Smithsonian Physical Tables (9th Revised Edition)

898 Pages·38.449 MB·English

by  Forsythe, W.E.

Smithsonian
Physical Tables

Ninth Revised Edition

Prepared by

WILLIAM ELMER FORSYTHE

Norwich, New York
2003
PREFACE TO THE NINTH REVISED EDITION
This edition of the Smithsonian Physical Tables consists of 901 tables giv-
ing data of general interest to scientists and engineers, and of particular inter-
est to those concerned with physics in its broader sense. The increase in size
over the Eighth Edition is due largely to new data on the subject of atomic
physics. The tables have been prepared and arranged so as to be convenient
and easy to use. The index has been extended. Each set of data given herein
has been selected from the best sources available. Whenever possible an expert
in each field has been consulted. This has entailed a great deal of correspond-
ence with many scientists, and it is a pleasure to add that, almost without
exception, all cooperated generously.
When work first started on this edition, Dr. E. U. Condon, then director of
the National Bureau of Standards, kindly consented to furnish any assistance
that the scientists of that institution were able to give. The extent of this help
can be noted from an inspection of the book. Dr. Wallace R. Brode, associate
director, National Bureau of Standards, gave valuable advice and constructive
criticism as to the arrangement of the tables.
D. H. Menzel and Edith Jenssen Tebo, Harvard University, Department of
Astronomy, collected and arranged practically all the tables on astronomy.
A number of experts prepared and arranged groups of related data, and
others either prepared one or two tables or furnished all or part of the data
for certain tables. Care has been taken in each case to give the names of those
responsible for both the data and the selection of it. A portion of the data was
taken from other published sources, always with the.consent and approval of
the author and publisher of the tables consulted. Due credit has been given in
all instances. Very old references have been omitted. Anyone in need of these
should refer to the Eighth Edition.
It was our intention to mention in this preface the names of all who took part
in the work, but the list proved too long for the space available. We wish,
however, to express our appreciation and thanks to all the men and women
from various laboratories and institutions who have been so helpful in con-
tributing to this Ninth Edition.
Finally, we shall be grateful for criticism, the notification of errors, and
new data for use in reprints or a new edition.
W. E. FORSYTHE
Astrophysical Observatory
Smithsonian Institution
January 1951
EDITOR’S NOTE
The ninth edition of the Physical Tables was first published in June 19.54.
In the first reprint (1956), the second reprint (1959), and the third (1964)
a few misprints and errata were corrected.
iii
TABLE 1.-TEMPERATURE CONVERSION TABLE
The numbers in boldface type refer to the temperature either in degrees Centigrade or Fahrenheit which it is desired to convert into the other sale.
If converting from degrees Fahrenheit to Centigrade, the equivalent will be be found in the column on the left, while if converting from degrees Centi-
grade to Fahrenheit the answer will be found in the columr! on the right.
- .
559.4 to 28 29 to 140 150 to a90 900 to 1650 1660 to 2410 2420 to 3000
/ C .r. . LI A. F 'C F c L F- ' ' c F
-273 -459.4 ... -1.67 29 84.2 66 150 302 482 900 1652 904 1660 302( 1327 2420 4388
-268 -450 ... -1.11 30 86.0 71 160 320 488 910 1670 910 1670 3031 1332 2430 4406
-262 -440 ... -0.56 31 87.8 77 170 338 493 920 1688 916 1680 30% 1338 2440 4424
-257 -430 ... 0 32 89.6 82 180 356 499 930 17Ot 92 1 1690 307d 1343 2450 4442
-251 -420 ... 0.56 33 91.4 88 190 374 504 940 1724 927 1700 309; 1349 2460 4464
-246 -410 ... 1.11 34 93.2 93 200 392 510 950 1742 932 1710 31 1( 1354 2470 4478
-240 -400 ... 1.67 35 95.0 99 210 410 516 960 176C 938 1720 3121 1360 2480 44%
-234 -390 ... 2.22 36 96.8 100 212 414 521 970 1778 943 1730 314 1366 2490 4514
-229 -380 ... 2.78 37 98.6 104 220 428 527 980 1796 949 1740 316' 1371 2500 4532
-223 -370 3.33 38 100.4 110 230 446 532 990 1814 954 1750 318; 1377 2510 4550
...
