Tải bản đầy đủ - 0 (trang)
Table 13. Formulas for Moments of Inertia, Radii of Gyration, and Weights of Various Shaped Solids

Table 13. Formulas for Moments of Inertia, Radii of Gyration, and Weights of Various Shaped Solids

Tải bản đầy đủ - 0trang

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



T A B L E 14.-LOGARITHMS



29



(continued)



P.P.

N



0



1



2



3



4



5



6



55



7404

7482

7559

7634

7709



7412

7490

7566

7642

7716



7419

7497

7574

7649

7723



7427

7505

7582

7657

7731



7435

7513

7589

7664

7738



7443

7520

7597

7672

7745



7451

7528

7604

7679

7752



7459 7466

7536 7543

7612 ,7619

7686 7694

7760 7767



7782

7853

7924

7993

8062



7789

7860

7931

8000

8069



7796

7868

7938

8007

8075



7803

7875

7945

8014

8082



7810

7882

7952

8021

8089



7818

7889

7959

8028

80%



7825

7896

7966

8035

8102



7832

7903

7973

8041

8109



8129

8195

8261

8325

8388



8136

8202

8267

8331

8395



8142

8209

8274

8338

8401



8149

8215

8280

8344

8407



8156

8222

8287

8351

8414



8162

8228

8293

8357

8420



8169

8235

8299

8363

8426



8451

8513

8573

8633

8692



8457

8519

8579

8639

8698



8463

8525

8585

8645

8704



8470

8531

8591

8651

8710



8476

8537

8597

8657

8716



8482

8543

8603

8663

8722



8751

8808

8865

8921

8976



8756

8814

8871

8927

8982



8762

8820

8876

8932

8987



8768

8825

8882

8938

8993



8774

8831

8887

8943

8998



9031

9085

9138

9191

9243



9036

9090

9143

9196

9248



9042

90%

9149

9201

9253



9047

9101

9154

9206

9258



9294

9345

9395

9445

9494



9299

9350

9400

9450

9499



9304

9355

9405

9455

9504



9542

9590

9638

9685

9731



9547

9595

9643

9689

9736



9777

9823

9868

9912

9956



9782

9827

9872

9917

9961



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



91

92

93

94

95



%



97

98

99



1



8



9



7474

7551

7627

7701

7774



i 2

1 2

1 2

1 2

1 1

1 1



3

2

2

2

2

2



4

3

3

3

3

3



s

4

4

4

4

4



7839

7910

7980

8048

8116



7846

7917

7987

8055

8122



1

1

1

1

1



1

1

1

1

1



2

2

2

2

2



3

3

3

3

3



4

4

3

3

3



8176

8211

8306

8370

8432



8182

8248

8312

8376

8439



8189

8254

8319

8382

8445



1

1

1

1

1



1

1

1

1

1



2

2

2

2

2



3

3

3

3

3



3

3

3

3

3



8488

8549

8609

8669

8727



8494

855s

8615

8675

8733



8500

8561

8621

8681

8739



8506

8567

8627

8686

8745



1



1



2



2



3



1

1

1



1 2

1 2

1 2



2

2

2



3

3

3



8779

8837

8893

8949

9004



8785

8842

8899

8954

9009



8791

8848

8904

8960

9015



8797

8854

8910

8965

9020



8802

8859

8915

8971

9025



1

1

1

1

1



1

1

1

1

1



2

2

2

2

2



2

2

2

2

2



3

3

3

3

3



9053

9106

9159

9212

9263



9058

9112

9165

9217

9269



9063

9117

9170

9222

9274



9069

9122

9175

9227

9279



9074

9128

9180

9232

9284



9079

9133

9186

9238

9289



1

1

1

1

1



1 2

1 2

1 2

1 2

1 2



2

2

2

2

2



3

3

3

3

3



9309

9360

9410

9460

9509



9315

9365

9415

9465

9513



9320

9370

9420

9469

9518



9325

9375

9425

9474

9523



9330

9380

9430

9479

9528



9335

9385

9435

9484

9533



9310

9390

9440

9489

9538



1

1

0

0

0



1 2

1 2

1 1

1 1

1 1



2

2

2

2

2



3

3

2

2

2



9552

9600

9647

9694

9741



9557

9605

9652

9699

9745



9562

9609

9657

9703

9750



9566

9614

9661

9708

9754



9571

9619

9666

9713

9759



9576

9624

9671

9717

9763



9581

9628

9675

9722

9768



9586

9633

9680

9727

9773



0



1



2



2



0

0



1

1



1 2

1 2



2

2



9786

9832

9877

9921

9965



9791

9836

9881

9926

9969



9795

9841

9886

9930

9974



9800 9805

9845 9850

9890 9894

9934 9939

9978 9983

(continued)



9809

9854

9899

9943

9987



9814

9859

9903

9948

9991



9818

9863

9908

9952

9996



0

0

0

0

0



1

1

1

1

1



1 2

1 2

1 2

1 2

1 2



2

2

2

2

2



SMlTHSONlAN PHYSICAL TABLES



i i Z 2 3



0



1



1

1



2



2



0 i i Z Z



TABLE 14.-LOGARITHMS



30

N



0



100



0000

0013



101

102

103

104

10S



106

107

108

109

110



111

112

113

114

115



116

117

118

119

120



121

122

123

124

125



126

127

128

129

130



131

132

133

134

135



136

137

138

139

140



141

142

143

144

145



146

147

148

149



(continued)



