 98727

0.99429 97454



7r



9.50285 01273



= 1.77245 38509



0.24857 49363



431 13



_

v7r

- 0.88622 69255

2



9.94754 49407



loglo2= 0.30102 99957



1

= 0.56418 95835

V7r



9.75142 50637



log2 = 0.69314 71806



_ - 1.12837 91671

v7r -



0.05245 50593



= 1.25331 41373



logelo = 2.30258 50930



10gio I O ~ , O =

C 9.63778



v7r



'



10gio.z = M . I o ~ ~ x



4;



0.09805 99385



logSx = log,x.logoe



d$



= 0.79788 45608



9.90194 00615



+ log,B



??

4 -- 0.78539 81634



9.89508 98814



= 1.14472 98858



v?r - 0.44311 34627

--



9.64651 49450



= 4.18879 02048



0.62208 86093



-5 - 1.08443 75514

\/2H



0.03520 45477



= l0g.x

log.7r

p



= 0.47693 62762 *



log p = 9.67846 03565

Probable error, modulus of precision.

SMITHSONIAN PHYSICAL TABLES



+7r



T A B L E 12.-FACTORI



26



ALS



P a r t 1.-Numerical

1

n:



n



1

2

3

4

5

6

7



8

9

10

11

12

13

14

15

16

17

18

19

20



1.

0.5

.I6666

.04166

.00833

0.00138

.00019

.WOO2

.OOOOO

.OOOOO

0.00000



66666

66666

33333

88888

84126

48015

27557

02755

00250

.ooooo 00020

.ooooo 00001

.00000 00000

.ooooo 00000

0.00000 00000

.ooooo 00000

.ooooo 00000

.ooooo 00000

.ooooo 00000



n:=



1.2.3.4



...n



n



1



66666

66666

33333

88888

98412

87301

31922

73192

52108

87675

60590

11470

00764

00047

00002

00000

00000

00000



66666

66666

33333

88888

69841

58730

39858

23985

38544

69878

43836

74559

71637

79477

81145

15619

00822

00041



66667

66667

33333

88889

26984

15873

90653

89065

17188

68099

82161

77297

31820

33239

72543

20697

06352

10318



8

130

2092

35568

6 40237

121 64510

2432 90200



3

36

399

4790

62270

71782

76743

27898

74280

37057

04088

81766



2

6

24

120

720

5040

40320

62880

28800

16800

01600

20800

91200

68000

88000

96000

28000

32000

40000



1

2

3

4

5

6

7

8

9

10

11

~~



12

13

14

15

16

17

18

19

20



Part 2.-Logarithmic

Logarithms of the products 1.2.3.. . . . ..rt, n from 1 to 100.

n



1



2

3

4

5

6



7



8



9

10

11



12

13

14

15

16



17

18

19

20

21



22

23

24

25



log (n!)



n



log (n!)



n



log (n!)



n



0.000000

0.301030

0.778151

1.380211

2.079181

2.857332

3.702431

4.605521

5.559763

6.559763

7.601156

8.680337

9.794280

10.940408

12.116500

13.320620

14.551069

15.806341

17.085095

18.386125

19.708344

21.050767

22.412494

23.792706

25.190646



log ( n 0



26



26.6056 19

28.036983

29.484141

30.946539

32.423660

33.915022

35.420172

36.938686

38.470165

40.014233

41.570535

43.138737

44.718520

46.309585

47.911645

49.524429

51.147678

52.781147

54.424599

56.077812

57.740570

59.412668

6 1.093909

62.784105

64.483075



51



66.190645

67.906648

69.630924

71.363318

73.103681

74.851869

76.607744

78.371 172

80.142024

81.920175

83.705505

85.4978%

87.297237

89.103417

90.916330

92.735874

94.561949

96.394458

98.233307

100.078405

101.929663

103.786996

105.650319

107.519550

109.394612



76



111.275425

113.161916

115.054011

116.951638

118.854728

120.763213

122.677027

124.596105

126.520384

128.449803

130.384301

132.323821

134.268303

136.217693

138.171936

140.130977

142.094765

144.063248

146.036376

148.014099

149.996371

151.983142

153.974368

155.970004

157.970004



27

28

29

30

31



32

33

34

35

36



37

38

39

40

41



42

43

44

45

46

47

48

49

50



-



SMlTHSONlAN PHYSICAL TABLES



52

53

54

55

56



57

58

59



60

61



62

63

64

65

66



67

68

69

70

71



71

73

74

75



77

78

79

80

81



82

83

84

85

86



87

88

89

90

91



92

93

94

95

96



97

98

99

100



27

T A B L E 13.-FORMULAS

F OR M O M E N T S O F I N E R T I A , R A D I I O F G Y R A T I O N , A N D

W E I G H T S O F VARIOUS SHAPED SOLIDS



I n cacli case the axis is supposed to traverse the center of gravity of the body. T h e axis is one of

symmetry. T h e mass of a unit of volume is zw.