-218 -360 3.89 39 102.2 116 240 464 538 1000 1832 960 1760 320( 1382 2520 4568
-212 -350 ...... 4.44 40 104.0 121 250 482 543 1010 185C 966 1770 321t 1388 2530 4586
-207 -340 ... 5.00 41 105.8 127 260 500 549 1020 1868 971 1780 3236 1393 2540 4604
-201 -330 5.56 42 107.6 I32 270 518 554 1030 1886 977 1790 3254 1399 2550 4622
...
-196 -320 6.11 43 109.4 138 280 536 560 1040 1904 982 1800 327; 1404 2560 4640
...
-190 -310 ... 6.67 44 111.2 143 290 554 566 1050 1922 988 1810 32% 1410 2570 4658
-184 -300 ... 7.22 45 113.0 149 300 572 571 1060 1940 993 1820 3301 1416 2580 4676
-179 -290 7.78 46 114.8 154 310 5% 577 1070 1958 999 1830 3326 1421 2590 4694
-173 -280 .,. 8.33 47 116.6 160 320 608 582 1080 1976 1004 1840 3344 1427 2600 4712
-169 -273 -459.4 8.89 48 118.4 16G 330 626 588 1090 1994 1010 1850 3362 1432 2610 4730
-168 -270 -454 9.44 49 120.2 171 340 644 593 1100 2012 1016 1860 338C 1438 2620 4748
-162 -260 -436 10.0 50 122.0 I77 350 662 599 1110 2030 1021 1870 3398 1443 2630 4766
-157 -250 -418 10.6 51 123.8 182 360 680 604 1120 2048 1027 1880 3416 1449 2640 4784
-151 -240 -400 11.1 52 125.6 I88 370 698 610 1130 2066 1032 1890 3434 1454 2650 4802
-146 -230 -382 11.7 53 127.4 193 380 716 616 1140 2084 1038 1900 3452 1460 2660 4820
-140 -220 -364 12.2 54 129.2 199 390 734 62 1 1150 21 02 1043 1910 3476 1466 2670 4838
-134 -210 -346 12.8 55 131.0 !04 400 752 627 1160 2120 1049 1920 3488 1471 2680 4856
-129 -200 -328 13.3 56 132.8 210 410 770 632 1170 2138 1054 1930 3506 1477 2690 4874
-123 -190 -310 13.9 57 134.6 216 420 788 638 1180 2156 1060 1940 3524 1482 2700 4892
-118 -180 -292 14.4 58 136.4 22 1 430 806 643 1190 2174 1066 1950 3542 1488 2710 4910
-112 -170 -274 15.0 59 138.2 !27 440 824 649 1200 2192 1071 1960 3560 1493 2720 4928
-107 -160 -256 15.6 60 140.0 232 450 842 654 1210 2210 1077 1970 3578 1499 2730 4946
--10 1 -150 -238 16.1 61 141.8 ?38 460 860 660 1220 2228 I082 1980 3596 I504 2740 4964
- 95.6 -140 -220 16.7 62 143.6 !43 470 878 666 1230 2246 1088 1990 3614 IS10 2750 4982
90.0 -130 -202 17.2 63 145.4 !49 480 896 671 1240 2264 1093 2000 3632 1516 2760 SO00
-
- 84.4 -120 -184 17.8 64 147.2 254 490 914 677 1250 2282 1099 2010 3650 1521 2770 5018
- 78.9 -110 -166 18.3 65 149.0 260 500 932 682 1260 2300 1104 2020 3668 1527 2780 5036
- 73.3 --10 0 -148 18.9 66 150.8 266 510 950 688 1270 2218 1110 2030 3686 1532 2790 5054
- 67.8 90 -130 19.4 67 152.6 271 520 968 693 1280 2336 1116 2040 3704 1538 2800 5072
- 62.2 -- 80 -112 20.0 68 154.4 277 530 986 699 1290 2354 1121 2050 3722 1543 2810 5090
- 56.7 - 70 --9 4 20.6 69 156.2 282 540 1004 704 1300 2372 1127 2060 3740 1549 2820 5108
- 51.1 - 60 - 76 21.1 70 158.0 288 550 1022 710 1310 2390 1132 2070 3758 1554 2830 5126
- 45.6 - 50 58 21.3 71 159.8 293 560 1040 716 1320 2408 1138 2080 3776 1560 2840 5144
