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