4



5



6



7



8



9



10



0086



0128

0170



0004 OOO9 0013

0048 0052 0056

OO90 0095 0099

0133 0137 0141

0175 0179 0183



0017

0060

0103

0145

0187



0022

0065

0107

0149

0191



0026

0069

0111

0154

0195



0030

0073

0116

0158

0199



0035

0077

0120

0162

0204



0039

0082

0124

0166

0208



0043

0086

0128

0170

0212



0212

0253

0294

0334

0374



0216

0257

0298

0338

0378



0220

0261

0302

0342

0382



0224

0265

0306

0346

0386



0228

0269

0310

0350

0390



0233

0273

0314

0354

0394



0237

0278

0318

0358

0398



0241

0282

0322

0362

0402



0245

0286

0326

0366

0406



0249

0290

0330

0370

0410



0253

0294

0334

0374

0414



0414

0453

0492

0531

0569



0418

0457

0496

0535

0573



0422

0461

0500

0538

0577



0426

0465

0504

0542

0580



0430 0434 0438

0469 0473 0477

0508 0512 0515

0546 0550 0554

0584 0588 0592



0-241 n44.5 n449



0519 0523 0527

0558 D561 0565

0596 0599 0603



0453

0492

0531

0569

0607



0686 0689 0693

0722 0726 0730

0759 0763 0766



0622

0660

0697

0734

0770



0626

0663

0700

0737

0774



0630

0667

0704

0741

0777



0633

0671

0708

0745

0781



0637

0674

0711

0748

0785



0641

0678

0715

0752

0788



0645

0682

0719

0755

0792



0799 0803

0835 0839

0871 0874

09M 0910

0941 0945



0806

0842

0878

0913

0948



0810 0813



0934



0795

0831

0867

0903

0938



0846

0881

0917

0952



0849

0885

0920

0955



0817

0853

0888

0924

0959



0821

0856

0892

0927

0962



0824

0860

0896

0931

0966



0828

0864

0899

0934

0969



0969

1004

1038

1072

1106



0973

1007

1041

1075

1109



0976

1011

1045

1079

1113



0980



0983

1017

1052

1086

1119



0986

1021

1055

1089

1123



0990

1024

1059

1092

1126



0993

i028

1062

1096

1129



0997

io3i

1065

1099

1133



inno



io14

1048

1082

1116



1069

1103

1136



1004

1038

1072

1106

1139



1139

1173

1206

1239

1271



1143

1176

1209

1242

1274



1146

1179

1212

1245

1278



1149

1183

1216

1248

1281



1153

1186

1219

1252

1284



1156

1189

1222

1255

1287



1159

1193

1225

1258

1290



1163

1196

1229

1261

1294



1166

1199

1232

1265

1297



1169

1202

1235

1268

1300



1173

1206

1239

1271

1303



1303

1335

1367

1399

1430



1307

1339

1370

1402

1433



1310

1342

1374

1405

1436



1313

1345

1377

1408

1440



1316

1348

1380

1411

1443



1319

1351

1383

1414

1446



1323

1355

1386

1418

1449



1326

1358

1389

1421

1452



1329

1361

1392

1424

1455



1332

1364

1396

1427

1458



1335

1367

1399

1430

1461



1461

1492

1523

1553

1584



1464

1495

1526

1556

1587



1467

1498

1529

1559

1590



1471

1501

1532

1562

1593



1474

1504

1535

1565

1596



1477

1508

1538

1569

1599



1480

1511

1541

1572

1602



1483

1514

1544

1575

1605



1486

1517

1547

1578

1608



1489

1520

1550

1581

1611



1492

1523

1553

1584

1614



1614

1644

1673

1703

1732



1617

1647

1676

1706

1735



1620

1649

1679

1708

1738



1623

1652

1682

1711

1741



1626

1655

1685

1714

1744



1629

1658

1688

1717

1746



1632

1661

1691

1720

1749



1635

1664

1694

1723

1752



1638

1667

1697

1726

1755



1641

1670

1700

1729

1758



1644

1673

1703

1732

1761



0607

0645

0682

0719

0755

0792

0828

0864

0899



1



2



3



0611 0615 0618

0618 0652 0656



(continued)



SMITHSONIAN PHYSICAL TABLES



048i 0484 0488



i0%



TABLE 14.-LOGARITHMS



(concluded)



31



N



0



1



2



3



5



6



7



8



9



1764

1793

1821

1850

1878



1767

1796

1824

1853

1881



1770

1798

1827

1855

1884



1772

1801

1830

1858

1886



1775

1804

1833

1861

1889



1778

1807

1836

1864

1892



10



150



1761

1790

1818

1847

1875



4



1781

1810

1838

1867

1895



1784

1813

1841

1870

1898



1787

1816

1844

1872

1901



1790

1818

1847

1875

1903



1903

1931

1959

1987

2014



1906

1934

1962

1989

2017



1909

1937

1965

1992

2019



1912

1940

1967

1995

2022



1915

1942

1970

i998

2025



1917 1920

1945 1948

197.3 1976



2028 2030



1923

1951

1978

2006

2033



1926

1953

1981

2009

2036



1928

1956

1984

2011

2038



1931

1959

1987

2014

2041



2041

2068

2095

2122

2148



2044

2071

2098

2125

2151



2047

2074

2101

2127

2154



2049

2076

2103

2130

2156



2052

2079

2106

2133

215Y



2055

2082

2109

2135

2162



2057

2084

2111

2138

2164



2060

2087

2114

2140

2167



2063

2090

2117

2143

2170



2066

2092

2119.

2146

2172



2068

2095

2122

2148

2175



2175

2201

2227

2253

2279



2177

2204

2230

2256

2281



2180

2206

2232

2258

2284



2183

2209

2235

2261

2287



2185

2212

2238

2263

2289



2188

2214

2240

2266

2292



2191

2217

2243

2269

2294



2193

2219

2245

2271

2297



2196

2222

2248

2274

2299



2198

2225

2251

2276

2302



2201

2227

2253

2279

2304



2304

2330

2355

2380

2405



2307

2333

2358

2383

2408



2310

2335

2360

2385

2410



2312

2338

2363

2388

2413



2315

2340

2365

2390

?415



2317

2343

2368

2393

2418



2320

2345

2370

2395

2420



2322

2348

2373

2398

2423



2325

2350

2375

2400

2425



2327

2353

2378

2403

2428



2330

2355

2380

2405

2430



2430

2455

2480

2504

2529



2433

2458

2482

2507

2531



2435

2460

2485

2509

2533



2438

2463

2487

2512

2536



2440

2465

2490

2514

2538



2443

2467

2492

2516

2541



2445

2470

2494

2519

2543



2448

2472

2497

2521

2545



2450

2475

2499

2524

2548



2453

2477

2502

2526

2550



2455

2480

2504

2529

2553



2553

2577

2601

2625

2648



2555

2579

2603

2627

2651



2558

2582

2605

2629

2653



2560

2584

2608

2632

2655



2562

2586

2610

2634

2658



2565

2589

2613

2636

2660



2567

2591

2615

2639

2662



2570

2594

2617

2641

2665



2572

2596

2620

2643

2667



2574

2598

2622

2646

2669



2577

260 1

2625

2648

2672



2672

2695

2718

2742

2765



2674

2697

2721

2744

2767



2676

2700

2723

2746

2769



2679

2702

2725

2749

2772



2681

2704

2728

2751

2774



2683

2707

2730

2753

2776



2686

2709

2732

2755

2778



2688

2711

2735

2758

2781



2690

2714

2737

2760

2783



2693

2716

2739

2762

2785



2695

2718

2742

2765

2788



2788

2810

2833

2856

2878



2790

2813

2835

2858

2880



2792

281.5

2838

2860

2882



2794

2817

2840

2862

2885



2797

2819

2842

2865

2887



2799

2822

2844

2867

2889



2801

2824

2847

2869

2891



2804

2826

2849

2871

2894



2806

2828

2851

2874

2896



2808

2831

2853

2876

2898



2810

2833

2856

2878

2900



2900

2923

2945

2967

2989



2903

2925

2947

2969

2991



2905

2927

2949

2971

2993



2907

2929

2951

2973

2995



2909

2931

2953

2975

2997



2911

2934

2956

2978

2999



2914

2936

2958

2980

3002



2916

2938

2960

2982

3004



2918

2940

2962

2984

3006



2920

2942

2964

2986

3008



2923

2945

2967

2989

3010



151

152

153

154

155



156

157

158

159

160



161

162

163

164

165



166

167

168

169

170



171

172

173

174

175



176

177

178

179

180



181

182

183

184

185



186

187

188

189

190



191

192

193

194

195



196

197

198

199



SMITHSONIAN PHYSICAL TABLES



Zoo0 2603



Radians



- G-%z



( TRIG O NO ME TRIC ) FU N C TI ON S *



T A B L E lB.-CIRCULAR



32

Degrees



Sines



5Fxz

.oooo 03



Cosines



Nat.



