DIMENSIONAL ANALYSIS 
 
DIMENSIONAL ANALYSIS 
 
 BY 
 
 P. W. BRIDGMAN 
 
 II 
 
 PEOFESSOE OF PHYSICS IN HAEVAED UNIVEESITY 
 
 NEW HAVEN 
 
 YALE^ UNIVERSITY PRESS 
 
 LONDON : HUMPHEEY MILFOED : OXFOED UNIVEESITY PEESS 
 
 MDCCCCXXII 
 
G 
 
 A 
 
 COPYRIGHT, 1922, BY 
 YALE UNIVERSITY PRESS 
 
PEEFACE 
 
 THE substance of the following pages was given as a series of five 
 lectures to the Graduate Conference in Physics of Harvard Univer- 
 sity in the spring of 1920. 
 
 The growing use of the methods of dimensional analysis in tech- 
 nical physics, as well as the importance of the method in theoretical 
 investigations, makes it desirable that every physicist should have 
 this method of analysis at his command. There is, however, nowhere 
 a systematic exposition of the principles of the method. Perhaps the 
 reason for this lack is the feeling that the subject is so simple that 
 any formal presentation is superfluous. There do, nevertheless, exist 
 important misconceptions as to the fundamental character of the 
 method and the details of its use. These misconceptions are so wide- 
 spread, and have so profoundly influenced the character of many 
 speculations, as I shall try to show by many illustrative examples, 
 that I have thought an attempt to remove the misconceptions well 
 worth the effort. 
 
 I have, therefore, attempted a systematic exposition of the princi- 
 ples underlying the method of dimensional analysis, and have illus- 
 trated the applications with many examples especially chosen to 
 emphasize the points concerning which there is the most common 
 misunderstanding, such as the nature of a dimensional formula, 
 the proper number of fundamental units, and the nature of dimen- 
 sional constants. In addition to the examples in the text, I have 
 included at the end a number of practise problems, which I hope 
 will be found instructive. 
 
 The introductory chapter is addressed to those who already have 
 some acquaintance with the general method. Probably most readers 
 will be of this class. I have tried to show in this chapter by actual 
 examples what are the most important questions in need of discus- 
 sion. The reader to whom the subject is entirely new may omit this 
 chapter without trouble. 
 
 I am under especial obligation to the papers of Dr. Edgar Buck- 
 ingham on this subject. I am also much indebted to Mr. M. D. Hersey 
 of the Bureau of Standards, who a number of years ago presented 
 Dr. Buckingham's results to the Conference in a series of lectures. 
 
 September, 1920. 
 
 497291 
 
CONTENTS 
 
 Chapter I. 
 
 Chapter II. 
 
 Chapter III. 
 
 Chapter IV. 
 
 Chapter Y. 
 
 Chapter VI. 
 
 Chapter VII. 
 
 Chapter VIII. 
 
 Problems 
 
 Index 
 
 Introductory .... 
 
 Dimensional Formulas .... 
 
 On the Use of Dimensional Formulas in 
 Changing Units 
 
 The H Theorem 
 
 Dimensional Constants and the Number of 
 Fundamental Units .... 
 
 Examples Illustrative of Dimensional Analy- 
 sis ....... 
 
 Applications of Dimensional Analysis to 
 Model Experiments. Other Engineering 
 Applications ..... 
 
 Applications to Theoretical Physics . 
 
 PAGE 
 
 1 
 
 17 
 
 28 
 36 
 
 48 
 66 
 
 81 
 
 88 
 
 107 
 
 111 
 
CHAPTER I 
 INTRODUCTORY 
 
 APPLICATIONS of the methods of dimensional analysis, to simple 
 problems, particularly in mechanics, are made by every student of 
 physics. Let us analyze a few such problems in order to refresh our 
 minds and get before us some of the questions which must be 
 answered in a critical examination of the processes and assumptions 
 underlying the correct application of the general method. 
 
 "We consider first the illustrative problem used in nearly every 
 introduction to this subject, that of the simple pendulum. Our 
 endeavor is to find, without going through a detailed solution of the 
 problem, certain relations which must be satisfied by the various 
 measurable quantities in which we are interested. The usual proce- 
 dure is as follows. We first make a list of all the quantities on which 
 the answer may be supposed to depend; we then write down the 
 dimensions of these quantities, and then we demand that these quan- 
 tities be combined into a functional relation in such a way that the 
 relation remains true no matter what the size of the units in terms 
 of which the quantities are measured. 
 
 Now let us try by this method to find how the time of swing of the 
 simple pendulum depends on the variables which determine the 
 behavior. The time of swing may conceivably depend on the length 
 of the pendulum, on its mass, on the acceleration of gravity, and on 
 the amplitude of swing. Let us write down the dimensions of these 
 various quantities, using for our fundamental system of units mass, 
 length, and time. In the dimensional formulas the symbols of mass, 
 length, and time will be denoted by capital letters, raised to proper 
 powers. Our list of quantities is as follows : 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 Time of swing, t T 
 
 Length of pendulum, 1 L 
 
 Mass of pendulum, m M 
 
 Acceleration of gravity, g LT~ 2 
 
 Angular amplitude of swing, 6 No dimensions. 
 
...DIMENSIONAL ANALYSIS 
 
 *^ *"*: A 
 
 1 We are' to 1 find t as a function of 1, m, g, and 0, such that the 
 
 functional relation still holds when the size of the fundamental 
 units is changed in any way whatever. Suppose that we have foupd 
 this relation and write 
 
 Now the dimensional formulas show how the various fundamental 
 units determine the numerical magnitude of the variables. The 
 numerical magnitude of the time of swing depends only on the size 
 of the unit of time, and is not changed when the units of mass or 
 length are changed. Hence if the equation is to remain true when 
 the units of mass and length are changed in any way whatever, the 
 quantities inside the functional sign on the right-hand side of the 
 equation must be combined in such a way that together they are also 
 unchanged when the units of mass and length are changed. In par- 
 ticular, they must be unchanged when the size of the unit of mass 
 alone is changed. Now the size of the unit of mass affects only the 
 magnitude of the quantity m. Hence if m enters the argument of 
 the function at all, the numerical value of the function will be 
 changed when the size of the fundamental unit of mass is changed, 
 and this change cannot be compensated by any corresponding 
 change in the values of the other quantities, for these are not 
 affected by changes in the size of the unit of mass. Hence the mass 
 cannot enter the functional relation at all. This shows that the 
 relation reduces to 
 
 Now 1 and g must together enter the function in such a way that 
 the numerical magnitude of the argument is unchanged when the 
 size of the unit of length is changed and the unit of time is kept 
 constant. That is, the change in the numerical value of 1 produced 
 by a change in the size of the unit of length must be exactly com- 
 pensated by the change produced in g by the same change. The 
 dimensional formula shows that 1 must be divided by g for this to 
 be accomplished. We now have 
 
 Now a change in the size of the fundamental units produces no 
 change in the numerical magnitude of the angular amplitude, be- 
 cause it is dimensionless, and hence may enter the unknown func- 
 tion in any way. But it is evident that 1/g must enter the function in 
 
INTRODUCTORY 3 
 
 such a way that the combination has the dimensions of T, since 
 these are the dimensions of t which stands alone on the left-hand 
 side of the equation. We see by inspection that 1/g must enter as 
 the square root in order to have the dimensions of T, and the final 
 result is to be written 
 
 where </> is subject to no restriction as far as the present analysis 
 can go. As a matter of fact, we know from elementary mechanics, 
 that < is very nearly a constant independent of 0, and is approxi- 
 mately equal to 2?r. 
 
 A question may arise in connection with the dimensions of 0. We 
 have said that it is dimensionless, and that its numerical magnitude 
 does not change when the size of the fundamental units of mass, 
 length, or time are changed. This of course is true, but it does not 
 follow that therefore the numerical magnitude of is uniquely 
 determined, as we see at once from the possibility of measuring in 
 degrees or in radians. Are we therefore justified in treating as a 
 constant and saying that it may enter the functional relation in any 
 way whatever? 
 
 Now let us discuss by the same method of analysis the time of 
 small oscillation of a small drop of liquid under its own surface 
 tension. The drop is to be thought of as entirely outside the gravita- 
 tional field, and the oscillations refer to periodic changes of figure, 
 as from spherical to ellipsoidal and back. The time of oscillation 
 will evidently depend on the surface tension of the liquid, on the 
 density of the liquid, and on the radius of the undisturbed sphere. 
 We have, as before, 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 Time of oscillation, t T 
 
 Surface tension, s MT~ 2 
 
 Density of liquid, d MLr 3 
 
 Radius of drop, r L 
 
 We are to find f such that 
 
 t = f (s, d, r) 
 
 where f is such that this relation holds true numerically whatever 
 the size of the fundamental units in terms of which t, s, d, and r are 
 
4 DIMENSIONAL ANALYSIS 
 
 measured. The method is exactly the same as for the pendulum 
 problem. It is obvious that M must cancel from the right-hand side 
 of the equation. This can occur only if s and d enter through their 
 quotient. Hence 
 
 t = f (s/d,r). 
 
 Now since L does not enter t, L cannot enter f . Hence s/d and r 
 must be combined in such a way that L cancels. Since L enters s/d 
 to the third power, it is obvious that s/d must be divided by the cube 
 of r in order to get rid of L. Hence 
 
 t f (s/dr 3 ). 
 
 Now the dimensions of s/dr 3 are T~ 2 . The function must be of 
 such a form that these dimensions are converted into T, which are 
 the dimensions of the left-hand side. Hence the final result is 
 
 t = Const V dr 8 /s. 
 
 That is, the time of oscillation is proportional to the three halves 
 power of the radius, to the square root of the density, and inversely 
 to the square root of the surface tension. This result is checked by 
 experiment. The result was given by Lord Rayleigh as problem 7 
 in his paper in Nature, 95, 66, 1915. 
 
 Now let us stop to ask what we meant when in the beginning we 
 said that the time of oscillation will ''depend" only on the surface 
 tension, density, and radius. Did we mean that the results are inde- 
 pendent of the atomic structure of the liquid, for example ? Every- 
 one will admit that surface tension is due to the forces between the 
 atoms in the surface layer of the liquid, and will depend in a way 
 too complicated for us at present to exactly express on the shape 
 and constitution of the atoms, and on the nature of the forces be- 
 tween them. If this is true, why should not all the elements which 
 determine the forces between the atoms also enter our analysis, for 
 they are certainly effective in determining the physical behavior? 
 "We might justify our procedure by some such answer as this. ' ' Al- 
 though it is true that the behavior is determined by a most com- 
 plicated system of atomic forces, it will be found that these forces 
 affect the result only in so far as they conspire to determine one 
 property, the surface tension." This implies that if we were to 
 measure the time of oscillation of drops of different liquids, differing 
 
INTRODUCTORY 5 
 
 as much as we pleased in atomic properties, we would find that all 
 drops of the same radius, density, and surface tension, executed 
 their oscillations in the same time. We add that the truth of this 
 reply may be checked by an appeal to experiment. But our critic 
 may not even yet be satisfied. He may ask how we were sure before- 
 hand that among the various properties of the liquids of which the 
 drop might be composed the surface tension was the only property 
 .affecting the time of oscillation. It may seem quite conceivable to 
 him that the time of oscillation might depend on the viscosity or 
 compressibility, and if we are compelled to appeal to experiment, 
 of what value is our dimensional analysis? To which we would be 
 forced to reply that we have indeed had a wider experimental 
 experience than our critic, and that there are conditions under 
 which the time of oscillation does depend on the viscosity or com- 
 pressibility in addition to the surface tension, but that it will be 
 found as a matter of experiment that if the radius of the drop is 
 made smaller and smaller there is a point beyond which the com- 
 pressibility will be found to play an imperceptibly small part, and 
 in the same way if the viscosity of the liquid is made smaller and 
 smaller, there will also be a point beyond which any further reduc- 
 tion of the viscosity will not perceptibly affect the oscillation time. 
 And we add that it is to such conditions as these that our analysis 
 applies. Instead of appealing to direct experiment to justify our 
 assertions, we might, since our critic is an intelligent critic, appeal 
 to that generalization from much experiment contained in the equa- 
 tions of hydrodynamics, and show by a detailed application of the 
 equations to the present problem that compressibility and viscosity 
 may be neglected beyond certain limiting conditions. 
 