- 40.0 - 40 --4 0 22.2 72 161.6 299 570 1058 72 1 1330 2426 1143 2090 3794 1566 2850 5162
- 34.4 - 30 22 22.8 73 163.4 304 580 1076 727 1340 2444 1149 2100 3812 1571 2860 5180
- 28.9 - 20 - 4 23.3 74 165.2 310 590 1094 732 1350 2462 1154 2110 3830 1577 2870 5198
- 23.3 10 14 23.9 75 167.0 316 600 1112 738 1360 2480 1160 2120 3848 1582 2880 5216
17.8 0 32 24.4 76 168.8 32 1 610 1130 743 1370 2498 1166 2130 3866 1588 2890 5234
-
- 17.2 1 33.8 25.0 77 170.6 327 620 1148 749 1380 2516 1171 2 140 3884 1593 2900 5252
- 16.7 2 35.6 25.6 78 172.4 332 630 1166 754 1390 2534 1177 2150 3902 1599 2910 5270
16.1 3 37.4 26.1 79 174.2 338 640 1184 760 1400 2552 1182 2160 3920 1604 2920 5288
-
- 15.6 4 39.2 26.7 80 176.0 343 650 1202 766 1410 2570 1188 2170 3938 1610 2930 5306
- 15.0 5 41.0 27.2 81 177.8 349 660 1220 771 1420 2588 1193 2180 3956 1616 2940 5324
- 14.4 6 42.8 27.8 82 179.6 354 670 1238 777 1430 2606 1199 2190 3974 1621 2950 5342
- 13.9 7 44.6 28.3 83 181.4 360 680 1256 782 1440 2624 1204 2200 3992 1627 2960 5360
- 13.3 8 46.4 28.9 84 183.2 366 690 1274 788 1450 2642 1210 2210 4010 1632 2970 5378
- 12.8 9 48.2 29.4 85 185.0 37 1 700 1292 793 1460 2660 1216 2220 4028 1638 2980 5396
- 12.2 I0 50.0 30.0 86 186.8 377 7 10 1310 799 1470 2678 1221 2230 4046 1643 2990 5414
11.7 11 51.8 30.6 87 188.6 382 720 1328 804 1480 2696 1227 2240 4064 1649 3000 5432
-
- 11.1 12 53.6 31.1 88 190.4 388 730 1346 810 1490 2714 1232 2250 4082
- 10.6 13 55.4 31.7 89 192.2 393 740 1364 816 1500 2732 1238 2260 4100
- 10.0 14 572 32.2 90 194.0 399 750 1382 821 1510 2750 1243 2270 4118 Interpolation
- 9.44 15 59.0 32.8 91 195.8 404 760 1400 827 1520 2768 1249 2280 4136 factor#
- 8.89 16 60.8 33.3 92 197.6 410 770 1418 832 1530 2786 1254 2290 4154
- 8.33 17 62.6 33.9 93 199.4 416 780 1436 838 1540 2804 1260 2300 4172
- 7.78 18 64.4 34.4 94 2012 421 790 1454 843 1550 2822 1266 2310 4190 c F
722 19 66.2 35.0 95 203.0 427 800 1472 849 1560 2840 1271 2320 4208 0.56 1 1.8
-
- 6.67 20 68.0 35.6 96 204.8 432 810 1490 854 1570 2858 1277 2330 4226 1.11 2 3.6
6.11 21 69.8 36.1 97 206.6 438 820 1508 860 1580 2876 1282 2340 4244 1.67 3 5.4
-
- 5.56 23 716 36.7 98 208.4 443 830 1526 866 1590 2894 1288 2350 4262 2.22 4 7.2
5.00 23 739 37.2 99 210.2 449 840 1544 871 1600 2912 1293 2360 4280 2.78 5 9.0
-
- 4.44 24 75.2 37.8 100 212.0 454 850 1562 877 1610 2930 1299 2370 4298 3.33 6 10.8
- 3.89 25 77.0 43 110 230 460 860 1580 882 1620 2948 1304 2380 4316 3.89 7 12.6
- 3.33 26 78.8 49 120 248 466 870 1598 888 1630 2966 1310 2390 4334 4...4 .4 . 8- 14.4
- 2.78 27 80.6 54 130 266 471 880 1616 893 1640 2984 1316 2400 4352 5.00 9 16.2
2.22 28 82.4 60 140 284 477 890 1634 899 1650 3002 1321 2410 4370 5.56 10 18.0
Ptcr-red by Alfred Sauveur; uud by the kind permiuion of bfr. Sanveur.