Log.



Tangents



.oooo



Cotangents



Nat.



Log.



343.77

171.89

114.59

85.940

68.750



2.5363

.2352

.0591

1.9342

.8373



90"OO'

50

40

30

20

10



1.5708

1.5679

1.5650

1.5621

1.5592

1.5563



.0175 8.2419

.OM4 .3089

.0233 .3669

11262 .4isi

.OZG ~ 6 3 8

.0320 SO53



57.290

49.104

42.%4

38.188

34.368

31.242



1.7581

.6911

.6331

S819

S362

.4947



89"OO'

50

40

30

20

10



1.5533

1.5504

1.5475

1.5446

1.5417

1.5388



.99949.9997

.9993 ,9997

.9992 ,9996

.9990 .9996

.9989 .9995

.9988 .9995



.03198.5431

.0378 S779

.0407 .6101

,0437 .6401

.0465 .6682

,0495 .6945



28.636

26.432

24.542

22.904

21.470

20.206



1.4569

.4221

.3899

.3599

.3318

.3055



88"00'

50

40

30

20

10



1.5359

1.5330

1.5301

1.5272

1.5243

1.5213



.0523 8.7188

. O W .7423

.0581 ,7645

.0610 ,7857

.0640 .8059

.0669 .8251



.99869.9994

,9985 9 9 3

.9983 .9993

.9981 .9992

.9980 .9991

,9978 ,9990



.05248.7194

.0553 ,7429

.0582 .7652

.0612 ,7865

.0641 ,8067

.0670 ,8261



19.081

18.075

17.169

16.350

15.605

14.924



1.2806

,2571

,2348

,2135

.1933

.1739



87"OO'

.

50

40

30

20

10



1.5184

1.5126

1.5097

1.5068

1.5039



0.0698 4"OO.'

0.0727 10

0.0756 20

0.0785 30

0.0814 40

0.0844 50



.06988.8436

.0727 ,8613

.0756 .8783

.0785 .8946

.0814 ,9104

,0843 ,9256



.9976 9.9989

.9974 .9989

.9971 ,9988

.9969 .9987

.9967 .9986

.9964 .9985



.06998.8446

.0729 ,8624

.0758 3795

.0787 .8960

,0816 .9118

,0846 ,9272



14.301

13.727

13.197

12.706

12.251

11.826



1.1554

.1376

,1205

.lo40

.0882

.0728



86"OO'

50

40

30

20

10



1.5010

1.4981

1.4952

1.4923

1.4893

1.4864



0.0873 5"OO'

0.0902 10

0.0931 20

0.0960 30

0.0989 40

0.1018 50



.08728.9403 .99629.9983

.9959 3982

I0629 ,9682 .9957 .9981

,0958 .981h ,9954 .9980

.0987 ,9945 ,9951 .9979

.lo169.0070 .9948 .9977



.0875 8.9420

,0904 .9563

.O934 -97.01

10963 ,9836

.0992 .9966

.lo229.0093



1 1.430 1.0580

11.059 .0437

10.712 .0299

10.385 ,0164

10.078 .0034

9.78820.9907



0.1047 6"OO'

0.1076 10



.lo45 9.0192

,1074 .0311

.1103 .0426

,1132 .0539

.1161 .0648

.1190 .0755



.9945 9.9976

.9942 .9975

.9939 .9973

.9936 ,9972

.9932 .9971

.9929 ,9969



.lo51 9.0216

,1080 .0336

,1110 .0453

,1139 .0567

.1169 ,0678

.1198 ,0786



9.51440.9784

9.2553 ,9664

9.0098 .9547

8.7769 .9433

8.5555 .9322

8.3450 .9214



84"OO'

50

40

30

20

10



0.1222 7"OO'

0.1251 10

0.1280 20

0.1309 30

0.1338 40

0.1367 50

0.1396 8"OO'

0.1425 10

0.1454 20

0.1484 30

0.1513 40

0.1542 50



.I2199.0859

.1248 ,0961

.1276 ,1060

.1305 ,1157

.1334 .1252

.1363 .1345

.1392 9.1436

.1421 .1525

.1449 .1612

.1478 ,1697

.1507 .1781

.1536 .1863



.9925 9.9968

,9922 ,9966

,9918 ,9964

.9914 ,9963

.9911 ,9961

,9907 .9959



.1228 9.0891

.1257 ,0995

.1287 ,1096

,1317 .1194

.1346 .1291

.1376 .1385

.1405 9.1478

.1435 .1569

.1465 ,1658

,1495 .1745

.1524 ,1831

.I554 .1915



8.1413 0.9109

7.9530 .9005

7.7704 ,8904

7.5958 ,8806

7.4287 .8709

7.2687 .8615



83"OO' 1.4486

50 1.4457

40 1.4428

30 1.4399

20 1.4370

10 1.4341



7.1154 0.8522

6.9682 .9431

6.8269 .8342

6.6912 .8255

6.5606 .8169

6.4348 ,8085



82"OO'

50

40

30

20

10



0.1571 9"OO'



.15649.1943 .9877 9.9946 ,15839.1997



6.31380.8003



81"OO' 1.4137



0.0000 0"OO'

0.0029

10

0.0058 20

0.0087 30

0.0116 40

0.0145 50



.OOL9 7.4637

.0058 ,7648

.0087 .9408

.01168.0658

.0145 .1627



1.oOoo .moo



1.0000 .OoOo

1.0000 .oooo

.9999 .oooo

.99f9 .moo



.0029 7.4637

.0058 ,7648

.0087 ,9409

.0116 8.0658

.0145 .1627



0.0175 1"OO'

0.0204 10

0.0233 20

0.0262 30

0.0291 40

0.0320 50



.0175 8.2419

.0204 .3088

.0233 .3668

.0262 ,4179

,0291 .4637

,0320 SO50



.99989.9999

.9998 .9999

.9997 .9999

.9997 .9999

.9996 .9998

.9995 .9998



0.0349 2"OO'

0.0378 10

0.0407 20

0.0436 30

0.0465 40

0.0495 50



.03498.5428

.0378 ,5776

.0407 ,6097

.0436 .6397

,0465 .6677

,0494 .6940



0.0524 3"OO'

0.0553

10

0.0582 20

0.0611 30

0.0640 40

0.0669 50



'



0.1105



20



0.1134

0.1164

0.1193



30

40

50



1.00000.0000



.nwi .9545



Nnt.



T.oa.



=z-



-



.99039.9958

,9899 .9956

.9894 .9954

,9890 ,9952

.9886 ,9950

.9881 ,9938

Nnt.



T.oa.



y

Sines



03



Kat.

r.og.

\-



Cotangents



* Taken from R. 0. Peirce's Short table of intcurals. Ginn S. Co.

(C O I l t ill lt cd

SMITHSONIAN PHYSICAL TABLES



0 3 E X )



Iiat.



1.5155



85"OO' 1.4835

50 1.4806

40 1.4777

A n 1.4748

20



ii4ii9



10 1.4690

1.4661

1.4632

1.4603

1.4574

1.4544

1.4515



1.4312

1.4283

1.1251

1.4224

1.4195

1.4166



LOR.