 We shall thus ultimately be able to satisfy our critic of the cor- 
 rectness of our procedure, but to do it requires a considerable back- 
 ground of physical experience, and the exercise of a discreet judg- 
 ment. The untutored savage in the bushes would probably not be 
 able to apply the methods of dimensional analysis to this problem 
 and obtain results which would satisfy us. 
 
 Now let us consider a third problem. Given two bodies of masses 
 m x and m 2 in empty space, revolving about each other in a spherical 
 orbit under their mutual gravitational attraction. We wish to find 
 how the time of revolution depends on the other variables. We make 
 a list of the various quantities as before. 
 
6 DIMENSIONAL ANALYSIS 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Mass of first body, n^ M 
 
 Mass of second body, m 2 M 
 
 Distance of separation, r L 
 
 Time of revolution, t T 
 
 These are evidently all the quantities physically involved, because 
 whenever we compel two bodies of masses 1% and m 2 to describe a 
 circular orbit about each other under their own gravitational attrac- 
 tion in empty space at a distance of separation r, we find that the 
 time of revolution is always the same, no matter what the material 
 of which the bodies are composed, or their past history, chemical, 
 dynamical, or otherwise. Now let us search for the functional rela- 
 tion, writing, 
 
 t=f (m^m^r). 
 
 We demand that this shall hold irrespective of the size of the funda- 
 mental units. A moment's examination confuses us, because the 
 left-hand side involves only the element of time, and the elements 
 of the right-hand side do not involve the time at all. Our critic at 
 our elbow now suggests, ' ' But you have not included all the elements 
 on which the result depends; it is obvious that you have left out 
 the gravitational constant." "But," say we, "how can this be? 
 The gravitational constant can look out for itself. Nature attends 
 to that for us. It is undeniable that two bodies of the masses m x and 
 m 2 when placed at a distance r apart always revolve in the same 
 time. We have included all the physical quantities which can be 
 varied." But our critic insists, and to oblige him we try the effect 
 of including the gravitational constant among the variables. We call 
 the gravitational constant G; it obviously has the dimensions 
 M" 1 L 3 T~ 2 , since it is defined by the equation of the force between 
 
 m 1 m 2 
 two gravitating bodies, force = G . A "constant" which has 
 
 dimensions and therefore changes in numerical magnitude when 
 the size of the fundamental units changes is called a ' ' dimensional ' ' 
 constant. We now have to find a function such that the following 
 relation is satisfied : 
 
 t = f (m lt m 2 , G,r). 
 
 Now this functional equation is not quite so easy to solve by inspec- 
 tion as the two previous ones, and we shall have to use a little 
 
INTRODUCTORY 7 
 
 algebra on it. Let us suppose that the function is expressed in the 
 form of a sum of products of the arguments. Then we know that if 
 the two sides of the equation are to remain equal no matter how the 
 fundamental units are changed in magnitude, the dimensions of 
 every one of the product terms on the right-hand side must be the 
 same as those of the left-hand side, that is, the dimensions must be 
 T. Assume that a typical product term is of the form 
 
 m mf GV r*. 
 This must have the dimensions of T. That is, 
 
 MM0 (M- 1 L 3 T- 3 )vL 5 = T. 
 "Writing down the conditions on the exponents gives 
 
 a + /3-y = 
 
 3y + 8 = 
 
 -2y = l 
 
 Hence 
 
 The values of a and ft are not uniquely determined, but only a 
 relation between them is fixed. This is as we should expect, because 
 we had only three equations of condition, and four unknown quan- 
 tities to satisfy them with. The relation between a and ft shows that 
 
 m and m 2 must enter in the form mT*[ ??i J , where there is no 
 
 \ m i/ 
 
 restriction on the value of x. Hence our unknown function is of the 
 form 
 
 f = 2 A, 
 
 G^ m M~ 
 
 where x and A x may have any arbitrary values. We may rewrite f, 
 by factoring, in the form 
 
 ~\ m 
 But now 2 AX ( - ) , if there is no restriction on x or A x , is 
 
 ( - 1 , if 
 
 \w 
 
8 DIMENSIONAL ANALYSIS 
 
 merely an arbitrary function of m /m , which we write as < [ ? J. 
 
 \ m i/ 
 Hence our final result is 
 
 L * ^ , C 
 
 That is, the square of the periodic time is proportional to the 
 cube of the distance of separation, and inversely as the gravita- 
 tional constant, other things being equal. 
 
 "We can, by other argument, find what the nature of the unknown 
 function </> is in one special case. Suppose a very heavy central body, 
 and a satellite so light that the two together revolve approximately 
 about the center of the heavy body. It is obvious that under these 
 conditions the time of revolution is independent of the mass of the 
 satellite, for if its mass is doubled, the attractive force is also 
 doubled, and double the force acting on double the mass leaves the 
 acceleration, and so the time of revolution, unaltered. Therefore 
 under these special conditions the unknown function reduces to a 
 constant, if we denote by n^ the mass of the satellite, and the rela- 
 tion becomes 
 
 t = Const r . 
 G*m< 
 
 This relation we know is verified by the facts of astronomy. 
 
 Our critic seems, therefore, to have been justified by the results, 
 and we should have included the gravitational constant in the 
 original list. We are nevertheless left with an uncomfortable feeling 
 because we do not see quite what was the matter with our argument, 
 and we are disturbed by the foreboding that at some time in the 
 future there may perhaps be a dimensional constant which we are 
 not clever enough to think of, and which may not proclaim the 
 impossibility of neglecting it in quite such uncompromising tones 
 as the gravitational constant in the example. "We are afraid that in 
 such a case we will get the incorrect answer, and not know it until 
 a Quebec bridge falls down. 
 
 Beside the matter of dimensional constants the last problem brings 
 up another question. "Why is it that we had to assume that the un- 
 known function could be represented as a sum of products of powers 
 of the independent variables? Certainly there 'are functions in 
 mathematics which cannot be represented in this way. Is nature to 
 
INTRODUCTORY 9 
 
 be arbitrarily restricted to a small part of the functions which one 
 of her own creatures is able to conceive ? 
 
 Consider now a fourth problem, treated by Lord Rayleigh in 
 Nature, vol. 15, 66, 1915. This is a rather famous problem in heat 
 transfer, treated before Rayleigh by Boussinesq. A solid body, of 
 definite geometrical shape, but variable absolute dimensions, is 
 fixed in a stream of liquid, and maintained at a definite tempera- 
 ture higher than the temperature of the liquid at points remote from 
 the body. It is required to find the rate at which heat is transferred 
 from the body to the liquid. As before, we make a list of the various 
 quantities involved, and their dimensions. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Rate of heat transfer, h HT- 1 
 
 Linear dimension of body, a L 
 
 Velocity of stream, v LT" 1 
 
 Temperature difference, 
 Heat capacity of liquid per 
 
 unit volume, c HL~ 3 Or 1 
 
 Thermal conductivity of liquid, k HL- 1 T- 1 0- 1 
 
 This is the first heat problem which we have met, and we have 
 introduced two new fundamental units, a quantity of heat (H), 
 and a unit of temperature (0). It is to be noticed that the unit of 
 mass does not enter into the dimensional formulas of any of the 
 quantities in this problem. If we desired, we might have introduced 
 it, dispensing with H in so doing. Now, just as in the last example, 
 we suppose that the rate of heat transfer, which is the quantity in 
 which we are interested, is expressed as a sum of products of powers 
 of the arguments, and we write one of the typical terms 
 
 Const a a 00 w c s k*. 
 
 As before, we write down the conditions on the exponents imposed 
 by the requirement that the dimensions of this product are the same 
 as those of h. We shall thus obtain four equations, because there 
 are four fundamental kinds of unit. The equations are 
 
 8 + c = 1 condition on exponent of H 
 p S e = condition on exponent of 
 a _|_y_3S_ o condition on exponent of L 
 y c = 1 condition on exponent of T 
 
 We have five unknown quantities and only four equations, so one 
 
10 DIMENSIONAL ANALYSIS 
 
 of the unknowns must remain arbitrary. Choose this one to be y, and 
 solve the equations in terms of y. This gives 
 
 =1-7 
 
 Hence the product term above becomes 
 Const a 0k 
 
 \ k / 
 
 The complete solution is the sum of terms of this type. As before, 
 there is no restriction on the constant or on y, so that all these terms 
 together coalesce into a single arbitrary function, giving the result 
 
 Hence the rate of heat transfer is proportional to the temperature 
 difference, but depends on the other quantities in a way not com- 
 pletely specifiable. Although the form of the function F is not 
 known, nevertheless the form of the argument of the function con- 
 tains very valuable information. For instance, we are informed 
 that the effect of changing the velocity of the fluid is exactly the 
 same as that of changing its heat capacity. If we double the velocity, 
 keeping the other variables fixed, we affect the rate of heat transfer 
 precisely as we would if we doubled the heat capacity of the liquid, 
 keeping the other variables constant. 
 
 This problem, again, is capable of raising many questions. One 
 of these questions has been raised by D. Riabouchinsky in Nature, 
 95, 591, 1915. We quote as follows: 
 
 In Nature of March 18 Lord Rayleigh gives this formula, 
 
 (Q rt T^X 
 ), considering heat, temperature, length, and time 
 
 as four " independent ' ' quantities. 
 
 If we suppose only three of these quantities are "really inde- 
 pendent, ' ' we obtain a different result. For example, if the tempera- 
 ture is defined as the mean kinetic energy of the molecules, the 
 principle of similitude* allows us only to affirm that 
 
 h = ka0F( -, c 
 
 yji *a 
 
 * Bayleigh and other English authors use this name for dimensional analysis. 
 
INTRODUCTORY \ 11 
 
 That is, instead of obtaining a result with an unknown function 
 of only one argument, we should have obtained a function of two 
 arguments. Now a function of two arguments is of course very much 
 less restricted in its character than a function of only one argument. 
 For instance, if the function is of the form suggested in two argu- 
 ments, it would not follow at all that the effect of changing the 
 velocity is the same as that of changing the heat capacity. Ria- 
 bouchinsky, therefore, makes a real point. 
 
 Lord Rayleigh replies to Riabouchinsky as follows on page 644 
 of the same volume of Nature. 
 
 The question raised by Dr. Riabouchinsky belongs rather to the 
 logic than the use of the principle of similitude, with which I was 
 mainly concerned. It would be well worthy of further discussion. 
 The conclusion that I gave follows on the basis of the usual Fourier 
 equations for the conduction of heat, in which temperature and heat 
 are regarded as sui generis. It would indeed be a paradox if the 
 further knowledge of the nature of heat afforded us by molecular 
 theory put us in a worse position than before in dealing with a 
 particular problem. The solution would seem to be that the Fourier 
 equations embody something as to the nature of heat and tempera- 
 ture which is ignored in the alternative argument of Dr. Ria- 
 bouchinsky. 
 
 This reply of Lord Rayleigh is, I think, likely to leave us cold. 
 Of course we do not question the ability of Lord Rayleigh to obtain 
 the correct result by the use of dimensional analysis, but must we 
 have the experience and physical intuition of Lord Rayleigh to 
 obtain the correct result also ? Might not perhaps a little examina- 
 tion of the logic of the method of dimensional analysis enable us to 
 tell whether temperature and heat are * ' really ' ' independent units 
 or not, and what the proper way of choosing our fundamental 
 units is ? 
 