nody



.\xis



Weight



Square of

radius of

gyration pus



Moment of

inertia I"



Sphcrc of radius r . ....... Diameter



4r7err3

-



2P

-



Spheroid of revolution. pcilar axis 217.equatorial tliameter 2r ............. Polar axis



87rwr5

15



4waP



8mwar'



3



15



2r'

-



3



4mwabc



Ellipsoid, axis 20. 26, 2 c . . Axis 20



3



Sphcrical shell. extcrnal ratlius r , internal r'. ...... 1)ianicter



4xw(rJ - r")

3



15



5



2 ( r6 - r")

5(rS-r")

2r2

3

r'

2

1J2

c2

4



15



4mr2dr



Circular cylinder, length 20, Longitudinal

radius r ...............

axis&



2mmr2



xzvar'



2rwabc



mhc(b2

2



Longitudinal

axis 20

I-ongi tudinal

axis



+ c2)



2rtcw ( r 2- 1")



m m ( r ' - r")



4 mwa rdr



4mwar'dr



Transverse

diameter



2rzwarZ



Elliptic cylinder. length 20, T r a n s v e r s e

axis 211

transvcrse axes 2a, 21). . .



2rwabc



Hollow circular cylintler.

length L o . external ra- T r a n s v e r s e

diameter

dius r . internal r ' . ......



a t ~ ~ l r ~ ( 3 r4a2)

'

6

mlahc ( 3c2 4a2)

G



2rzva(r2 - r'2)



Ditto, insensi1)ly thin. thick- T r a n s v e r s e

diameter

ness dr ...............



4rtuardr



Rectangular prism, dimensions 20. 2h, 21.. . . . . . . . . Axis2a



+ cz



8mv ( r5 - r")

8rten;'dr

3



Longitudinal

axis 2a



5



+



Imwabc ( b2 c 2 )

bP

-~



Ditto. insensibly thin, radius r . tliickncss d r . . . . . Iliameter



Elliptic cylinder. Icngth 20,

transverse axes 217. 2c. ..

Hollow circular cylinder.

length 20, external radius r , internal r ' . ......

Ditto, insensibly thin, thickness d r . . . . . . . . . . . . . . .

Circular cylindcr, length 20.

radius r . . . . . . . . . . . . . . .



5



+



r2



+ r"

2



+

+



r w a ( 2r3



+ 43 a2r)dr



r2



-



a2



2+3



+

+



+



8zuabc (1)'

c2)

b2 c2

3

3

Khombic prism, length 20,

2ZWflbC( b2 C 2 )

b2 c2

diagonals 2h. 2c.. ...... Axis 2a

4wabc

3

6

2wabc(c2

2a2)

c2

a2

Ilitto ................... Diagonal 21)

4wabc

+3

3

6

F u r iurt!icr niathematical data see Smithsonian Mathematical Tables, Becker and Van Orstrand

( Hyperbolic. Circiilar and Exponential Functions) ; Smithsonian Mathematical Formulae and Tables

of Elliptic Fulictl,iiss. Adams and Hippisley ; Smithsonian Elliptic Functions Tables. Spenceley ;

Smitlisonian Logaritlimic Tahles. Spenceley and Epperson ; Functionentafeln. Jahnke und Emde (xtgx,

x-ltgs. Roots of Traiir;cciitlcntal Equations, a bi and rcRi, Exponentials, Hyperbolic Functions,

8wabc



+



+



11 /":

2



9 drr,



+



dtr,



dii,



Fresnel



Integral,



Gamma



Function,



Gauss



Integral



\



~ - ~ ' d . Pearson

t-.

Function c - ~ " "



sin' c"'d.r,



Elliptic Integrals and Functions, Spherical and



Cylindrical Functions, etc.). For further references see under Tables, Mathematical, in the 16th ed.

Encyclopaedia Britannica. See also Carr's Synopsis of P u r e Mathematics and Mellor's Higher Mathematics for Students of Chemistry and Physics.

SMITHSONIAN PHYSICAL TABLES



TABLE 14.-LOGARITHMS



28



P.P.