L7_J



Tangents



De-



grees



Radians



T A B L E 15.-CIRCULAR

Radians



Degrees



Sines

Nat.



Log.



(TRIGONOMETRIC) FUNCTIONS (continued)

Cosines

Nat.



Log.



Tangents

Nat.



Log.



33



Cotangents



5icxT



0.1571 9'00'

0.1600

10

0.1629

20

0.1658

30

0.1687

40

0.1716

50



.1564 9.1943

.I593 .2022

.1622 .2100

.1650 2176

.1679 .2251

.I708 .2324



.98779.9946

,9872 .9944

.9868 .9942

.9863 .9940

.9858 3938

.9853 .9936



.15849.1997

.1614 .2078

.1644 ,2158

.1673 ,2236

.1703 .2313

.1733 2389



6.31380.8003

6.1970 ,7922

6.0844 ,7842

5.9757 .7764

5.8708 ,7687

5.7694 ,7611



81"OO'

50

40

30



0.1745 10"OO'

0.1774

10

0.1804

20

0.1833

30

0.1862

40

0.1891

50



.1736 9.2397

.1765 .2468

.1794 .2538

.I822 .2606

.1851 2674

.1880 .2740



.98489.9934

.9843 .9931

.9838 .9929

.9833 .9927

.9827 ,9924

.9822 .9922



.I763 9.2463

,1793 ,2536

.1823 .2609

.1853 .2680

.1883 2750

.1914 .2819



5.67130.7537

5.5764 .7464

5.4845 .7391

5.3955 .7320

5.3093 .7250

5.2257 .7181



80"OO' 1.3%3

50 1.3934

40 1.3904

30 1.3875

20 1.3846

10 1.3817



0.1920 1"OO'

0.1949

10

0.1978

20

0.2007

30

0.2036

40

0.2065

so



.19089.2806

.1937 2870

.1965 -29.34

.1%4 .2997

,2022 .3058

2051 .3119



.9816 9.9919

,9811 .9917

.9805 9 1 4

.9799 .9912

.9793 .9909

.9787 .9907



.19449.2887

,1974 .2953

.2004 .3020

,2035 .3085

.2065 ,3149

.2095 .3212



5.14460.7113

5.0658 .7047

4.9894 .6980

4.9152 ,6915

4.8430 .6851

4.7729 .6788



79"OO' 1.3788

50 1.3759

40 1.3730

30 13.701

20 1.3672

10 1.3643



0.2094 2"OO'

0.2123

10

20

0.2153

0.2182

30

0.2211

40

0.2240

50



2079 9.3179

2108 .3238

2136 .3296

2164 .3353

2193 .3410

.2221 .3466



.9781 9.9904

.9775 ,9901

.9769 ,9899

.9763 .98%

.9757 .9893

.9750 .9890



.2126 9.3275

.2156 .3336

.2186 .3397

2217 .3458

2247 ,3517

.2278 .3576



4.70460.6725

4.6382 .6664

4.5736 .6603

4.5107 .6542

4.4494 ,6483

4.3897 .6424



78"OO'

50

40

30



0.2269 13"W

i._

n

0.238

.

0.2327

20

0.2356

30

0.2385

40

50

0.2414



.22509.3521

,2278 .3575

2306 ,3629

2334 .3682

.2363 .3734

.2391 .3786



0.2443 14"OO'

0.2473

10

0.2502

20

0.2531

30

0.2560

40

0.2589

50



.24199.3837

.2447 .3887

.2176 ,3937

.2504 .3986



.97039.9869

.96% ,9866

.9689 .9863

,9681 .9859

:2532 .4035 .%74 .9856

.2560 .4083 ,9667 .9853



.2493 9.3968

.2524 .4021

,2555 .4074

.2586 .4127

.2617 .4178

.2648 .4230



4.01080.6032

3.9617 .5979

3.9136 .5926

3.8667 ,5873

3.8208 3 2 2

3.7760 S770



76"OO'

50

40

30



0.2618 15"OO'

0.2647

10

0.2676

20

0.2705

30

0.2734

40

0.2763

50



.25889.4130

.2616 .4177

,2644 .4223

2672 .4269

.2700 .4314

,2728 .4359



.96599.9849

.9652 .9846

.9644 ,9843

.9636 ,9839

.9628 .9836

.9621 .9832



2679 9.4281

,2711 ,4331

.2742 .4381

.2773 .4430

2805 .4479

.2836 .4527



3.7321 0.5719

3.6891 S669

3.6470 S619

3.6059 S570

3.5656 ,5521

3.5261 S473



75"OO'

50

40

30



0.2793 16"OO'

0.2822

10

0.2851

20

0.2880

30

3.2909

40

1.2938

50



2756 9.4403

-2784

__. - . .-4447

. . ..

.2812 .4491

.2840 .4533

.2868 .4576

2896 .4618



.96139.9828

.9605 .9825

.9596 .9821

.9588 .9817

.9580 .9814

.9572 .9810



.28679.4575

.2899 .4622

.2931 .4669

. 2 w .4%

.2994 .4762

.3026 .4808



3.4874 0.5425

3.4495 .5378

3.4124 .5331

3.3759 ~ 2 8 4

3.3402 S238

3.3052 S192



74"OO' 1.2915

50 1.2886

40 1.2857

30 1.2828

20 1.2799

10 1.2770



0.2967 17"OO'

0.29%

10

0.3025

20

0.3054

30

0.3083

40

50

0.3113



2924 9.4659

2952 .4700

.2979 .4741

.3007 .4781

.3035 .4821

.3062 .4861



.95639.9806

.9555 .9802

.9546 .9798

.9537 .9794

.9528 .9709

.9520 .9786



.30579.4853

.3089 ,4898

.3121 .4943

.3153 .4987

.3185 .SO31

.3217 SO75



3.27090.5147

3.2371 S102

3.2041 .SO57

3.1716 .SO13

3.1397 .4969

3.1084 .4925



73"OO' 1.2741

50 1.2712

40 1.2683

30 1.2654

20 1.2625

10 1.2595



0.3142 18"OO'



.30909.4900 .95119.9782 .32499.5118 3.07770.4882



72"OO' 1.2566



Nat.



Log.



-ELF



.9744'9.9887 ,23099.3634 4.33150.6366

-9737 -9884 2339 3691 4.2747 ,6309

.2370 .3748 4.2193 ,6252

,9724 .9878 .2401 .3804 4.1653 .6196

.9717 ,9875 2432 .3859 4.1126 .6141

.9710 .9872 .2462 ,3914 4.0611 .6086



3736 :9881



Nat.



Log.



sir;es



-Nat.



Cotangents



(continued)

SMITHSONIAN PHYSICAL TABLES



Log.



Nat.



20



10



20



10



1.4137

1.4108

1.4079

1.4050

1.4021

1.3992



1.3614

1.3584

1.3555

1.3526

1.3497

1.3468



77"OO' 1.3439

50 1.3410

40 1.3381

30 1.3352

20 1.3323

10 1.3294



20



10



20



10



1.3265

1.3235

1.3206

1.3177

1.3148

1.3119

1.3090

1.3061

1.3032

1.3003

1.2974

1.2945



Log.