 Beside the prime question of the proper number of units to choose 
 in writing our dimensional formulas, this problem of heat transfer 
 raises many others also of a more physical nature. For instance, 
 why are we justified in neglecting the density, or the viscosity, or 
 the compressibility, or the thermal expansion of the liquid, or the 
 absolute temperature? We will probably find ourselves able to 
 justify the neglect of all these quantities, but the justification will 
 involve real argument and a considerable physical experience with 
 physical systems of the kind which we have been considering. The 
 
12 DIMENSIONAL ANALYSIS 
 
 problem cannot be solved by the philosopher in his armchair, but 
 the knowledge involved was gathered only by someone at some time 
 soiling his hands with direct contact. 
 
 Finally, we consider a fifth problem, of somewhat different char- 
 acter. Let us find how the electromagnetic mass of a charge of elec- 
 tricity uniformly distributed throughout a sphere depends on the 
 radius of the sphere and the amount of the charge. The charge is 
 considered to be in empty space, so that the amount of the charge 
 and the radius of the sphere are the only variables. We apply the 
 method already used. The dimensions of the charge (expressed in 
 electrostatic units) we get from the definition, which states that the 
 numbers measuring the magnitude of two charges shall be such that 
 the force between them is equal to their product divided by the 
 square of the distance between them. "We accordingly have the 
 following table. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Charge, e M* L* T' 1 
 
 Radius of sphere, r L 
 
 Electromagnetic mass, m M 
 
 We now write 
 
 m = f (e,r) 
 
 and try to find the form of f so that this relation is independent of 
 the size of the fundamental units. It is obvious that T cannot enter 
 the right-hand side of the equation, since it does not enter the left, 
 and since T enters th right-hand side only through e, e cannot 
 enter. But if e does not enter the right-hand side of the equation, M 
 cannot enter either, because M enters only into e. Hence we are 
 left with a contradiction in requirements which -shows that the 
 problem is impossible of solution. But here again our Mephistophe- 
 lean critic suggests that we have left out a dimensional constant. 
 We demur ; our system is in empty space, and how can empty space 
 require dimensional constants? But our critic insists that empty 
 space does have properties, and when we push him, suggests that 
 light is propagated with a definite and characteristic velocity. So 
 we try again, including the velocity of light, c, of dimensions LT~ X , 
 and now we have 
 
 m = f (e,r,c). 
 
INTRODUCTORY 13 
 
 We now no longer encounter the previous difficulty, but imme- 
 diately, with the help of our experience with more complicated 
 examples, find the solution to be 
 
 m = Const . 
 re 8 
 
 This formula may be verified from any book on electrodynamics, 
 and our critic is again justified. We worry over the matter of the 
 dimensional constant, and ultimately take some comfort on recol- 
 lecting that c is also the ratio of the electrostatic to the electromag- 
 netic units, but still it is not very clear to us why this ratio should 
 enter. 
 
 On reflecting on the solutions of the problems above, we are 
 troubled by yet another question. Why is it that an equation which 
 correctly describes a relation between various measurable physical 
 quantities must in its form be independent of the size of the funda- 
 mental units? There does not seem to be any necessity for this in 
 the nature of the measuring process itself. An equation is a descrip- 
 tion of a phenomenon, or class of phenomena. It is a statement in 
 compact form that if we operate with a physical phenomenon in 
 certain prescribed ways so as to obtain a set of numbers describing 
 the results of the operations, these numbers will satisfy a certain 
 equation when substituted into it. For instance, let us suppose our- 
 selves in the position of Galileo, trying to determine the law of fall- 
 ing bodies. The material of our observation is all the freely falling 
 bodies available at the surface of the earth. We use as our instru- 
 ments of measurement a certain unit of length, let us say the yard, 
 and a certain unit of time, let us say the minute. With these instru- 
 ments we operate on all falling bodies according to definite rules. 
 That is, we obtain all the pairs of numbers we can by associating 
 for any and all of the bodies the distance which it has fallen from 
 rest with the interval of time which has elapsed since it started to 
 fall. And we make a great discovery in the observation that the 
 number expressing the distance of fall of any body, no matter what 
 its size or physical properties or the distance it has fallen, is always 
 a fixed constant factor times the square of the number expressing 
 the corresponding elapsed time. The numbers which we have ob- 
 tained by measurement to fit into this relationship were obtained 
 with certain definite sized units, and our description is a valid 
 
14 DIMENSIONAL ANALYSIS 
 
 description, and our discovery is an important discovery even under 
 the restriction that distance and time are to be measured with the 
 same particular units as those which we originally employed. 
 We can write our discovery in the form of an equation 
 
 s = Const t 2 . 
 
 Now an inhabitant of some other country, who uses some other 
 system of units equally as unscientific as the yard and the minute, 
 hears of our discovery, and tries our experiments with his measur- 
 ing instruments. He verifies our result, except that he must use a 
 different factor of proportionality in the equation. That is, the 
 constant depends on the size of the units used in the measurements, 
 or in other words, is a dimensional constant. 
 
 The verification of our discovery by an inhabitant of another 
 country is reported to us, and we retire to contemplate. We at 
 length offer the comment that this is as it should be, and that it 
 could not well be otherwise. We offer to predict in advance just 
 how the constant should be changed to fit with any system of meas- 
 urement, and on being asked for details, make the sophisticated 
 suggestion that we so change the constant as to exactly neutralize 
 any change in the numbers representing the length or the time, so 
 that we will still have essentially the same equation as before. In 
 particular, if the unit of length is made half as large as originally, 
 so that the number measuring a certain distance of fall is now 
 twice as large as it was formerly, we multiply the constant by 2 so 
 as to compensate for the factor 2 by which otherwise the left-hand 
 side of the equation would be too large. Similarly if the unit of time 
 is made three times as long as formerly, so that the number express- 
 ing the duration of a certain free fall becomes only % of its 
 original value, then we will multiply the constant by 9 to com- 
 pensate for the factor %, by which otherwise the right-hand side of 
 the equation would be too small. In other words, we give to the con- 
 stant the dimensions of plus one in length, and minus two in time, 
 and so obtain a formula valid no matter what the size of the 
 fundamental units. 
 
 This experience emboldens us, and we try its success with other 
 much more complicated systems. For instance, we make observations 
 of the height of the tides at our nearest port, using a foot rule to 
 measure the height of the water, 'and a clock graduated in hours as 
 
INTRODUCTORY 15 
 
 the time-measuring instrument. As the result of many observations, 
 we find that the height of water may be represented by the formula 
 
 h = 5 sin 0.5066 t. 
 
 We now write this in a form to which any other observer using 
 any other system of units may also fit his measurements by intro- 
 ducing two dimensional constants into the formula, which takes 
 the form 
 
 h G! = 5 sin 0.5066 C 2 1, 
 
 where C x has the dimensions of If" 1 , and C 2 has the dimensions 
 of T- 1 . 
 
 This result immediately suggests a generalization. Any equation 
 whatever, no matter what its form, which correctly reproduces the 
 results of measurements made with any particular system of units 
 on any physical system, may be thrown into such a form that it will 
 be valid for measurements made with units of different sizes, by the 
 simple device of introducing as a factor with each observed quantity 
 a dimensional constant of dimensions the reciprocal of those of the 
 factor beside which it stands, and of such a numerical value that in 
 the original system of units it has the value unity. 
 
 Of course it may often happen that the form of the equation is 
 such that two or more of these dimensional constants coalesce into a 
 single factor. The first example above of the falling body is one of 
 this kind. The general rule just given would have led to the intro- 
 duction of two dimensional constants, one with s on the left-hand 
 side of the equation, and the other with t 2 on the right-hand side 
 of the equation, but by multiplying up, these two may be combined 
 into a single constant. 
 
 Our query is therefore answered, and we see that every equation 
 can be put in such a form that it holds no matter what the size of 
 the fundamental units, but we are left in a greater quandary than 
 ever with regard to dimensional constants. May there not be new 
 dimensional constants appropriate to every new kind of problem, 
 and how can we tell beforehand what the dimensional constants will 
 be ? If we cannot tell beforehand what dimensional constants enter 
 a problem, how can we hope to apply dimensional analysis? The 
 dimensional situation thus appears even more hopeless than at first, 
 for we could see a sort of reason why the gravitational constant 
 should enter the problem of two revolving bodies, and could even 
 
16 DIMENSIONAL ANALYSIS 
 
 catch a glimmer of reasonableness in the entrance of the velocity 
 of light into the problem of electromagnetic mass, but it is cer- 
 tainly difficult to discover reasonableness or predictableness if 
 dimensional constants can be used indiscriminately as factors by the 
 side of every measured quantity. 
 
 In our consideration of the problems above we have also made one 
 more observation that calls for comment. We have noticed that every 
 dimensional formula of every measurable quantity has always 
 involved the fundamental units as products of powers. Is this neces- 
 sary, or may there be other kinds of dimensional formulas for quan- 
 tities measured in other ways, and if so, how will our methods apply 
 to such quantities? 
 
 To sum up, we have met in this introductory chapter a number 
 of important questions which we must answer before we can hope 
 to use the methods of dimensional analysis with any certainty that 
 our results are correct. These questions are as follows. 
 
 First and foremost, when do dimensional constants enter, and 
 what is their form? 
 
 Is it necessary that the dimensional formula of every measured 
 quantity be the product of powers of the fundamental kinds of unit ? 
 
 What is the meaning of quantities with no dimensions? 
 
 Must the functions descriptive of phenomena be restricted to the 
 sum of products of powers of the variables? 
 
 What kinds of quantity should we choose as the fundamentals in 
 terms of which to measure the others? In particular, how many 
 kiixds of fundamental units are there ? Is it legitimate to reduce the 
 number of fundamental units as far as possible by the introduction 
 of definitions in accord with experimental facts? 
 
 Finally, what is the criterion for neglecting a certain kind of 
 quantity in any problem, as for example the viscosity in the heat 
 flow problem, and what is the character of the result which we will 
 get, approximate or exact? And if approximate, how good is the 
 approximation ? 
 
CHAPTER II 
 DIMENSIONAL FORMULAS 
 
 IN the introductory chapter we considered some special problems 
 which raised a number of questions that must be answered before 
 we can hope to really master the method of dimensional analysis. 
 Let us now begin the formal development of the subject, keeping 
 these questions in mind to be answered as we proceed. 
 
 The purpose of dimensional analysis is to give certain informa- 
 tion about the relations which hold between the measurable quanti- 
 ties associated with various phenomena. The advantage of the 
 method is that it is rapid; it enables us to dispense with making 
 a complete analysis of the situation such as would be involved in 
 writing down the equations of motion of a mechanical system, for 
 example, but on the other hand it does not give as complete infor- 
 mation as might be obtained by carrying through a detailed analysis. 
 
 Let us in the first place consider the nature of the relations be- 
 tween the measurable quantities in which we are interested. In deal- 
 ing with any phenomenon or group of phenomena our method is 
 somewhat as follows. We first measure certain quantities which we 
 have some reason to expect are of importance in describing ^e \/ 
 phenomenon. These quantities which we measure are of different 
 kinds, and for each different kind of quantity we have a different 
 rule of operation by which we measure it, that is, associate the 
 quantity with a number. Having obtained a sufficient array of 
 numbers by which the different quantities are measured, we search 
 for relations between these numbers, and if we are skillful and 
 fortunate, we find relations which can be expressed in mathematical 
 form. We are usually interested preeminently in one of the 
 measured quantities and try to find it in terms of the others. Under 
 such conditions we would search for a relation of the form 
 
 *i = f (x 2 ,x 3 ,x 4 , etc.) 
 
 where x 15 x 2 , etc., stand for the numbers which are the measures 
 of particular kinds of physical quantity. Thus x x might stand for 
 
18 DIMENSIONAL ANALYSIS 
 
 the number which is the measure of a velocity, x 2 may stand for the 
 number which is the measure of a viscosity, etc. By a sort of short- 
 hand method of statement we may abbreviate thi& long-winded 
 description into saying that x is a velocity, but of course it really 
 is not, but is only a number which measures velocity. 
 