Contents
(For detailed breakdown of tables, see index.)

Front Matter i
Temperature Conversion Table (Table 1) ii
Preface to the Ninth Revised Edition iii
Introduction 1
Units of Measurem ent 1
Conversion Factors and Dimensional Formu lae 2
Some Fundamental Definitions (Tabl e 2) 4
Part 1. Geometrical and Mechanica l Units 4
Part 2. Hea t Units 7
Part 3. Electrical and Magnetic Units 10
Fundamental Standards (Table 3) 13
Part 1. Selection of Fundamental Qu antities 13
Part 2. Some Proposed Systems o f Units 15
Part 3. Electrical and Magneti c Units 16
Part 4. The Ordinary and the Ampere-turn Magnetic Units 18
The New (1948) System of Electric Units (Table 6) 19
Relative Magnitude of the Old International Electrical Units and t he
New 1948 Absolute Electrical Units (Table 5) 20
Relative Values of the Three Systems of Electrical Units (Table 6) 20
Conversion Factors for Units of Energy (Table 7) 21
Former Electrical Equivalents (Table 8) 22
Some Mathematical Tables (Tables 9-15) 23-36
Treatment of Experimental Data (Tables 16-25) 37-45
General Physical Constants (Tables 26-28) 46-55
Common Units of Measurement (Tables 29-36) 56-69
Constants for Temperature Measurement (Tables 37-51) 70-78
The Blackbody and its Radiant Energy (Tables 52-57) 79-86
Photometry (Tables 58-77) 87-97
Emissivities of a Number of Materials (Tables 78-84) 98-101
Characteristics of Some Light-source Materials, and Some Light
Sources (Tables 85-102) 102-111
Cooling by Radiation and Convection (Tables 103-110) 112-116
Temperature Characteristics of Materials (Tables 111-125) 117-130
Changes in Freezing and Boiling Points (Tables 126-129) 131-135
Heat Flow and Thermal Conductivity (Tables 130-141) 136-144
Thermal Expansion (Tables 142-146) 145-154
Specific Heat (Tables 147-158) 155-164
Latent Heat (Tables 159-164) 165-167
Thermal Properties of Saturated Vapors (Tables 165-170) 168-178
Heats of Combustion (Tables 171-183) 179-186
Physical and Mechanical Properties of Materials (Tables 184-209) 187-228
Characteristics of Some Building Materials (Tables 210-217) 229-231
Physical Properties of Leather (Tables 218-223) 232-233
Values of Physical Constants of Different Rubbers (Tables 224-229) 234-238
Characteristics of Plastics (Tables 233-236) 239-240
Properties of Fibers (Tables 233-236) 241-245
Properties of Woods (Tables 237-240) 246-258
Temperature, Pressure, Volume, and Weight Relations of Gases and
Vapors (Tables 241-253) 259-267
Thermal Properties of Gases (Tables 254-260) 268-277
The Joule-Thomson Effect in Fluids (Tables 261-267) 278-281
Compressibility (Tables 268-280) 282-290
Densities (Tables 281-295) 291-305
Velocity of Sound (Tables 296-300) 306-308
Acoustics (Tables 301-310A) 309-317
Viscosity of Fluids and Solids (Tables 311-338) 318-336
Aeronautics (Tables 339-346A) 337-353
Diffusion, Solubility, Surface Tension, and Vapor Pressure
(Tables 347-369) 354-374
Various Electrical Characteristics of Materials (Tables 370-406) 375-396
Electrolytics Conduction (Tables 407-415) 397-403
Electrical and Mechanical Characteristics of Wire (Tables 416-428) 404-420
Some Characteristics of Dielectrics (Tables 429-452) 421-433
Radio Propagation Data (Tables 453-465) 434-450
Magnetic Properties of Materials (Tables 466-494) 451-467
Geomagnetism (Tables 495-512) 468-502