Tangents



Degrees



Radians



34

Radians



T A B L E 15.-CIRCULAR

Degrees



( T R I G O N O M E T R I C ) F U N C T I O N S (continued)



Sines

Nat.



Log.



Cosines

Nat.



Tangents



Cotangents



Log.



0.3142 18"OO'

0.3171

10

0.3200

20

0.3229

30

0.3258

40

0.3287

50



.3090 9.4900

.3118 .4939

.3145 .4977

.3173 ,5015

,3201 .5052

.3228 .5090



.9511 9.9782

,9502 .9778

,9492 .9774

.9483 ,9770

.9474 .9765

.9465 .9761



.32499.5118

.3281 .5161

,3314 .5203

.3346 .5245

.3378 .5287

.3411 .5329



3.07770.4882

3.0475 .4839

3.0178 .4797

2.9887 .4755

2.9600 ,4713

2.9319 .4671



72"OO'

50

40

30

20

10



1.2566

1.2537

1,2508

1.2479

1.2450

1.2421



0.3316 19"OO'

0.3345

10

0.3374

20

0.3403

30

0.3432

40

0.3462

50



.32569.5126

.3283 .5163

.3311 ,5199

.3338 ,5235

.3365 5270

,3393 .5306



-94559.9757

19446 -:97si

,9436 ,9748

.9426 .9743

.9417 ,9739

.9407 .9734



3443 9.5370

.3476 __541.1.

.3508 .5451

.3541 .5491

.3574 .5531

.3607 .5571



2.90420.4630

2.8770 .4589

n2 .4549

2.85"._._

2.8239 .4509

2.7980 ,4469

2.7725 .4429



71"OO'

50

49

36

20



1.2392

1.2363

123.~



0.3491 2O"OO'

0.3520

10

0.3549

20

0.3578

30

0.3607

40

0.3636

50



.34209.5341

.3448 .5375

.3475 .5409

,3502 .5443

.3529 .5477

,3557 S510



-93979.9730

.9387 .9725

,9377 ,9721

,9367 ,9716

.9356 .9711

,9346 .9706



.3640 9.5611

-3673 .5659

,3706 .5689

,3739 ,5727

.3772 .5766

,3805 .5804



2.7475 0.4389

2.7228 .4350

2.6985 .43ii

2.6746 ,4273

2.6511 .4234

2.6279 .4196



7O"OO' 1.2217

SO

1.2188

40 1.2159

30 1.2130

20 1.2101

10 1.2072



0.3665 21"OO'

0.3694

10

20

0.3723

0.3752

30

0.3782

40

0.3811

50



.35849.5543

.3611 .5576

,3638 ,5609

.3665 ,5641

.3692 ,5673

.3719 .5704



.93369.9702

.9325 .9697

.9315 .9692

.9304 .9687

,9293 .9682

.9283 .9077



,38399.5842

.3872 .5879

.3906 ,5917

.3939 .5954

.3973 .5991

.4006 .6028



2.6051 0.4158

2.5826 .4121

2.5605 .4083

25386 ,4046

2.5172 .4009

2.4960 .3972



69"OO'

50

40

30

20

10



1.2043

1.2014

1.1985

1.1956

1.1926

1.1897



0.3840 22"OO'

0.3869

10

0.3898

20

0.3927

30

0.3956

40

0.3985

50



.37469.5736

.3773 5767

:38@ .5798

,3827 .5828

.3854 .5859

.3881 ,5889



.92729.9672

.9261 .9667

-9250 -9661

....

.9239 .9656

.9228 .9651

,9216 .9646



.4040 9.6064

.4074 ,6100

.4108 6136

.4142 .6172

.4176 .6208

.4210 .6243



2.4751 0.3936

2.4545 .3900

2.4342 .3864

2.4142 .3828

2.3945 .3792

2.3750 .3757



68"OO'

50

40

30

20

10



1.1868

1.1839

1.1810

1.1781

1.1752

1.1723



0.4014 23"OO'

0.4043

10

0.4072

20

0.4102

30

0.4131

40

0.4160

50



,39079.5919

.3934 .5948

,3961 .5978

.3987 .6007

.4014 .6036

.4041 .6065



.9205 9.9640

.9194 .9635

.9182 .9629

.9171 .9624

.9159 .9618

,9147 .9613



.42459.6279

.4279 .6314

.4314 .6348

.4348 .638,3

.4383 .6417

,4417 .6452



2.35590.3721

2.3369 .3686

2.3183 .3652

2.2998 .3617

2.2817 .3583

2.2637 .3548



67"W 1.1694

50 1.1665

40 1.1636

30 1.1606

20 1.1577

10 1.1548



9.4189 24"OO'

oki8

10

0.4247

20

0.4276

30

0.4305

40

0.4334

50



.40679.6093

,4094 .6121

.4120 .6149

.4147 .6177

.4173 .6205

.4200 .6232



.91359.9607

.9124 .9602

.9112 .9596

.9100 .9590

.9088 .9584

.9075 .9579



.44529.6486

.4487 .6520

.4522 .6553

.4557 .6587

.4592 .6620

.4628 .6654



2.2460 6.3514

2.2286 .3480

2.2113 .3447

2.1943 .3413

2.1775 .3380

2.1609 .3346



66"OO'

50

40

30

20

10



1.1432

1.1403

1.1374



0.4363 25"OO'

0.4392

10

0.4422

20

0.4451

30

0.4480

40

0.4509

50



.42269.6259

.4253 .6286

.4279 .6313

.4305 .6340

.4331 .6366

.4358 .6392



9063 9.9573

90.51 .9567

9038 .9561

.9026 .9555

9013 .9549

.9001 .9543



,46639.6687

.4699 ,6720

.4734 .6752

.4770 ,6785

.4806 .6817

.4841 ,6850



2.14450.3313

2.1283 .3280

2.1123 .3248

2.0965 .3215

2.0809 .3183

2.0655 .3150



65"OO'

50

40

30

20

10



1.1345

1.1316

1.1286

1.1257

1.1228

1.1199



0.4538 26"W

0.4567

10

0.4596

20

0.4625

30

0.4654

40

0.4683

50

0.4712 27"OO'



.43849.6418

.4410 .6444

,4436 .6470

,4462 ,6495

.4488 .6521

.4514 .6546

.45409.6570



.89889.9537

.8975 .9530

.8962 .9524

,8949 .9518

3936 .9512

.8923 .9505

.8910 9.9499



.4877 9.6882

,4913 .6914

,4950 .6946

.4986 ,6977

.5022 .7009

.5059 .7040

.SO95 9.7072



2.05030.3118

2.0353 ,3086

2.0204 .3054

2.0057 .3023

1.9912 2991

1.9768 .2960

1.9626 0.2928



64'00'

50

40

30



Nat.



Nat.



Nat.



Nat.



Log.



-2L-



SMITHSONIAN PHYSICAL TABLES



Log.



Sines



Log.



Cotangents



- -



Log.