 Now the first observation which we make with regard to a func- 
 tional relation like the above is that the arguments fall into two 
 groups, depending on the way in which the numbers are obtained 
 physically. The first group of quantities we call primary quantities. 
 These are the quantities which, according to the particular set of 
 rules of operation by which we assign numbers characteristic of 
 the phenomenon, are regarded as fundamental and of an irreducible 
 simplicity. Thus in the ordinary systems of mechanics, the funda- 
 mental quantities are taken as mass, length, and time. In any 
 functional relation such as the above, certain arguments of the 
 function may be the numbers which are the measure of certain 
 lengths, masses, or times. Such quantities we will agree to call 
 primary quantities. 
 
 In the measurement of primary quantities, certain rules of opera- 
 tion must be set up, establishing the physical procedure by which it 
 is possible to measure a length in terms of a particular length which 
 we choose as the unit of length, or a time in terms of a particular 
 interval of time selected as the standard, or in general, it is charac- 
 teristic of primary quantities that there are certain rules of proce- 
 dure by which it is possible to measure any primary quantity 
 directly in terms of units of its own kind. Now it will be found that 
 we always make a tacit requirement in selecting the rules of opera- 
 tion by which primary quantities are measured in terms of quanti- 
 ties of their own kind. This requirement for measurement of length, 
 for example, is that if a new unit of length is chosen let us say half 
 the length of the original unit, then the rules of operation must be 
 such that the number which represents the measure of any particu- 
 lar concrete length in terms of the new unit shall be twice as large 
 as the number which was its measure in terms of the original unit. 
 Very little attention seems to have been given to the methodology of 
 systems of measurement, and I do not know whether this charac- 
 teristic of all our systems of measurement has been formulated or 
 not, but it is evident on examination of any system of measurement 
 in actual use that it has these properties. The possession of this 
 property involves a most important consequence, which is that the 
 
DIMENSIONAL FORMULAS 19 
 
 ratio of the numbers expressing the measures of any two concrete 
 lengths, for example, is independent of the size of the unit with 
 which they are measured. This consequence is of course at once 
 obvious, for if we change the size of the fundamental unit by any 
 factor, by hypothesis we change the measure of every length by the 
 reciprocal of that factor, and so leave unaltered the ratio of the 
 measures of any two lengths. This means that the ratio of the 
 lengths of any two particular objects has an absolute significance 
 independent of the size of the units. This may be put into the con- 
 verse form, as is evident on a minute 's reflection. If we require that 
 our system of measurement of primary kinds of quantity in terms 
 of units of their own kind be such that the ratio of the measures of 
 any two concrete examples shall be independent of the size of the 
 unit, then the measures of the concrete examples must change 
 inversely as the size of the unit. , 
 
 Besides primary quantities, there is another group of quantities 
 which we may call secondary quantities. The numerical measures 
 of these are not obtained by some operation which compares them 
 directly with another quantity of the same kind which is accepted 
 as the unit, but the method of measurement is more complicated and 
 roundabout. Quantities of the second kind are measured by making 
 measurements of certain quantities of the first kind associated with 
 the quantity under consideration, and then combining the measure- 
 ments of the associated primary quantities according to certain rules 
 which give a number that is defined as the measure of the secondary 
 quantity in question. For example, a velocity as ordinarily defined 
 is a secondary quantity. We obtain its measure by measuring a 
 length and the time occupied in traversing this length (both of 
 these being primary quantities), and dividing the number measur- 
 ing the length by the number measuring the time (or dividing the 
 length by the time according to our shorthand method of state- 
 ment). 
 
 Now there is a certain definite restriction on the rules of opera- 
 tion which we are at liberty to set up in defining secondary quan- 
 tities. We make the same requirement that we did for primary quan- 
 tities, namely, that the ratio of the numbers measuring any two 
 concrete examples of a secondary quantity shall be independent of 
 the size of the fundamental units used in making the required 
 primary measurements. That is, to say that one substance is twice 
 as viscous as another, for example, or that one automobile is travel- 
 
20 DIMENSIONAL ANALYSIS 
 
 ling three times as rapidly as another, has absolute significance, 
 independent of the size of the fundamental primary units. 
 
 This requirement is not necessary in order to make measurement 
 itself possible. Any rules of operation will serve as the basis of a 
 system of measurement by which numbers may be assigned to 
 phenomena in such a way that the particular aspect of the phenome- 
 non on which we are concentrating attention is uniquely denned by 
 the number in conjunction with the rules of operation. But the 
 requirement that the ratio be constant, or we may say the require- 
 ment of the absolute significance of relative magnitude, is essential 
 to all the systems of measurement in scientific use. In particular, 
 it is an absolute requirement if the methods of dimensional analysis 
 are to be applied to the results of the measurements. Dimensional 
 analysis cannot be applied to systems which do not meet this 
 requirement, and accordingly we consider here only such systems. 
 
 It is particularly to be noticed that the line of separation between 
 primary and secondary quantities is not a hard and fast one im- 
 posed by natural conditions, but is to a large extent arbitrary, and 
 depends on the particular set of rules of operation which we find 
 convenient to adopt in defining our system of measurement. For 
 instance, in our ordinary system of mechanics, force is a secondary 
 quantity, and its measure is obtained by multiplying a number 
 which measures a mass and the number which measures an accelera- 
 tion (itself a secondary quantity). But physically, force is perfectly 
 well adapted to be used as a primary quantity, since we know what 
 we mean by saying that one force is twice another, and the physical 
 processes are known by which force may be measured in terms of 
 units of its own kind. It is the same way with velocities ; it is pos- 
 sible to set up a physical procedure by which velocities may be 
 added together directly, and which makes it possible to measure 
 velocity in terms of units of its own kind, and so to regard velocity 
 as a primary quantity. It is perhaps questionable whether all kinds 
 of physical quantity are adapted to be treated, if it should suit our 
 convenience, as primary quantities. Thus it is not at once obvious 
 whether a physical procedure could be set up by which two viscosi- 
 ties could be compared directly with each other without measuring 
 other kinds of quantity. 
 
 But this question is not essential to our progress, although of 
 great interest in itself, and need not detain us. The facts are simply 
 these. The assigning of numerical magnitudes to measurable quanti- 
 
DIMENSIONAL FORMULAS 21 
 
 ties involves some system of rules of operation such that the quanti- 
 ties fall into two groups, which we call primary and secondary. 
 
 We have stated that the requirement of the absolute significance 
 of relative magnitude imposes definite restrictions on the operations 
 by which secondary quantities may be measured in terms of primary 
 quantities. Let us formulate this restriction analytically. We call 
 the primary quantities in terms of which the secondary quantity 
 are measured a, ft, y, etc. Measurements of the primary quantities 
 are combined in a certain way to give the measure of the secondary 
 quantity. We represent this combination by the functional symbol f, 
 
 putting the secondary quantity = f (a, /?, y, ) . Now if there 
 
 are- two concrete examples of the secondary quantity, the associated 
 primary quantities have different numerical magnitudes. Let us 
 denote the set associated with the first of the concrete examples by 
 the subscript 1, and that with the second set by the subscript 2. 
 
 Then f (c^, /? 15 y 1? ) will be the measure of the first concrete 
 
 example, and f (a 2 , /? 2 , y 2 , ) will be the measure of the second. 
 
 We now change the size of the fundamental units. We make the 
 unit in terms of which a is measured 1/xth as large. Then, as we 
 have shown, the number measuring a will be x times as large, or xa. 
 In the same way make the unit measuring /? 1/yth as large, and the 
 measuring number becomes y/?. Since our rule of operation by which 
 the numerical measure of the secondary quantity is obtained from 
 the associated primary quantities is independent of the size of the 
 primary units, the number measuring the secondary quantity now 
 becomes f {xa, y/?, ) . The measures of the two concrete exam- 
 ples of the secondary quantity will now be f (xa , y^, ) and 
 
 f (xa 2 ,y/? 2 , ). 
 
 Our requirement of absolute significance of relative magnitude 
 now becomes analytically 
 
 This relation is to hold for all values of a 1? ft, , a 2 , ft, 
 
 andx, y, z, . 
 
 We desire to solve this equation for the unknown function f. 
 Rewrite in the form 
 
 f (**,, yft, ) = f (xa,, yft, - - ) X *< a "&' * 
 
22 DIMENSIONAL ANALYSIS 
 
 Differentiate partially with respect to x. Use the notation f x to 
 denote the partial derivative of the function with respect to the first 
 argument, etc. Then we obtain 
 
 % f i (**!, yft, ---- ) = o a fi (xo 2 , y/? 2 , - -) X f "" 
 Now put x, y, z, etc., all equal to 1. Then we have 
 
 > f (/&, ---- ) f (a,, ft, ---- )* 
 
 This is to h61d for all values of a^ /3 1? ---- and a 2 , 2 , - 
 Hence, holding a 2 , /? 2 , ---- fast, and allowing a x , j8 1? ---- to vary, 
 we have 
 
 a dt 
 
 - g- == Const, 
 
 or 
 
 JL df _ Const 
 
 T ^ " ~v~> 
 
 which integrates to 
 
 The factor C is a function of the other variables /?, y, ---- . 
 
 The above process may now be repeated, differentiating partially 
 successively with respect to y, z, etc., and integrating. The final 
 result will obviously be 
 
 f =C 
 
 where a, b, c, - - and C are constants. 
 
 Every secondary quantity, therefore, which satisfies the require- 
 ment of the absolute significance of relative magnitude must be 
 expressible as some constant multiplied by arbitrary powers of the 
 primary quantities. We have stated that it is only secondary quanti- 
 ties of this kind which are used in scientific measurement, and no 
 other kind will be considered here. 
 
 We have now answered one of the questions of the introductory 
 chapter as to why it was that in the dimensional formulas the 
 fundamental units always entered as products of powers. 
 
 It is obvious that the operations by which a secondary quantity 
 is measured in terms of primary quantities are defined mathemati- 
 cally by the coefficient C, and by the exponents of the powers of 
 
DIMENSIONAL FORMULAS 23 
 
 the various primary quantities. For the sake of simplicity, the 
 coefficient is almost always chosen to be unity, although there is no 
 necessity in such a choice. There are systems in use in which the 
 factor is not always chosen as unity. Thus the so-called rational and 
 the ordinary electrostatic units differ by a factor V^w- Any differ- 
 ences in the numerical coefficient are not important, and are always 
 easy to deal with, but the exponents of the powers are a matter of 
 vital importance. The exponent of the power of any particular 
 primary quantity is by definition the ' ' dimension ' ' of the secondary 
 quantity in that particular primary quantity. 
 
 The "dimensional formula" of a secondary quantity is the aggre- 
 gate of the exponents of the various primary quantities which are 
 involved in the rules of operation by which the secondary quantity 
 is measured. In order to avoid confusion, the exponents are asso- 
 ciated with the symbols of the primary quantities to which they 
 belong, that symbol being itself written as raised to the power in 
 question. 
 
 For example, a velocity is measured by definition by dividing a 
 certain length by a certain time (do not forget that this really means 
 dividing the number which is the measure of a certain length by 
 the number which measures a certain time). The exponent of length 
 is therefore plus one, and the exponent of time is minus one, and 
 the dimensional formula of velocity is LT" 1 . In the same way a force 
 is defined in the ordinary Newtonian mechanical system as mass 
 times acceleration. The dimensions of force are therefore equal to 
 mass times the dimensions of acceleration. The dimensions of accel- 
 eration are obtained from its definition as time rate of change of 
 velocity to be LT~ 2 , which gives for the dimensions of force MLT~ 2 . 
 
 It is to be noticed that the dimensions of any primary quantity 
 are by a simple extension of the definition above merely the dimen- 
 sional symbol of the corresponding primary quantity itself. 
 