Magneto-optic Effects (Tables 513-521) 503-508
Optical Glass and Optical Crystals (Tables 522-555) 509-534
Transmission of Radiation (Tables 556-573) 535-548
Reflection and Absorption of Radiation (Tables 574-592) 549-556
Rotation of Plane of Polarized Light (Tables 593-597) 557-560
Media for Determinations of Refractive Indices with the Microscope
(Tables 598-601) 561
Photography (Tables 602-609) 562-567
Standard Wavelengths and Series Relations in Atomic Spectra
(Tables 610-624) 568-585
Molecular Constants of Diatomic Molecules (Tables 625-625a) 586-591
The Atmosphere (Tables 626-630) 592-595
Densities and Humidities of Moist Air (Tables 631-640) 596-605
The Barometer (Tables 641-648) 606-613
Atmospheric Electricity (Tables 649-653) 614-617
Atomic and Molecular Data (Tables 654-659) 618-624
Abundance of Elements (Tables 660-668) 625-629
Colloids (Tables 669-682) 630-634
Electron Emission (Tables 683-689) 635-637
Kinetic Theory of Gases (Tables 690-696) 638-624
Atomic and Molecular Dimensions (Tables 697-712) 643-650
Nuclear Physics (Tables 713-730) 651-671
Radioactivity (Tables 731-758) 672-691
X-rays (Tables 759-784) 692-705
Fission (Tables 785-793) 706-709
Cosmic Rays (Tables 794-801) 710-713
Gravitation (Tables 802-807) 714-718
Solar Radiation (Tables 808-824) 719-727
Astronomy and Astrophysics (Tables 825-884) 728-771
Oceanography (Tables 885-899) 772-779
The Earth's Rotation: Its Variation (Table 900) 780
General Conversion Factors (Table 901) 781-785
Index 787
lNTRODUCTION
UNITS OF MEASUREMENT
The quantitative measure of anything is expressed by two factors-one,
a certain definite amount of the kind of physical quantity measured, called the
unit; the other, the number of times this unit is taken. A distance is stated
as 5 meters. The purpose in such a statement is to convey an idea of this dis-
tance in terms of some familiar or standard unit distance. Similarly quantity
of matter is referred to as so many grams ; of time, as so many seconds, or
minutes, or hours.
The numerical factor definitive of the magnitude of any quantity must de-
pend on the size of the unit in terms of which the quantity is measured. For
example, let the magnitude factor be 5 for a certain distance when the mile is
used as the unit of measurement. A mile equals 1,760 yards or 5,280 feet. The
numerical factor evidently becomes 8,800 and 26,400, respectively, when the
yard or the foot is used as the unit. Hence, to obtain the magnitude factor for
a quantity in terms of a new unit, multiply the old magnitude factor by the ratio
of the magnitudes of the old and new units ; that is, by' the number of the new
units required to make one of the old.
The different kinds of quantities measured by physicists fall fairly definitely
into two classes. In one class the magnitudes may be called extensive, in the
other, intensive. To decide to which class a quantity belongs, it is often helpful
to note the effect of the addition of two equal quantities of the kind in question.
If twice the quantity results, then the quantity has extensive (additive) mag-
nitude. For instance, two pieces of platinum, each weighing 5 grams, added
together weigh 10 grams; on the other hand, the addition of one piece of
platinum at 100" C to another at 100" C does not result in a system at 200" C.