Tangents



10



i:2505

1.2275

1.2246



1.1519

1.1490



1,1461



1.1170

1.1141

1.1112

1.1083

20 1.1054

10 1.1025

63"OO' 1.0996

Degrees



Radians



T A B L E 15.-CIRCULAR

Radians



Degrees



Sines

Nat.



Log.



( T R I G O N O M E T R I C ) F U N C T I O N S (continued)

Cosines

Nat.



Log.



Tangents

Nat.



Log.



35



Cotangents

Nat.



Log.



0.4712 27"OO'

0.4741

10

0.4771

20

0.4800

30

40

0.4829

0.4858

50



.45409.6570

,4566 A595

,4592 .6620

.4617 ,6644

.4643 .6668

.4669 .6692



,89109.9409

,8897 .9492

,8884 ,9486

,8870 .9479

.8857 ,9473

3843 .9466



.SO959.7072

S132 ,7103

S169 .7134

,5206 ,7165

,5243 .7196

S280 .7226



1.96260.2928

1.9486 ,2897

1.9347 .2866

1.9210 .2835

1.9074 .2804

1.8940 ,2774



63"OO'

50

40

30

20

10



1.0996

1.0966

1.0937

1.0908

1.0879

1.0850



0.4887 28"OO'

0.4916

10

0.4945

20

0.4974

30

0.5003

40

0.5032

SO



,46959.6716

,4720 .6740

,4746 6763

,4772 ,6787

.4797 ,6810

.4823 ,6833



,88299.9459

,8816 ,9453

3802 ,9446

.8788 .Y439

3774 .9432

,8760 .9425



.5317 9.7257

,5354 .7287

5392 ,7317

,5430 ,7348

,5467 ,7378

,5505 ,7408



1.88070.2743

1.8676 ,2713

1.8546 .2683

1.8418 ,2652

1.8291 ,2622

1.8165 .2592



62"OO'

50

40

30

20

10



1.0821

1.0792

1.0763

1.0734

1.0705

1.0676



0.5061 29"OO'

0.5091

10

0.5120

20

0.5149

30

0.5178

40

0.5207

50



,48489.6856

,4874 .6878

,4899 .6901

.4924 A923

.4950 .6946

.4975 ,6968



,87469.9418

,8732 ,9411

,8718 ,9404

,8704 ,9397

3689 .9390

,8675 ,9383



,55439.7438

,5581 ,7467

S619 ,7497

,5658 ,7526

5696 ,7556

S735 .7585



1.80400.2562

1.7917 ,2533

1.7796 ,2503

1.7675 ,2474

1.7556 ,2444

1.7437 .2415



61"OO'

50

40

30

20

10



1.0647

1.0617

1.0588

1.0559

1.0530

1.0501



0.5236 30"00'

0.5265

10

0.5294

20

0.5323

30

0.5352

40

50

0.5381



S O 0 0 9.6990

,5025 .7012

SOSO ,7033

.SO75 .7055

,5100 .7076

S125 .7097



3660 9.9375

,6646 ,9368

,6631 ,9361

,8616 .9353

,8601 .9346

,8587 ,9338



S774 9.7614

3312 ,7644

SS51 .7673

,5890 .7701

,5930 .7730

SY69 ,7759



1.7321 0.2386

1.7205 ,2356

1.7090 ,2327

1.6977 ,2299

1.6864 ,2270

1.6753 .2241



6O"OO' 1.0472

50 1.0443

40 1.0414

30 1.0385

20 1.0356

10 1.0327



0.5411 31"OO'

0.5440

10

0.5469

20

0.5498

30

0.5527

40

0.5556

50



,51509.7118

S175 .7139

S200 .7160

S225 ,7181

,5250 .7201

S275 .7222



3572 9.9331

.8557 ,9323

23542 .9315

A526 .9308

.8511 .9300

I3496 ,9292



.60099.7788

.6048 ,7816

,6088 ,7845

,6128 .7873

.6168 .7902

.6208 .7930



1.66430.2212

1.6534 2184

1.6426 ,2155

1.6319 ,2127

1.6212 .2098

1.6107 ,2070



59"OO'

50

40

30

20

10



0.5585 32"OO'

0.5614

10

0.5643

20

0.5672

30

0.5701

40

50

0.5730



S299 9.7242

5324 .7262

.5348 .7282

S373 ,7302

,5398 ,7322

.5422 .7342



3480 9.9284

.9276

.8450 ,9268

3434 .9260

3418 .9252

3403 .9244



,62499.7958

,6289 ,7986

.6330 3014

.6371 ,8042

.6412 3070

h453 .SO97



1.60030.2042

1.5900 2014

i15798 3 8 6

1.5697 ,1958

1.5597 ,1930

1.5497 ,1903



58"W 1.0123

50 1.0094

40 1.0065

30 1.0036

20 1.0007

10 0.9977



0.5760 33'00'

10

0.5789

0.5818

20

0.5847

30

0.5876

40

0.5905

50



S446 9.7361

S471 .7380

,5495 .7400

S519 .7419

S544 ,7438

S568 .7457



.83879.9236

3371 ,9228

,8355 .9219

3339 .9211

3323 .9203

3307 .9194



.6494 9.8125

.6536 3153

.6577 .81N

,6619 .8208

.6661 3235

.6703 A263



1.53990.1875

1.5301 .1847

i.5204 .is20

1.5108 ,1792

1.5013 .1765

1.4919 .1737



57"OO'

50

40

30

20

10



0.5934 34"CO'

0.5963

10

0.5992

20

0.6021

30

0.6050

40

0.6080

50



S592 9.7476

,5616 .7494

S640 .7513

.5664 .7531

5688 .7550

S712 .7568



,82909.9186

3274 .9177

3258 .9169

,8241 .9160

,8225 .9151

3208 .9142



.67459.8290

,6787 3317

,6830 ,8344

.6873 ,8371

.6916 ,8398

.6959 3425



1.48260.1710

1.4733 .1683

1.4641 .1656

1.4550 .1629

1.4460 .1602

1.4370 .1575



56"OO' 0.9774

so 0.9745

30

20

10



0.9687

0.9657

0.9628



0.6109 35"OO'

0.6138

10

0.6167

20

0.6196

30

0.6225

40

0.6254

50



,57369.7586

S760 .7604

S783 .7622

.5807 .7640

,5831 ,7657

S854 .7675



3192 9.9134

,8175 ,9125

,8158 ,9116

.8141 .9107

,8124 .9098

3107 .9089



.70029.8452

.7046 3479

.7089 ,8506

.7133 ,8533

,7177 .8559

,7221 .8586



1.3281 0.1548

1.4193 ,1521

1.4106 .1494

1.4019 .1467

1.3934 ,1441

1.3848 .1414



55"00'



0.9599

0.9570

0.9541

0.9512

0.9483

0.9354



0.6283 36"OO'



.8465



,58789.7692 3090 9.9080 .72659.8613 1.37640.1387

Nat.



Log.



coi,s



Nat.



Log.



SiAes



Nat.



Cotangents



(continucd)

SMITHSONIAN PHYSICAL TABLES



Log.



Nat.