 It is particularly to be emphasized that the dimensions of a pri- 
 mary quantity as defined above have no absolute significance what- 
 ever, but are defined merely with respect to that aspect of the rules 
 of operation by which we obtain the measuring numbers associated 
 with the physical phenomenon. The dimensional formula need not 
 even suggest certain essential aspects of the rules of operation. For 
 example, in the dimensional formula of force as mass times accelera- 
 tion, the fact is not suggested that force and acceleration are vectors, 
 and the components of each in the same direction must be com- 
 
24 DIMENSIONAL ANALYSIS 
 
 pared. Furthermore, in our measurements of nature, the rules of 
 operation are in our control to modify as we see fit, and we would 
 certainly be foolish if we did not modify them to our advantage 
 according to the particular kind of physical system or problem with 
 which we are dealing. We shall in the following find many problems 
 in which there is an advantage in choosing our system of measure- 
 ment, that is, our rules of operation, in a particular way for the 
 particular problem. Different systems of measurement may differ 
 as to the kinds of quantity which we find it convenient to regard 
 as fundamental and in terms of which we define the others, or they 
 may even differ in the number of quantities which we choose as 
 fundamental. All will depend on the particular problem, and it is 
 our business to choose the system in the way best adapted to the 
 problem in hand. 
 
 There is therefore no meaning in saying "the" dimensions of a 
 physical quantity, until we have also specified the system of meas- 
 urement with respect to which the dimensions are determined. This 
 is not always kept clearly in mind even by those who in other condi- 
 tions recognize the relative nature of a dimensional formula. As for 
 example, Buckingham in Phys. Rev. 4, 357, 1914, says: "... Mr. 
 Tolman's reasoning is based on the assumption that absolute tem- 
 perature has the dimensions of energy, and this assumption is not 
 permissible." Tolman, 1 in a reply, admitted the correctness of this 
 position. My position in this matter would be that Mr. Tolman has 
 a right to make the dimensions of temperature the dimensions of 
 energy if it is compatible with the physical facts (as it seems to be) 
 and if it suits his convenience. 
 
 This view of the nature of a dimensional formula is directly 
 opposed to one which is commonly held, and frequently expressed. 
 It is by many considered that a dimensional formula has some 
 esoteric significance connected with the ' ' ultimate nature " of an 
 object, and that we are in some way getting at the ultimate nature 
 of things in writing their dimensional formulas. Such a point of 
 view sees something absolute in a dimensional formula and attaches 
 a meaning to such phrases as "really" independent, as in Ria- 
 bouchinsky's comments on Lord Rayleigh's analysis of a certain 
 problem in heat transfer. For this point of view it becomes impor- 
 tant to find the "true" dimensions, and when the "true" dimen- 
 sions are found, it is expected that something new will be suggested 
 about the physical properties of the system. To this view it is 
 
DIMENSIONAL FORMULAS 25 
 
 repugnant that there should be two dimensional formulas for the 
 same physical quantity. Often a reconciliation is sought by the 
 introduction of so-called suppressed dimensions. Such speculations 
 have been particularly fashionable with regard to the nature of 
 the ether, but so far as I know, no physical discovery has ever fol- 
 lowed such speculations; we should not expect there would if the 
 view above is correct. 
 
 In the appendix of this chapter are given a number of quotations 
 characteristic of this point of view, or others allied to it. 
 
26 DIMENSIONAL ANALYSIS 
 
 APPENDIX TO CHAPTER II 
 
 QUOTATIONS ILLUSTRATING VAKIOUS COMMON POINTS OF VIEW 
 WITH REGARD TO THE NATURE OF DIMENSIONAL FORMULAS 
 
 R. C. TOLMAN, Phys. Rev. 9 : 251, 1917. 
 
 . . . our ideas of the dimensions of a quantity as a shorthand re- 
 statement of its definition and hence as an expression of its essential 
 physical nature. 
 
 A. W. RUCKER, Phil. Mag. 27 : 104, 1889. 
 
 IN the calculation of the dimensions of physical quantities we not 
 
 infrequently arrive at indeterminate equations in which two or more 
 
 unknowns are involved. In such cases an assumption has to be made, 
 
 and in general that selected is that one of the quantities is an ab- 
 
 stract number. In other words, the dimensions of that quantity are 
 
 suppressed. 
 
 The dimensions of dependent units which are afterwards deduced 
 from this assumption are evidently artificial, in the sense that they 
 do not necessarily indicate their true relations to length, mass, and 
 time. They may serve to test whether the two sides of an equation 
 are correct, but they do not indicate the mechanical nature of the 
 derived units to which they are assigned. On this account they are 
 often unintelligible. 
 
 W. W. WILLIAMS, Phil. Mag. 34 : 234, 1892. 
 
 That these systems (i.e., the electrostatic and the electromagnetic) 
 are artificial appears when we consider that each apparently ex- 
 presses the absolute dimensions of the different quantities, that is, 
 their dimensions only in terms of L, M, and T ; whereas we should 
 expect that the absolute dimensions of a physical quantity could be 
 expressed in only one way. Thus from the mechanical force between 
 two poles we get 
 
 and this, being a qualitative as well as a quantitative relation, 
 involves the dimensional equality of the two sides. ... In this way 
 we get two different absolute dimensions for the same physical 
 quantity, each of which involves a different physical interpreta- 
 tion. . . . 
 
 The dimensional formula of a physical quantity expresses the 
 numerical dependence of the unit of that quantity upon the funda- 
 mental and secondary units from which it is derived, and the indices 
 of the various units in the formula are termed the dimensions of the 
 
DIMENSIONAL FORMULAS 27 
 
 quantity with respect to those units. When used in this very re- 
 stricted sense, the formulae only indicate numerical relations be- 
 tween the various units. It is possible, however, to regard the matter 
 from a wider point of view, as has been emphasized by Professor 
 Riicker in the paper referred to. The dimensional formulae may be 
 taken as representing the physical identities of the various quanti- 
 ties, as indicating, in fact, how our conceptions of their physical 
 nature (in terms, of course, of other and more fundamental con- 
 ceptions) are formed, just as the formula of a chemical substance 
 indicates its composition and chemical identity. This is evidently 
 a more comprehensive and fundamental view of the matter, and 
 from this point of view the primitive numerical signification of a 
 dimensional formula as merely a change ratio between units becomes 
 a dependent and secondary consideration. 
 
 The question then arises, what is the test of the identity of a 
 physical quantity ? Plainly it is the manner in which the unit of that 
 quantity is built up (ultimately) from the fundamental units L, M, 
 and T, and not merely the manner in which its magnitude changes 
 with those units. 
 
 That the dimensional formulae are regarded from this higher 
 standpoint, that is, regarded as being something more than mere 
 1 ' change ratios ' ' between units, is shown by the fact that difficulties 
 are felt when the dimensions of two different quantities, e.g., couple 
 and work, happen to be the same. 
 
 S. P. THOMPSON, Elementary Lessons in Electricity and Magnetism, 
 p. 352. 
 
 It seems absurd that there should be two different units of elec- 
 tricity. 
 
 R. A. FESSENDEN, Phys. Rev. 10 : 8, 1900. 
 
 The difference between the dimensional formula and the qualita- 
 tive formula or quality of a thing is that, according to the defini- 
 tions of the writers quoted above, the dimensions "are arbitrary, " 
 are "merely a matter of definition and depend entirely upon the 
 system of units we adopt, " whilst the quality is an expression of 
 the absolute nature, and never varies, no matter what system of 
 units we adopt. For this to be true, no qualities must be suppressed. 
 
 REFERENCES 
 
 (1) R. C. Tolman, Phys. Rev. 6, 1915, p. 226, footnote. 
 
CHAPTER III 
 
 ON THE USE OF DIMENSIONAL FORMULAS IN 
 CHANGING UNITS 
 
 WE saw in the last chapter how to obtain the dimensional formula 
 of any quantity in terms of the quantities which we chose by defini- 
 tion to make fundamental. Our method of analysis showed also the 
 connection between the numerical magnitude of the derived quantity 
 and the fundamental quantities. Thus if length enters to the first 
 power in the dimensional formula, we saw that the number measur- 
 ing that quantity is doubled when the unit of length is halved, or 
 the numerical measures are inversely as the size of the unit, raised 
 to the power indicated in the dimensional formula. 
 
 Let us consider a concrete example. What is a velocity of 88 feet 
 per second when expressed in miles per hour ? The dimensional for- 
 mula of a velocity is LT" 1 . Now if our unit of length is made larger 
 in the ratio of a mile to a foot, that is, in the ratio of 5280 to 1, the 
 velocity will be multiplied by the factor 1/5280, because length 
 enters in the dimensional formula to the first power. And similarly, 
 if the unit of time is made larger in the ratio of the hour to the 
 second, that is, in the ratio of 3600 to 1, the velocity will be multi- 
 plied by the factor 3600, because time enters the dimensional for- 
 mula to the inverse first power. To change from feet per second to 
 miles per hour we therefore multiply by 3600/5280, and in this 
 particular case the result is 88 X 3600/5280, or 60 miles per hour. 
 
 Now the result of these operations may be much contracted and 
 simplified in appearance by a sort of shorthand. We write 
 
 mile 
 
 sec. 1 sec. _J_ h 
 
 3600 
 
 88 3600 mile 
 5280 hour 
 
 = 60^?. 
 hour 
 
FORMULAS IN CHANGING UNITS 29 
 
 A little reflection, considering the relation of the dimensional 
 formula to the operations by which we obtain the measuring number 
 of any physical quantity, will at once show that this procedure is 
 general, and that we may obtain any new magnitude in terms of 
 new units from the old magnitude by using the dimensional formula 
 in precisely the same way. This method of use of the dimensional 
 formula is frequently very convenient, and is the simplest and most 
 reliable way of changing units with which I am acquainted. 
 
 In treating the dimensional formula in this way we have endowed 
 it with a certain substantiality, substituting for the dimensional 
 symbol of the fundamental unit the name of the concrete unit 
 employed, and then replacing this concrete unit by another to which 
 it is physically equivalent. That is, we have treated the dimensional 
 formula as if it expressed operations actually performed *on physical 
 entities, as if we took a certain number of feet and divided them by 
 a certain number of seconds. Of course, we actually do nothing of 
 the sort. It is meaningless to talk of dividing a length by a time; 
 what we actually do is to operate with numbers which are the meas- 
 ure of these quantities. We may, however, use this shorthand method 
 of* statement, if we like, with great advantage in treating problems 
 of this sort, but we must not think that we are therefore actually 
 operating with the physical things in any other than a symbolical 
 way. 1 
 
 This property of the dimensional formula of giving the change in 
 the numerical magnitude of any concrete example when the size of 
 the fundamental units is changed makes possible a certain point 
 of view with regard to the nature of a dimensional formula. This 
 view has perhaps been expressed at greatest length by James Thom- 
 son in B. A. Rep. 1878, 451. His point of view agrees with that taken 
 above in recognizing that it is meaningless to say literally that a 
 velocity, for instance, is equal to a length divided by a time. We 
 cannot perform algebraic operations on physical lengths, just the 
 same as we can never divide anything by a physical time. James 
 Thomson would prefer, instead of saying velocit/ = length/time, 
 to say at greater length 
 
 - Change ratio of velocity = Change ratio of length 
 
 Change ratio of time 
 
 Of course Thomson would not insist on this long and clumsy expres- 
 sion in practise, but after the 'matter is once understood, would 
 allow us to write a dimensional formula in the accustomed way. 
 
30 DIMENSIONAL ANALYSIS 
 
 This point of view seems perfectly possible, and as far as any 
 results go, it cannot be distinguished from that which I have 
 adopted. However, by regarding the symbols in the dimensional for- 
 mula as reminders of the rules of operation which we used physi- 
 cally in getting the numerical measure of the quantity, it seems to 
 me that we are retaining a little closer contact with the actual 
 physics of the situation than when we regard the symbols as repre- 
 senting the factors used in changing from one set of units to another, 
 which after all is a more or less sophisticated thing to do, and which 
 is not our immediate concern when first viewing a phenomenon. 
 