Volume, entropy, energy may be taken as typical of extensive magnitudes;
density, temperature and magnetic permeability, of intensive magnitudes.
The measurement of quantities having extensive magnitude is a compara-
tively direct process. Those having intensive magnitude must be correlated
with phenomena which may be measured extensively. In the case of tempera-
ture, a typical quantity with intensive magnitude, various methods of measure-
ment have been devised, such as the correlation of magnitudes of temperature
with the varying lengths of a thread of mercury.
Fundamental units.-It is desirable that the fewest possible fundamental
unit quantities should be chosen. Simplicity should regulate the choice-
simplicity first, psychologically, in that they should be easy to grasp mentally,
and second, physically, in permitting as straightforward and simple definition
as possible of the complex relationships involving them. Further, it seems de-
sirable that the units should be extensive in nature. It has been found possible
to express all measurable physical quantities in terms of five such units : first,
geometrical considerations-length, surface, etc.-lead to the need of a length ;
second, kinematical considerations-velocity, acceleration, etc.-introduce
time ; third, mechanics-treating of masses instead of immaterial points-in-
SMITHSONIAN PHYSICAL TABLES 1
2
troduces matter with the need of a fundamental unit of mass ; fourth, electrical,
and fifth, thermal considerations require two more such quantities. The dis-
covery of new classes of phenomena may require further additions.
As to the first three fundamental quantities, simplicity and good use sanction
the choice of a length, L, a time interval, T, and a mass, M. For the measure-
ment of electrical quantities, good use has sanctioned two fundamental quan-
tities-the dielectric constant, K, the basis of the “electrostatic” system, and
the magnetic permeability, p, the basis of the “electromagnetic” system. Be-
sides these two systems involving electrical considerations, there is in common
use a third one called the “absolute” system, which will be referred to later.
For the fifth, or thermal fundamental unit, temperature is generally ch0sen.l
Derived units.-Having selected the fundamental or basic units-namely,
a measure of length, of time, of mass, of permeability or of the dielectric
constant, and of temperature-it remains to express all other units for physical
quantities in terms of these. Units depending on powers greater than unity of
the basic units are called “derived units.” Thus, the unit volume is the volume
of a cube having each edge a unit of length. Suppose that the capacity of some
volume is expressed in terms of the foot as fundamental unit and the volume
number is wanted when the yard is taken as the unit. The yard is three times
as long as the foot and therefore the volume of a cube whose edge is a yard is
3 x 3 x 3 times as great as that whose edge is a foot. Thus the given volume
will contain only 1/27 as many units of volume when the yard is the unit of
length as it will contain when the foot is the unit. To transform from the foot
as old unit to the yard as new unit, the old volume number must be multiplied
by 1/27, or by the ratio of the magnitude of the old to that of the new unit of
volume. This is the same rule as already given, but it is usually more con-
venient to express the transformations in terms of the fundamental units
directly. In the present case, since, with the method of measurement here
adopted, a volume number is the cube of a length number, the ratio of two units
of volume is the cube of the ratio of the intrinsic values of the two units of
length. Hence, if I is the ratio of the magnitude of the old to that of the new
unit of length, the ratio of the corresponding units of volume is k. Similarly
the ratio of two units of area would be 12, and so on for other quantities.
CONVERSION FACTORS AND DIMENSIONAL FORMULAE
For the ratio of length, mass, time, temperature, dielectric constant, and
permeability units the small bracketed letters, [ 1 J , [ m], [ t], [ 01, [ K], and [ p]
will be adopted. These symbols will always represent simple numbers, but the
magnitude of the number will depend on the relative magnitudes of the units
the ratios of which they represent. When the values of the numbers represented
by these small bracketed letters as well as the powers of them involved in any
particular unit are known, the factor for the transformation is at once obtained.
Thus, in the above example, the value of 1 was 1/3, and the power involved
in the expression for volume was 3; hence the factor for transforming from
cubic feet to cubic yards was P or 1/33 or 1/27 These factors will be called
conversion factors.