1.0297

1.0268

1.0239

1.0210

1.0181

1.0152



0.9948

0.9919

0.9890

0.9861

0.9832

0.9803



40 0 3 i i



SO

40

30

20

10



54"OO' 0.9425



Log.



Tangents



Degrees



Radians



36

Radians



T A B L E 15.-CIRCULAR

Degrees



Sines

Nat.



Log. ,



( T R I G O N O M E T R I C ) F U N C T I O N S (concluded)

Cosines

Tangents

*

*

Nat.

Log.

Nat.

Log.



Cotangents



0.6283 36"OO'

10

0.6312

0.6341

20

0.6370

30

0.6400

40

50

0.6429



S878 9.7692

,5901 .7710

5925 ,7727

,5948 ,7744

,5972 .7761

,5995 .7778



,80909.9080

3073 .9070

.8056 .9061

3039 .9052

3021 ,9042

.8004 9033



.7265 9.8613

.7310 .8639

,7355 .8666

.7400 .8692

,7445 .8718

.7490 3745



1.37640.1387

1.3680 .I361

1.3597 .1334

1.3514 .I308

1.3432 .1282

1.3351 .I255



54"OO'

50

40

30

20

10



0.9425

0.9396

0.9367

0.9338

0.9308

0.9279



0.6458 37"OO'

0.6487

10

0.6516

20

0.6545

30

0.6574

40

0.6603

50



.6018 9.7795

,6041 .7811

.6M5 .7828

.6088 ,7844

.6111 .7861

A134 ,7877



.79869.9023

.7969 .9014

.7951 .9004

,7934 .8995

.7916 .8985

.7898 ,8975



,75369.8771

.7581 .8797

,7627 2-824

.7673 3850

.7720 3876

.7766 3902



1.32700.1229

1.3190 ,1203

1.3111 .I176

1.3032 ,1150

1.2954 .1124

1.2876 .lo98



53"W

50

40

30



0.9250



10



0.9221

0.9192

0.9163

0.9134

0.9105



0.6632 38"OO'

0.6661

10

0.6690

20

0.6720

30

0.6749

40

0.6778

50



,61579.7893

.6180 ,7910

,6202 .7926

.6225 ,7941

.6248 ,7957

,6271 .7973



.7880 9.8965

.7862 ,8955

,7844 3945

.7826 ,8935

,7808 2925

.7790 ,8915



'7813 9.8928

.7860 3954

,7907 ,8980

,7954 ,9006

,8002 ,9032

,8050 .9058



1.27990.1072

1.2723 .lo46

1.2647 .lo20

1.2572 ,0994

1.2497 .0968

1.2423 .0942



52"OO'

50

40

30

20

10



0.9076

0.9047

0.9018

0.8988

0.8959

0.8930



0.6807

~

~ 3

.9

.0

.0

.0

.'

0.6836

10

0.6865

20

0.6894

30

0.6923

40

0.6952

50



6293 9.7989

.6316 -:SO04

,6338 3020

.6361 .SO35

.6383 ,8050

,6406 3066



,77710.8905

,7753 ,8895

'.7735 3884

.7716 .8874

,7698 ,8864

,7679 3853



,80989.9084

A146 .9110

3195 ,9135

,8243 ,9161

,8292 .9187

.8342 ,9212



1.23490.0916

1.2276 .0890

1.2203 ,0865

1.2131 .0839

1.2059 .0813

1.1988 .0788



51"W 0.8901

50

40

30

20

10



0.8872

0.8843

0.8814

0.8785

0.8756



0.6981 4O"OO'

0.7010

i.n.

... .-.

0.7039

20

0.7069

30

40

0.7098

50

0.7127



.64289.8081 ,76609.8843

-6450 A096 . 7 w .8832

3472 i i i i

.8821

.6494 3125 .7604 .8810

.6517 3140 ,7585 ,8800

,6539 .8155 .7566 ,8789



3391 9.9238

.844i .9264

i849I Z89

.8541 .9315

.8591 ,9341

3642 ,9366



1.1918 0.0762

1.1847 07.36



1.1708 .0685

1.1640 ,0659

1.1571 .0634



50"00'

50

40

30



'10



0.8727

0.8698

0.8668

0.8639

0.8610

0.8581



0.7156 41"OO'

0.7185

10

0.7214

20

0.7243

30

0.7272

40

50

0.7301



.6561 9.8169

,6583 ,8184

.6604 ,8198

.6626 .8213

.6648 ,8227

.6670 .8241



.75479.8778

,7528 ,8767

.7528

.7509

.7509 3756

,7490 .8745

.7470 .8733

.7451 3722



,86939.9392

,8744 .9417

.8744

,8796 .9443

,8847 .9468

,8899 .9494

A952 .9519



1.1504 0.0608

1.1436 .058.3

,0583

i.1369

.OSS7

1.1369 .0557

1.1303 .0532

1.1237 ,0506

1.1171 .0481



49"W

50

40

30

20

10



0.8552

0.8523

0.8494

0.8465

0.8436

0.8407



0.7330 42"OO'

0.7359

10

0.7389

20

0.7418

30

0.7447

40

0.7476

50



.6691 9.8255

,6713 3269

.6734 3283

.6756 ,8297

.6777 A311

.6799 .8324



.7431 9.8711

.7412 3699

.7392 3688

.7373 3676

.7353 ,8665

.7333 ,8653



,90049.9544

,9057 ,9570

,9110 .9595

.9163 .9621

.9217 .9646

.9271 ,9671



1.1106 0.0450

1.1041 .0430

1.0977 ,0405

1.0913 ,0379

1.0850 ,0354

1.0786 .0329



48"OO'

50

40

30



0.8378

0.8348

0.8319

0.8290

0.8261

0.8232



0.7505 43'00'

0.7534

10

0.7563

20

0.7592

30

0.7621

40

0.7650

50



.68209.8338

.6841 3351

.6862 ,8365

.6884 3378

.6905 .8391

,6926 .8405



.73149.8641

.7294 A629

.7274 3618

.7254 3606

.7234 ,8594

.7214 .8582



.93259.9697

.9380 .9722

,9435 .9747

.9490 ,9772

.9545 .9798

.9601 .9823



1.07240.0303

1.0661 .0278

1.0599 .0253

1.0538 .0228

1.0477 .0202

1.0416 .0177



47'00'

50

40

30



0.7679 44"OO'

n77n9

10

. -.

_.

0.7738

20

0.7767

30

0.7796

40

0.7825

50



.69479.8418

.6967 3431

.6988 ,8444

.7009 A457

.7030 ,8469

.7050 3482



.71930.8569

.7173 .8557

,7153 .8545

.7133 ,8532

.7112 ,8520

.7092 .8507



.96579.9848

,9713 .9874

,9770 ,9899

.9827 .9924

,9884 .9949

.9942 ,9975



1.03550.0152

1.0295 .0126

1.0235 .0101

1.0176 .0076

1.0117 .0051

1.0058 :OO25



46"OO'

50

40

30

20



0.7854 45"OO'



.7071 9.8495



1.00000.0000



45"Oo' 0.7854



~~



Nat.



Log.



Cosines



SMITHSONIAN PHYSICAL TABLES



III77S :oiii



- ,70719.8495 1.0000 0.0000



Nat.