 Beside the sort of change of unit considered above, in which we 
 change merely the sizes of the fundamental units, there is another 
 sort of change of unit to be considered, in which we pass from one 
 system of measurement to another in which the fundamental units 
 are not only different in size, but different in character. 2 Thus, for 
 example, in our ordinary system of units of Newtonian mechanics 
 we regard mass, length, and time as the fundamental units, but it is 
 well known that we might equally regard force, length, and time 
 as fundamental. We may therefore expect to encounter problems of 
 this sort: how shall we express a kinetic energy of 10 gm cm 2 sec~ 2 
 in a system in which the units are the dyne, the cm, and the sec ? 
 
 There are obviously two problems involved here. One is to find 
 the dimensional formula of kinetic energy in terms of force, length, 
 and time, and the other is to find the new value of the numerical 
 coefficient in that particular system in which the unit of force is 
 the dyne, the unit of length the centimeter, and the unit of time the 
 second. 
 
 The transformed dimensional formula is obtained easily if we 
 observe the steps by which we pass from one system to the other. 
 The transition is of course to be made in such a way that the two 
 systems are consistent with each other. Thus if force is equal to mass 
 times acceleration in one system, it is still to be equal to mass times 
 '. acceleration in the other. If this were not so, we would be concerned 
 merely with a formal change, and the thing which we might call 
 force in the one system would not correspond to the same physical 
 complex in the other. This relation of force and mass in the two 
 systems is maintained by an application of simple algebra. In the 
 first system we define force as mass times acceleration, and in the 
 second we define mass as force divided by acceleration. Thus in 
 each system the secondary quantity is expressed in terms of the 
 
FORMULAS IN CHANGING UNITS 31 
 
 fundamental quantities of the system, and the two systems are 
 consistent. 
 
 The correct relation between the dimensional formulas in the two 
 systems is to be maintained simply by writing down the dimensional 
 formulas in the first system, and then inverting these formulas by 
 solving for the quantities which are to be regarded as secondary in 
 the second system. In the special case considered, we would have 
 the following dimensional formulas : 
 
 In the first system Force = MLT~ 2 
 
 In the second system Mass = FL -1 T 2 . 
 
 The transformation of the numerical coefficient is to be done 
 exactly as in the example which we have already considered by 
 treating the dimensional symbols as the names of concrete things, 
 and replacing the one to be eliminated by its value in terms of the 
 one which is to replace it. Thus the complete work associated with 
 the problem above is as follows : 
 
 Igm (lcm)*_ 
 
 (i860)' 
 
 We have to find the transformation equation of 1 gm. into terms 
 of dynes, cm, and sec. Now 
 
 1 dyne = 1 ^ lcm 
 (1 sec) 9 
 
 Hence 
 
 sec) a 
 
 1 cm 
 and substituting 
 
 1Q Igm (1cm) 8 __ 1Q 1 dyne (1 sec) 8 x (1 cm) 8 
 (1 sec) 8 1 cm (1 sec) 8 
 
 = 10 dyne cm, 
 
 which of course is a result which we immediately recognize as true. 
 
 Let us consider the general case in which we are to change from 
 a system of units in which the fundamental quantities are X 1; X 2 , 
 and X 3 to a system in which the fundamentals are Y , Y 2 , and Y 3 . 
 
 We must first have the dimensional formulas of Y t , Y 2 , and Y 3 in 
 terms of X 1? X 2 , and X 3 . Let us suppose the dimensions of Y t are 
 a t , a 2 , and a 3 , those of Y 2 , b 1? b 2 , and b 3 , and those of Y 3 , c t , c 2 , and c 3 
 
32 
 
 DIMENSIONAL ANALYSIS 
 
 in X lf X 2 , and X 3 respectively. Then in any concrete case we may 
 write 
 
 where the C's are numerical coefficients. These equations are to be 
 solved for the X's in terms of the Y's. This may be done conven- 
 iently by taking the logarithm of the equations, giving, 
 
 a log X t + a 2 log X 2 + a 3 log X 3 = log C 1 Y x 
 b x log X t + b 2 log X 2 + b 3 log X 3 = log C 2 Y 2 
 Cj. log X x + c 2 log X 2 + c 3 log X 3 = log C 3 Y 3 
 
 These are algebraic equations in the logarithms, and may be 
 solved immediately. The solution for X is 
 
 X = 
 
 _L_ A c 8 c 3 ^_ A a 2 a 3 _^_ A 
 
 ' ' y (C a Y a ) a8aj * i (C 3 Y 3 ) lb2bal ' 
 
 In this solution A stands for the determinant of the exponents. 
 
 a 2 
 b 
 
 The values of X 2 and X 3 are to be obtained from the value for X x 
 by advancing the letters. 
 
 Now let us consider an example. It is required to find what a 
 momentum of 15 tons (mass) miles/hour becomes in that system 
 whose fundamental units are the "2 Horsepower," the "3 ft per 
 sec," and the "5 ergs." This ought to be sufficiently complicated. 
 Introduce the abbreviations: 
 
 Y 2 for the "3 ft per sec" 
 Y 3 for the "5 ergs." 
 
 1 Ib (force) 1 ft 
 
 Y = 2 H.P. 2 X 33000 - ~ ~ 
 
 1mm 
 
 In the first place we have to change Ibs (force) to Ibs (mass). 
 
 
FORMULAS IN CHANGING UNITS 33 
 
 Now a pound force is that force which imparts to a mass of one 
 pound an acceleration of 32.17 ft/sec 2 . 
 Hence 
 
 1 Ib (mass) 1 ft 
 
 1 Ib force = 32.17 -r- 
 
 (Isec) 2 
 
 and 
 
 Ib mass 1 ft 1ft 
 
 (1 sec) 3 1 min 
 
 Y 1= = 66000 x 32.17 
 
 (i sec)' 
 
 = 32.17 X 66000 ^TF ton (^ mi) 3 
 dnnnr hour) 3 -fa hour 
 
 = 2.962 x 10 4 i ^^ t 
 hour 8 
 
 or, writing in the standard form 
 
 3.380 X 10~ 5 Y! = ton 1 mi 2 hour- 3 . 
 
 Y 3 3 ^A-7 mi _ ^5 mi 
 sec -.rirW hour 22 hour 7 
 
 Again 
 
 or 
 
 .4889 Y 2 = ton mi 1 hour" 1 . 
 And again 
 
 v _ , _ ,.1 gin (1cm) 3 _ 1.103 x lO^tons (6.214 x 10- 6 mi) 2 
 
 x o ergs o D . 
 
 (1 sec)' ( hour)' 
 
 or 
 
 3.622 X 10 8 Y 3 = tons 1 mi 2 hour~ 2 . 
 
 We rewrite these to obtain our system of equations in the stand- 
 ard form 
 
 3.380 x 10- 5 Y, - ton 1 mi 3 hour- 3 
 
 .4889 Y 2 = ton mi 1 hour- 1 
 
 3.622 xlO 8 Y 3 = ton 1 mi 2 hour- 3 . 
 
 We now solve for the ton, mile, and hour in terms of Y 1? Y 2 , and 
 Y 3 . We first find the determinant of the exponents. 
 
 A = 
 
 12-3 
 01-1 
 122 
 
 __ - 
 
 This is pleasingly simple. 
 
34 DIMENSIONAL ANALYSIS 
 
 The general scheme of solution above now gives 
 
 1 ton = (3.380 x H)- 6 Y,) (.4889 Y a )- 3 (3.622 x 10 8 Y 3 )' 
 1 mile = ( " )-> ( " )+ ( " ) +1 
 
 lhour = ( " )'(")'( " ) + S 
 
 or simplifying, 
 
 1 ton = 15.16 x 10 8 Y a - 2 Y 3 
 1 mile = 5.223 x 10 ia Y^ 1 Y 3 Y s 
 1 hour = 1.069 X 10 13 Y,- 1 Y 3 . 
 And finally 
 
 tons mi _ 15.16 x 5.223 x 10 ao Y,- a Y, Y,- 1 Y, Y 3 
 
 hour 1.069 x 10 13 Y^' Y 3 
 
 = 1.112 X 10 10 Y,,- 1 Y 3 
 5 ergs 
 
 - L112 >< 10 ' ts/s 
 
 which is the answer sought. It is to be noticed that the result in- 
 volves only two of the new kind of unit instead of three, the "2 
 H.P. ' ' having dropped out. This of course will not in general be the 
 case. It might at first sight appear that we might take advantage of 
 this fact and eliminate some of the computation, but on examination 
 this turns out not to be the case, for each of the numerical factors 
 connecting the ton, the mile, and the hour with the new units is seen 
 to be involved in the final result. 
 
 There are two things to be noticed in connection with the above 
 transformations. In the first place it is not always possible to pass 
 from a system of one kind of units to a system of another kind, but 
 there is a certain relation which must be satisfied. This is merely the 
 condition that the equations giving one set of units in terms of the 
 other shall have a solution. This condition is the condition that the 
 transformed equations, after the logarithms have been taken, shall 
 also have a solution, and this is merely the condition that a set of 
 algebraic equations have a solution. This condition is that the deter- 
 minant of the coefficients of the algebraic equations shall not vanish. 
 Since the coefficients of the algebraic equations in the logarithms are 
 the exponents of the original dimensional formulas, the condition is 
 that the determinant of the exponents of the dimensional formulas 
 for one system of units in terms of the other system of units shall 
 not vanish. 
 
 In attempting any such transformation as this, the first thing is 
 
FORMULAS IN CHANGING UNITS 35 
 
 to find whether it is a possible transformation, by writing down the 
 determinant of the exponents. If this vanishes, the transformation 
 is not possible. This means that one of the new kinds of unit in 
 terms of which it is desired to build up the new system of measure- 
 ment is not independent of the others. Thus in the example, if 
 instead of the ' ' 5 erg ' ' as the third unit of the new system we had 
 chosen the ' ' 5 dyne, ' ' we would have found that the determinant of 
 the exponents vanishes, and the transformation would not have been 
 possible. This is at once obvious from other considerations. For a 
 horse power is a rate of doing work, and is of the dimensions of the 
 product of a force and a velocity, and the second unit was of the 
 dimensions of a velocity, so that the proposed third unit, which was 
 of the dimensions of a force, could be obtained by dividing the first 
 unit by the second, and would therefore not be independent of them. 
 The second observation is that the new system of units to which 
 we want to transform our measurement must be one in which there 
 are the same number of kinds of fundamental unit as in the first 
 system. If this is not true, we shall find that, except in special cases, 
 there are either too few or too many equations to allow a solution for 
 the new units in terms of the old. In the first case the solution is 
 indeterminate, and in the second no solution exists. 
 
 REFERENCES 
 
 (1) D. L. Webster, Sci. 46, 187, 1917. 
 A. Lodge, Nat. 38, 281, 1888. 
 
 (2) A. Buchholz, Ann. Phys. 51, 678, 1916. 
 
CHAPTER IV 
 THE n THEOREM 
 
 IN the second chapter we saw that the dimensional formulas of all 
 the quantities with which we shall have to deal are expressible as 
 products of powers of the fundamental quantities. Let us see what 
 inferences this enables us to draw about the forms of the relations 
 which may hold between the various measurable quantities con- 
 nected with a natural phenomenon. 
 
 We also saw in the second chapter that at least sometimes the 
 functional relation will involve certain so-called dimensional con- 
 stants as well as measurable quantities. We met two examples of 
 dimensional constants, namely, the gravitational constant, and the 
 velocity of light in empty space, and we assigned dimensional 
 formulas to these constants. Now it is most important to notice that 
 these two dimensional constants had dimensional formulas of the 
 type proved to be necessary for the measurable quantities, namely, 
 they were expressible as products of powers of the fundamental 
 quantities. This is no accident, but it is true of all the dimensional 
 constants with which we shall have to deal. The proof can best be 
 given later when we have obtained a little clearer insight into the 
 nature of a dimensional constant. A certain apparent exception, the 
 so-called logarithmic constant, will also be dealt with later. We may 
 remark here, however, that one class of dimensional constant must 
 obviously be of this form. We saw that if we start with an empirical 
 equation which experimentally has been found to be true from 
 measurements with a particular set of units, this equation can be 
 made to hold for all sizes of the units by the device of introducing 
 as a factor with each measurable quantity a dimensional constant of 
 dimensions the reciprocal of those of the measured quantity. Since 
 the dimensions of every measured quantity are products of powers, 
 the dimensions of the reciprocal must also be products of powers, 
 and the theorem is proved for this restricted class of dimensional 
 constants. We will for the present accept as true the statement that 
 all dimensional constants have this type of dimensional formula. 
 