N



0



oooo



1



2



3



04.1 0492

0086 0128

0453

0SSi



4



5



6



1



8



9



1



2



3



4



5



0212

0607

0969

1303

1614



0253

0645

1004

1335

1644



0294

0682

1038

1367

1673



0334

0719

1072

1399

1703



0374

0755

1106

1430

1732



4

4

3

3

3



8

8

7

6

6



12

11

10

10

9



17

15

14

13

12



21

19

17

16

15



0414

0792

1139

1461



0828 0864 0899

1173. 1206 1239

1492 1523 1553



0170

0569

0934

1271

1584



1761

204 1

2304

2553

2788



1790

2068

2330

2577

2810



1818

2095

2355

2601

2833



1847

2122

2380

2625

2856



1875

2148

2405

2648

2878



1903

2175

2430

2672

2900



1931

2201

2455

2695

2923



1959

2227

2480

2718

2945



1987

2253

2504

2742

2967



2014

2279

2529

2765

2989



3

3

2

2

2



6

5

5

5

4



8 11 14

8 11 13

7 10 12

7 912

7 911



3010

3222

3424

3617

3802



3032

3243

3444

3636

3820



3054

3263

3464

3655

3838



3075

3284

3483

3674

3856



3096

3304

3502

3692

3874



3118

3324

3522

3711

3892



3139

3315

3541

3729

3909



3160

3365

3560

3747

3927



3181

3385

3579

3766

3945



3201

3404

3598

3784

3962



2

2

2

2

2



4

4

4

4

4



6

6

6

5

5



8 11

8 10

8 10

7 9

7 9



3979

4150

4314

4472

4624



3997

4166

4330

4487

4639



4014

4183

4346

4502

4654



4031

4200

4362

4518

4669



4048

4216

4378

4533

4683



4055

4232

4393

4548

4698



4082

4249

4409

4564

4713



4771

4914

5051

5185

5315



4786

4928

506s

5198

5328



4800

4942

5079

5211

5340



4814

4955

5092

5224

5353



4829 4843 4857

4969 4983 4997



sios sii9 si3i



5237 5250 5263

5366 5378 5391



4871 4886 4900

5011 5024 5038

Siis 5159 5172

5276 5289 5302

5403 5416 5428



1

1

1

1

1



3

3

3

3

3



4

4

4

4

4



6

6

5

5

5



7

7

7

6

6



36

37

38

39



5441

5563

5682

5798

5911



5453

5575

5694

5809

5922



5465

5587

5705

5821

5933



5478

5589

5717

5832

5944



5490

5611

5729

5843

5955



5502

5623

5740

is55

5966



5514

5635

5752

5866

5977



5527

5647

5763

5877

5988



5539

5658

5775

5888

5999



5551

5670

5786

5899

6010



1

1

1



2

2

2



4

4

3



5

5

5



6

6

6



1



2



3



4



6



40

41

42

43

44



6021 6031

6138

6128

.~~~

6232 6243

6335 6345

6435 6444



6042

6149

6253

6355

6454



6053

6160

6263

6365

6464



6064

6170

6274

6375

6474



6075

6180

6284

6385

6484



6085

6191

6294

6395

6493



6096

6201

6304

6405

6503



6107

6212

6314

6415

6513



6117

6222

6325

6425

6522



1

1

1

1

1



2

2

2

2

2



3

3

3

3

3



4

4

4

4

4



5

5

5

5

5



45

46

47

48

49



6532

6628

6721

6812

6902



6542

6637

6730

6821

6911



6551

6646

6739

6830

6920



6561

6656

6749

6839

6928



6571

6665

6758

6848

6937



6580

6675

6767

6857

6946



6590

6684

6776

6866

6955



6599

6693

6785

6875

6964



6609

6702

6794

6884

6972



6618

6712

6803

6893

6981



1



2



3



4



4



50



6990

7076

7160

7243

7324



6998 7007

7084 7093

7168 7177

7251 7259

7332 7340



7016

7ioi

7185

7267

7348



7024

7110

7193

7275

7356



7033 7042

7118 7126

7202 7210

7284 7292

7364 7372

(continued)



7050

7135

7218

7300

7380



7059

7143

7226

7308

7388



7067

7152

7235

7316

7396



1



2



3



3



4



10



11

12

13

14

15



16

17

18

19

20



21

22

23

24

25



26

27

28

29

30



31

32

33

34

35

~~



51

52

53

54



SMITHSONIAN PHYSICAL TABLES



4099 4116 4133



426s 428I 4298



4425 4440 4456

4579 4594 4609

4728 4742 4757



12556