1 Because of its greater psychological and physical simplicity, and the desirability that
the unit chosen should have extensive magnitude, it has been proposed to choose as the
fourth fundamental quantity a quantity of electrical charge, e. The standard units of electri-
cal charge would then be the electronic charge. For thermal needs, entropy has been pro-
posed. While not generally so psychologically easy to grasp as temperature, entropy is of
fundamental importance in thermodynamics and has extensive magnitude. (Tolman, R. C.,
The measurable quantities of physics, Phys. Rev., vol. 9, p. 237, 1917.)
SMlTHSONlAN PHYSICAL TABLES
3
To find the symbolic expression for the conversion factor for any physical
quantity, it is sufficient to determine the degree to which the quantities, length,
mass, time, etc., are involved. Thus a velocity is expressed by the ratio of the
number representing a length to that representing an interval of time, or
[L/T],a nd acceleration by a velocity number divided by an interval-of-time
number, or [LIT2]a,n d so on, and the corresponding ratios of units must
therefore enter in precisely the same degree. The factors would thus be for
the just-stated cases, [Z/t] and [1/t2]. Equations of the form above given for
velocity and acceleration which show the dimensions of the quantity in terms of
the fundamental units are called dimensional equations. Thus [El= [ML2T-']
will be found to be the dimensional equation for energy, and [ML2T2]th e
dimensional formula for it. These expressions will be distinguished from the
conversion factors by the use of bracketed capital letters.
In general, if we have an equation for a physical quantity,
Q = CLaMbTc,
where C is a constant and L, M, T represent length, mass, and time in terms
of one set of units, and it is desired to transform to another set of units in terms
of which the length, mass, and time are L1,M 1, T1,w e have to find the value of
L,/L, M,/M, 1',/T, which, in accordance with the convention adopted above,
will be 1, m, t, or the ratios of the magnitudes of the old to those of the new
units.
Thus L,=Ll, M,=Mnz, T,=Tt, and if Ql be the new quantity number,
Ql= CL,ahllbTIC,
= CLalaMbmbTCtCQ=la mbtc,
or the conversion factor is [lambtc]a, quantity precisely of the same form as
the dimension formula [LaMbTC].
Dimensional equations are useful for checking the validity of physical equa-
tions. Since physical equations must be homogeneous, each term appearing in
theni must be dimensionally equivalent. For example, the distance moved by
+
a uniformly accelerated body is s=n,t +atz. The corresponding dimensional
+
equation is [L]= [ (L/T)1 '3 [ ( L/T2)T 2]e,a ch term reducing to [L].
Dimensional considerations may often give insight into the laws regulating
physical phenomena.2 For instance, Lord Rayleigh, in discussing the intensity
of light scattered from small particles, in so far as it depends upon the wave-
length, reasons as follows :
The object is to compare the intensities of the incident and scattered ray; for these will
clearly be proportional. The number (i) expressing the ratio of the two amplitudes is a
function of the following quantities:-V, the volume of the disturbing particle; r, the
distance of the point under consideration from it; A, the wavelength; c, the velocity of
propagation of light ; D and D', the original and altered densities : of which the first three
depend only on space, the fourth on space and time, while the fifth and sixth introduce the
consideration of mass. Other elements of the problem there are none, except mere numbers
and angles, which do not depend upon the fundamental measurements of space, time, and
mass. Since the ratio i, whose expression we seek, is of no dimensions in mass, it follows
at once that D and D' occur only under the form D : D', which is a simple number and may
therefore be omitted. It remains to find how i varies with V,r , A, c.
Now, of these quantities, c is the only one depending on time ; and therefore, as i is of no
dimensions in time, c cannot occur in its expression. We are left, then, with V,r , and A ; and
from what we know of the dynamics of the question, we may be sure that i varies directly as
V and inversely as Y, and must therefore be proportional to V t A?, V being of three di-
Buckingham, E., Phys. Rev., vol. 4, p. 345, 1914 ; also Philos. Mag., vol. 42, p. 696, 1921.
Philos. Mag., ser. 4, voI. 41, p. 107, 1871. See also Robertson, Dimensional analysis,
Gen. Electr. Rev., vol. 33, p. 207, 1930.
SMITHSONIAN PHYSICAL TABLES