THE H THEOREM 37 
 
 Now let us suppose that we have a functional relation between 
 certain measured quantities and certain dimensional constants. We 
 shall suppose that the dimensional formulas of all these quantities 
 are known, including the dimensional constants. We shall further- 
 more suppose that the functional relation is of such a form that it 
 remains true formally without any change in the form of the func- 
 tion when the size of the fundamental units is changed in any way 
 whatever. An equation of such a form we shall call a "complete" 
 equation. 1 We have seen that it is by no means necessary that an 
 equation should be a complete equation in order to be a correct and 
 adequate expression of the physical facts, although the contrary 
 statement is almost always made, and is frequently made the basis 
 of the proof of the principle of dimensional homogeneity of ' ' physi- 
 cal ' ' equations. Although every adequate equation is not necessarily 
 complete, we have seen that every adequate equation can be made 
 complete in a very simple way, so that the assumption of complete- 
 ness is no essential restriction in our treatment, although it makes 
 necessary a more careful examination of the question of dimensional 
 constants. 
 
 The assumption of the completeness of the equation is absolutely 
 essential to the treatment, and in fact dimensional analysis applies 
 only to this type of equation. It is to be noticed that the changes of 
 fundamental unit contemplated in the complete equation are re- 
 stricted in a certain sense. We may change only the size of the 
 fundamental units and not their character. Thus, for example, a 
 complete equation which holds for all changes in the size of the 
 fundamental units as long as these units are units of mass, length, 
 and time, no longer is true, and in fact becomes meaningless in 
 another system of units in which mass, force, length, and time are 
 taken as fundamental. 
 
 With this preliminary, let us suppose that we have a complete 
 equation in a certain number of measurable quantities and dimen- 
 sional constants, valid for a certain system of fundamental units. 
 Since we are concerned only with the dimensional formulas of the 
 quantities involved, we need not distinguish in our treatment the 
 measurable quantities from the dimensional constants. We will 
 
 denote the variables by a, /?, y, to n quantities, and suppose 
 
 there is a functional relation 
 
38 DIMENSIONAL ANALYSIS 
 
 The expanded meaning of this expression is that if we substitute 
 into the functional symbol the numbers which are the measures of 
 the quantities a, ft etc., the functional relation will be satisfied. We 
 use a interchangeably for the quantity itself and for its numerical 
 measure, as already explained. Now the fact that the equation is a 
 complete equation means that the functional relation continues to 
 be satisfied when we substitute into it the numbers which are the 
 numerical measure of the quantities a, ft ---- in a system of meas- 
 urement whose fundamental units differ in size from those of the 
 fundamental system in any way whatever. Now we have already 
 employed a method, making use of the dimensional formulas, for 
 finding how the number measuring a particular quantity changes 
 when the size of the fundamental units changes. This was the sub- 
 ject of the second chapter. Let us call the fundamental units m^ m 2 , 
 m 3 , etc., to m quantities, and denote by a x , a 2 , a 3 , ---- etc., the 
 dimensions of a, by ft, ft>, ft$, ---- etc., the corresponding dimen- 
 sions of ft etc., in m 1? m 2 , m 3 , etc., respectively. 
 
 We now decrease the size of the fundamental units m^ m 2 , etc., 
 by the factors x ly x 2 , etc. Then the numerical measures of a, ft etc., 
 in terms of the new units, which we will call a 1 , /P, etc., are, as 
 proved in Chapter III, given by 
 
 j8-=x{.xf. ---- ft 
 
 Now since the equation <f> (a ---- ) =0 is a complete equation, 
 it must still hold when a 1 , ft 1 , ---- etc., are substituted for a, ft, 
 ---- . That is 
 
 or 
 
 *(X?1X. ---- a,X?ixf.- -ft ---- ) = 0. 
 
 This equation must hold for all values of x , x 2 , etc. 
 
 Now differentiate successively partially with respect to x , x 2 , 
 etc., and after the differentiation put all the x's equal to 1. Then we 
 obtain the following set of equations : 
 
THE H THEOREM 39 
 
 Consider now the first of these equations. Introduce new inde- 
 pendent variables 
 
 j_ j_ 
 
 a" = a% ft" ft? i, - - - etc. 
 
 The equation now becomes 
 
 ^*+ + 0r/ *+:.. a 
 
 do" 3/2" 
 
 The solution of this equation is well known, it being a special case 
 covered by Euler's theorem. The solution is the most general homo- 
 geneous function of the zeroth order in the variables a", /?", 
 
 etc. Now by a function homogeneous of the zeroth order we mean 
 such a function that its numerical value is unchanged when all the 
 arguments are multiplied by the same constant factor, which may 
 be entirely arbitrary. It follows therefore that the arguments of the 
 
 homogeneous function must be the original arguments a", ft", 
 
 etc., grouped in such a way that they are unchanged in value when 
 all of the original arguments are multiplied by any factor, the same 
 for all. Now the analysis of Chapter II applies here immediately. 
 The analysis of that chapter applied to the case of each of the fun- 
 damental units being multiplied by a different arbitrary factor. 
 The case here is a special form of the previous case, since we here 
 require that the result shall be unchanged when each of the argu- 
 ments is multiplied by the same arbitrary factor. It follows that the 
 
 most general way in which the arguments a", ft", etc., can 
 
 be grouped is as products of powers. If we substitute the special 
 requirements into such a product of powers, we see at once that the 
 
 sum of the exponents of a", ft", etc., in any such product of 
 
 powers must be zero. Let us write such a product in the form 
 
 (a")". . (j8")". 
 
 Then we have the condition 
 
40 DIMENSIONAL ANALYSIS 
 
 Substituting now the values of a, /?, ---- etc., in terms of a 
 ---- etc., the typical product takes the form 
 
 It is at once obvious that there are n-1 independent products of 
 this form, since there are n quantities a , b , ---- etc., and these 
 are subject to only one restriction. The arbitrary function has, there- 
 fore, n-1 independent arguments. 
 
 ai 
 
 Consider now the dimensions of the product a ^ ---- in the 
 fundamental unit m^ Since the dimensions of a in m are a l5 the 
 
 ai bi f 
 
 dimensions of a! in n^ are a x . Similarly the dimensions of /? ^ inn^ 
 areb-t.But'the exponents satisfy the condition a + b -] ----- = 0, 
 so that we see at once that the products which are the arguments of 
 the arbitrary function must all be dimensionless in n^. 
 
 The second of the equations A, by precisely the same argument 
 imposes the additional restriction that the arguments of the arbi- 
 trary function be products of the arguments which are dimension- 
 less in m 2 . In imposing this additional restriction, the number of 
 independent arguments of the arbitrary function is diminished 
 from n 1 to n 2. 
 
 In the same way the remaining equations of A demand that the 
 products shall be dimensionless in m 3 , m 4 , etc., until finally we find 
 that the products must be dimensionless in all the fundamental 
 units. At the same time, the number of independent products, which 
 serve as the arguments of the unknown function, are cut down in 
 number to n m. 
 
 Hence we have the final result. // the equation <f> (a, /?, y, ---- ) 
 = is to be a complete equation, the solution has the form 
 
 F(n 1? n 2 , ---- )=o 
 
 where the n's are the nm independent products of the arguments 
 a, /?, ---- et c., which are dimensionless in the fundamental units. 
 The result stated in this form is known as the H theorem, and 
 seems to have been first explicitly stated by Buckingham, 4 although 
 an equivalent result had been used by Jeans, 3 without so explicit a 
 statement. 
 
THE H THEOREM 41 
 
 The solution in the form above may be solved explicitly for any 
 one of the products, giving the equivalent form of result 
 
 a = p**-f* 4>(n 2 ,n 3 ,- -) 
 
 where the x 's are such that a ft~ x i y~*2 is dimensionless. 
 
 The result in this form embodies the mathematical statement of 
 the principle of dimensional homogeneity. For the arbitrary func- 
 tion on the right-hand side is a function of arguments each of which 
 is of zero dimensions, so that every term of the resulting function 
 must itself be dimensionless. Every term of this function is to be 
 multiplied by a term of the same dimensions as the left-hand side of 
 the equation, with the result that every term on the right-hand side 
 has the same dimensions as the left-hand side. The terms may now 
 be rearranged in any way that we please, but whatever the rear- 
 rangement, the dimensions of all terms will remain the same. This 
 is known as the principle of dimensional homogeneity. 
 
 The attempt is often made to give an off-hand proof of the princi- 
 ple of dimensional homogeneity from the point of view which 
 regards a dimensional formula as an expression of the "essential 
 physical nature" of a quantity. Thus it is said that an equation 
 which is an adequate expression of the physical facts must remain 
 true no matter how the fundamental units are changed in size, for 
 a physical relationship cannot be dependent on an arbitrary choice 
 of units, and if the equation is to remain true for all choices of units 
 the dimensions of each term must be the same, for otherwise we 
 would have quantities of different physical natures put equal to 
 each other. For instance, we could not according to this view have 
 a quantity of the dimensions of a length on the one side of an equa- 
 tion equal to a quantity of the dimensions of an area on the other 
 side, for it is absurd that an area should be equal to a length. 2 The 
 criticism of this point of view should be obvious after what has 
 been said about an equation merely being an equation between 
 numbers which are the numerical measures of certain physical 
 quantities. 
 
 It is to be most carefully noticed that the work above was subject 
 to a most important tacit restriction at the very outset. In putting 
 < (a, (3, ) = it was tacitly assumed that this is the only rela- 
 tion between a, ft etc., and that the partial derivatives may 
 
 be computed in the regular way on this assumption. If a, ft y, etc., 
 are connected by other relations than </> (a, ft ) = 0, then the 
 
42 DIMENSIONAL ANALYSIS 
 
 analysis above does not hold, and the results are no longer true. For 
 it is not true in general that an equation which is a complete equa- 
 tion, that is, an equation which remains true when the size of the 
 fundamental units is changed, is dimensionally homogeneous. Such 
 an equation is dimensionally homogeneous of necessity only when 
 there is no other numerical relation between the variables than that 
 defined by the equation itself. Consider as an example a falling 
 body. Let v be its velocity, s the distance of fall, t the time of fall, 
 and g the acceleration of gravity. Now these quantities are related, 
 and there is more than one equation of connection, because both v 
 and s are fixed when t and g are given. The relations connecting 
 these quantities are v = gt, and s = %gt 2 . In the light of the above 
 we would expect that a complete equation connecting v, s, g, and t 
 need not be dimensionally homogeneous. An example can be given 
 immediately, namely, 
 
 v + s = gt + y 2 gt 2 . 
 
 This is obviously a complete equation in that it is true and remains 
 true no matter how the fundamental units of length and time are 
 changed in size. We may, if we please, write from these elements 
 an equation which is very much more unusual and offensive in 
 appearance, such as 
 
 r . S -f gtl sinh(s-*gt 2 ) 
 
 v sin = gt cosh (v gt). 
 
 L v J 
 
 This again is a complete equation; it is not dimensionally homo- 
 geneous, and also offends our preconceived notions of what is possi- 
 ble in the way of transcendental functions. 
 
 The possibility of equations like those just considered is in itself 
 a refutation of the intuitional method of proof of the principle of 
 dimensional homogeneity sometimes given. 
 