Log.



Sines



Nat.



Log.



Cotangents



Nat.



Log.



Tangents



20



20



20



10



20



10



10



De.

grees



0.8203

0.8174

0.8145

0.8116

0.8087

0.8058

0.8029

0.7999

0.7970

0.7941

0.7912

0.7883

Radians



T A B L E 16,-METHODS



OF AVERAGING DATA



LVlien a number of measurements are made of any quantity variatioils \vill he found.

The question is: \Vhat is tlie best represcnt3t:ve value for the quantity thus mea.wrctl :

and how shall the precision oi the iiieasiireiiiciits be stated? The arithmetic iiicaii of all

the readings is generally taken a s the hest value. T o tell soiiictliiiiK almut tlie Iwecision

of the final result any one of five measures of variation which arc tliscu.sctl i n hooks dealiiip

with this subject may be given. These measures of deviation arc':

fi = probable error

a = the average deviation (from the arithmetic nicaiil

u = the standard deviation

1/11 = the reciprocal of tlie modulus of precisicm

k / z v = the reciprocal of the "precision constant"



Of these precision indexes the standard deviation. u. is most easily computctl. For the

set of observed values .rl, .r2..x,, of equal weight. the u for a siiiqlc observation is given hy

Z(.r - .r)?



=



$1



and for the mean by

u=



-1



-:4

=



v I1



~



4



-(.I'

r



I Z(.r-.r)?



--(X--.i-)*



Il(11-1)



I,.r):



-



v



112



The ratios of these precision indexes to one another for a iiornial (or Gaussian)

distribution are :

p : t i : u : 1 11 : i: 'it' : : 0.376936 : 1 *\;

: \,-)

: 1.000 : \'r

or roughly as p : n : u : 1/11 : k i t ' : : 7 : 8 : 10 : 14 : 25

Most experimental data can be represented by an equation of sonic form. One (it' thc

recommended methods for determining the coefficients oi such equations is the use of a

least-squares solution. This means that an attempt is niadc t o find values icir the coefficients

such that the sum of the squares of the deviations oi tlie cxpcriniciital points irom the

resulting curve has the least possible value. Certain tahles arc of help in making such

solutions (Tables 16-26), and reference shouitl be made to books or pspers on this suhjcct

for their use.

Xri example of one niethod of finding tlic cocfiicients of such sclcctctl equations (based

on "Treatment of Experimental Data," by \Vorthing and Gefiner. published h>- \\-ilcy.

1933) follows.

P a r t 1,-Least



squares adjustment of measurements of linearly related quantities



Let, Q,, Q,. . . QI. be the k adjusted, but initially unkno\vn, values of the lincarly related

quantities. Let S,,

S2.. . S,,

be I i (> 1:) measured values oi Q's or o i linear combinations

of two or more, Q's.

Let &, &. . . A , $ be the adjustments or corrections that m i s t be applied to tlic iiieasii~cd

s ' s to yield consistent least-squares values for the Q's. See below for a simple illustration.

As O / J S C ~ P * O ~ ~cqircitiorrs

OII

we have



................................



+.



+



n,,Q, /bQ2 . ./:,,QI. - s,,= A,,

of which n i . O i . . . k i are constants. whose values are ircquciitly

1. - 1. or 0.

~ r r s foriiied. For equally \veiglitccl

From the observation equations k riorrrrol c . ~ ] l / ~ i l i ~ are

observed values of S.they arc

r ~ r o l l c ) Ir o i n i i ~ . r 1 7 , c - i ~ ~ ~ 2. . illt/;,ic)k- rll,.yii = o

">,lli1Qj [Fil!$lQ2i! ! ~ , c , l c : l. +. . I / l , / : i l ~ ) I . - r 1 7 , S j l = 0

(2)



+

+



+



+



+.



.......................................................

r k i ~ i l ( _+

) l r/:ir~ilQ,+ i i ~ i ~ ~ ; +.

l c ).. .i/:rl:ilch

l

-rk,sil =0



SMITHSONIAN PHYSICAL TABLES



OF AVERAGING D A T A (continued)



T A B L E 16.-METHODS



38



of which, as representative bracketed L 1 coefficients, we have

[atat] = alal azaz a3a3 .. .a,,a,

[acbrl= a161 a262 a3b3 . .anBn

[ a l X l l = a,X, azXz aaX3 . .anXn



+ + +

+ + +.

+ + +.



(3)



.....................................

[ktatl = klal + hzaz + k3a3+ ...knan

Solutions of equation (2) yield'the least-squares adjusted values of



Qi,



Qz...ex.



For unequally weighted values of X , that is wl, wz,. . .wnfor X , X z . . .Xn, the rrornral

equations become



+

+



[ w t a ~ a t l Q ~ [wtatbtlQ2

[ ~ i b c a < l Q i [wtDtbtlQZ



+ [zvtatctIQ3+.. . [ W C ~ ~ -~ ~[wcatXtl

I Q X =0

+ [ ~ , b t ~ i l Q+..

3 .[ ~ r b t l ~ ~ l [Q~ ~t b t X i=

l 0



(4)



.....................................................................



+



+



+.



~w~k~aclQ

[ wl l h ~ b t l Q z I W ~ ~ ~ C ..~twthtktlQkI Q ~

of which



[ w t a d = zphalal

[zv~acbtl

= walbl



+ w z ~ a+z w3a3a3+ ...wnana8,

+ zfia2bY+ w3a3b3+. . .twnanbn



IwtkcXtl = 0



(5)



............................................

[wtk+atl= wlklal + wIkza2+ w3ksar+. . .wnknan

The weights wl, m . . .w,, associated with the Xi, X Z . . . X , and with the successive observation equations are taken as inversely proportional to the squares of the probable

errors (or of the standard deviations) of the corresponding X's. It is customary to take

simple rounded numbers for the proportional values. A precise set of 28, 50, 41, and 78

may be rounded to 3, 5, 4, and 8.

As a simple application, consider the elevations of stations B, C, and D above A. Let

those elevations in order be Q1, Q2,and Q3. Let the quantities measured and the observed

elevations be such as to yield the following observation equations :



Qz - Q 3 -12 ft = A5

Qi - Q3 - 5 ft = A6

Th coefficients al, b ~ and

,

are obvious. Substitution



are seen to be 1, 0, and 0. The values of the other coefficients

equation (2) yields for the normal equations

3Qz- Q a Q36ft=O

(7)

- Qi 3 Q 2 - Q 3 - 39 ft = O

- Qi - Q z 3Q3 13 ft = 0



+



+



+



Solutions of equation ( 7 ) yield 91 ft, 174 ft, and 44 ft for the elevations of B, C, and D

above A.

P a r t 2.-Least-squares



+ bx,



equations of the type y = a

observed (x,y) values



to represent a series of



For equally weighted pairs of (x,y) of which the errors of measurement are associated

with the determinations of the y's



of which



SMITHSDNIAN PHYSICAL TABLES



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Table 13. Formulas for Moments of Inertia, Radii of Gyration, and Weights of Various Shaped Solids

Tải bản đầy đủ ngay(0 tr)

×