 The equation v + s = gt + %gt 2 reminds one of the procedure 
 used in vector analysis, in which three scalar equations may be 
 replaced by a single vector equation. Obviously we may add together 
 any number of complete equations and obtain a result which remains 
 true. And provided that the dimensions of the original equations 
 were all different, the resulting compound equation (complete but 
 not dimensionally homogeneous) may be decomposed, like the vector 
 equation, into a number of simpler equations, by picking out the 
 parts with the same dimensions. I do not know whether this method 
 
THE H THEOREM 
 
 43 
 
 of throwing the results into a compact form can ever be made to 
 yield any practical advantages or not. 
 
 Let us now return to the first form in which we put the result 
 above, namely, 
 
 FCn^n,,- -)=0. 
 
 Consider the n 's and how they are formed from the variables. "Write 
 a typical n in the form 
 
 The a, b, c, etc., are to be so chosen that this is dimensionless. 
 Substituting now the dimensional symbols of a, /?, ---- etc., gives 
 as many equations of condition between a, b, c, etc., as there are 
 kinds of fundamental unit. The equations are 
 
 a 2 a + 
 
 I 
 
 = 
 
 There are m equations, each with n terms. Now the theory of the 
 solutions of such sets of equations may be found in any standard 
 work on algebra. In general, n will be greater than m. Under these 
 conditions there will in general be n m independent sets of solu- 
 tions. That is, there will in general be n m independent dimension- 
 less products, and the arbitrary function F will be a function of 
 n m variables. 
 
 In certain special cases this conclusion will have to be modified. 
 If, for instance, n = m, there will in general be no solution, but 
 there may be in the special case that the determinant of the ex- 
 ponents 
 
 vanishes. 
 
 Furthermore, there may be more than n m independent solutions 
 if it should happen that all the m-rowed determinants of the ex- 
 ponents vanish. This, of course, will not very often occur, but we 
 shall meet at least one example later. 
 
44 DIMENSIONAL ANALYSIS 
 
 In the general case, where there are n m independent solutions, 
 it is generally possible to select n m of the quantities a, b, c, etc., in 
 any convenient way, assign to them n m sets of independent values, 
 and solve for the remaining quantities, thus obtaining n m sets of 
 values which determine n m dimensionless products. Sometimes 
 this is not possible, and the particular set of the quantities a, b, c, 
 etc., to which arbitrary values can be assigned cannot be chosen with 
 complete freedom. This occurs when certain determinants chosen 
 from the array of the exponents vanish. We will not stop here to 
 develop a general theory, but let the exceptions take care of them- 
 selves, as it is always easy to do in any special problem. 
 
 It is to be noticed that the n theorem does not contain anything 
 essentially new, and does not enable us to treat any problems which 
 we could not already have handled by the methods of the introduc- 
 tion. The advantage of the theorem is one of convenience ; it places 
 the result in a form in which it can be used with little mental effort, 
 and in a form of a good deal of flexibility, so that the results of the 
 dimensional analysis may be exhibited in a variety of forms, de- 
 pending on the variables in which we are particularly interested. In 
 this way it has very -important advantages. 
 
 The result of this dimensional analysis places no restrictions 
 whatever on the form of the functions by which the results of experi- 
 ments may be expressed, but the restriction is on the form of the 
 arguments only. However complicated the function, if it is one 
 which satisfies the fundamental requirements of the theory as de- 
 veloped above, it must be possible to rearrange the terms in such a 
 way that it appears as a function of dimensionless arguments only. 
 Now in using the theorem we are nearly always interested in ex- 
 pressing one of the quantities as a function of the others. This is 
 done by solving the function for the particular dimensionless 
 product in which the variable in question is located, and then 
 multiplying that dimensionless product (and of course the other 
 side of the equation as well) by the reciprocal of the other quanti- 
 ties which are associated with it in the dimensionless product. The 
 result is that on the one side of the equation the variable stands 
 alone, while on the other side is a product of certain powers of some 
 of the other variables multiplied into an arbitrary function of the 
 other dimensionless products. This arbitrary function may be tran- 
 scendental to the worst degree ; there is absolutely no restriction on 
 it, but its arguments are dimensionless. This agrees with the result 
 
THE H THEOREM 45 
 
 of common experience in regard to the nature of the possible func- 
 tional relations. We have come to expect that any argument which 
 appears under the sign of a transcendental function must be a 
 dimensionless argument. This is usually expressed by saying that it 
 makes no sense to take the hyperbolic sine, for example, of a time, 
 but the only thing of which we can take the sinh is a number. 5 Now 
 although the observation is correct which remarks that the argu- 
 ments of the sinh functions which appear in our analysis are usually 
 dimensionless, the reason assigned for it is not correct. There is no 
 reason why we should not take the sinh of the number which meas- 
 ures a certain interval of time in hours, any more than we should 
 not take the number which counts the number of apples in a peck. 
 Both operations are equally intelligible, but the restrictions imposed 
 by the II theorem are such that we seldom see written the sinh of a 
 dimensional quantity, and even if we should, it would be possible by 
 a rearrangement of terms, as already explained, to get rid of the 
 transcendental function of the dimensional argument by coalescing 
 two or more such functions into a sinh of a single dimensionless 
 argument. Thus it is perfectly correct to write the equation of a 
 falling body in the form 
 
 sinh v = sinh gt, 
 
 but no one would do it, because this form is more complicated than 
 that obtained by taking the sinh" 1 of both sides. The equation above 
 might be rewritten 
 
 sinh v cosh gt cosh v sinh gt 0, 
 
 in which form the rearrangement to get rid of the transcendental 
 function of a dimensional argument is not so immediate, particu- 
 larly if one is rusty on his trigonometric formulas. But the last 
 form is perfectly adapted for numerical computation, in the sense 
 that it will always give the correct result, and still holds when the 
 size of the fundamental units is changed. 
 
 There is a corollary to these remarks about transcendental func- 
 tions with respect to the exponents of powers. It is obvious that in 
 general we cannot have an exponent which has dimensions. If such 
 appears, it is possible to combine it with others in such a way that 
 the dimensionality is lost. But there is absolutely no restriction 
 whatever imposed as to numerical exponents ; these may be integral, 
 or fractional, or incommensurable. It is often felt that the dimen- 
 
46 DIMENSIONAL ANALYSIS 
 
 sional formula of a quantity should not involve the fundamental 
 quantities to fractional powers. 6 This is a part of the view that 
 regards a dimensional formula as an expression of operations on 
 concrete physical things, and this point of view finds it hard to 
 assign a meaning to the two-thirds power of a time, for example. 
 But it seems to me just as hard to assign a physical meaning to a 
 minus second power of a time, and the possibility of such exponents 
 is admitted by everyone. 
 
 The II theorem as given contains all the elements of the situation. 
 But in use there is a great deal of flexibility in the choice of the 
 arguments of the function, as is suggested by the fact that it is 
 possible to choose the independent solutions of a set of algebraic 
 equations in a great number of ways. The way in which the inde- 
 pendent solutions are chosen determines the form of the dimension- 
 less products, and the best form for these will depend on the par- 
 ticular problem. We shall in chapter VI treat a number of concrete 
 examples which will illustrate how the products are to be chosen in 
 special cases. 
 
 REFEEENCES 
 
 (1) E. Buckingham, Phys. Rev. 4, 345, 1914. 
 
 (2) Routh. 
 
 (3) J. H. Jeans, Proc. Roy. Soc. 76, 545, 1905. 
 
 (4) E. Buckingham, Reference 1, also Jour. Wash. Acad. Sci. 4, 
 347, 1914. 
 
 (5) E. Buckingham, Ref. 1, page 346. 
 
 ' ' Such expressions as log Q or sin Q do not occur in physical 
 equations ; for no purely arithmetical operator, except a simple 
 numerical multiplier, can be applied to an operand which is not 
 itself a dimensionless number, because we cannot assign any 
 definite meaning to the result of such an operation. ' ' 
 
 See also in this connection page 266 of the following. 
 
 (6) S. P. Thompson, Elementary Lessons in Electricity and 
 Magnetism, p. 352. 
 
 ' ' It also seems absurd that the dimensions of a unit of elec- 
 tricity should have fractional powers, since such quantities as 
 M and L* are meaningless. ' ' 
 
 W. Williams, Phil. Mag. 34, 234, 1892. 
 
 ' ' So long, however, as L, M, and T are fundamental units, we 
 cannot expect fractional powers to occur. . . . Now all dynami- 
 cal conceptions are built up ultimately in terms of these three 
 
THE n THEOREM 47 
 
 ideas, mass, length, and time, and since the process is syntheti- 
 cal, building up the complex from the simple, it becomes ex- 
 pressed in conformity with the principles of Algebra by integral 
 powers of L, M, and T. . . . Obviously if mass, length, and time 
 are to be ultimate physical conceptions, we cannot give inter- 
 pretations to fractional powers of L, M, and T, because we can- 
 not analyze the corresponding ideas to anything simpler. We 
 should thus be unable, according to any physical theory, to 
 give interpretations to formulae involving fractional powers 
 of the fundamental units. ' ' 
 
CHAPTER V 
 
 DIMENSIONAL CONSTANTS AND THE NUMBER 
 OF FUNDAMENTAL UNITS 
 
 THE essential result which we have obtained in the II theorem is in 
 the restriction which it places on the number of arguments of the 
 arbitrary function. The fewer the arguments, the more restricted 
 the function, and the greater our information about the answer. 
 Thus if the problem is such that there are four variables, and three 
 fundamental kinds of unit, our analysis shows that there is only one 
 dimensionless product, which we can determine, and that some func- 
 tion of this product is zero. This is equivalent to saying, in this 
 special case, that the product itself is some constant, and we have 
 complete information as to the nature of the solution, except for the 
 numerical value of the constant. This was the nature of the solution 
 which we found for the pendulum problem. If it had not been for 
 the dimensional analysis, any conceivable relation between the four 
 arguments might have been possible, and we should have had abso- 
 lutely no information about the solution. Similarly, if there are two 
 more variables than fundamental kind of quantity, there will be 
 two dimensionless products. The solution is an arbitrary function 
 of these two products put equal to zero, which may be solved for one 
 of the products as a function of the other. This was the case with the 
 heat transfer problem already treated. It certainly gives more infor- 
 mation to know that the solution is of this form than merely to 
 know that there is some function of the five variables which van- 
 ishes, which was all that we could say before we applied our 
 analysis. 
 
 It is to our advantage, evidently, that the number of arguments 
 which are to be connected by the functional relation should be as 
 small as possible. Now the variables which enter the functional rela- 
 tion to which our analysis has been applied comprise all the varia- 
 bles which can change in numerical magnitude under the conditions 
 of the problem. These variables are of two kinds. First are the 
 
DIMENSIONAL CONSTANTS 49 
 
 physical variables, which are the measures of certain physical quan- 
 tities, and which may change in magnitude over the domain to 
 which our result is to apply. The numbers measuring these physical 
 quantities may also change when the size of the fundamental units 
 changes. In the second place, there may be other arguments of the 
 nature of coefficients in the equation which do not change in numeri- 
 cal magnitude when the physical system alone changes, but which 
 change in magnitude when the size of the fundamental measuring 
 units changes. It is these which we have called dimensional con- 
 stants. Now in any actual case we are interested only in the physical 
 problem, and are interested in finding a relation between the physi- 
 cally variable quantities. The dimensional constants are to be 
 regarded as an evil, to be tolerated only if they make possible more 
 information about the physical variables. 
 
 "We thus see that the n theorem applies to the aggregate of physi- 
 cal variables and dimensional constants, whereas we are interested 
 primarily in the physical variables alone. If the number of dimen- 
 sional constants is so great that the number of arguments of the 
 arbitrary function allowed by the II theorem is equal to or greater 
 than the number of physical variables alone, then we are no better 
 off after applying our n theorem than before. Now we have already 
 seen that in the worst possible case the number of dimensional con- 
 stants cannot exceed the number of physical variables, for any 
 empirical equation can be made complete by the introduction of a 
 dimensional constant with each physical variable. Furthermore, it 
 is almost always true that the number of physical variables is equal 
 to or greater than the number of fundamental units. Hence, if the 
 number of dimensional constants is equal the number of physical 
 variables, the number of dimensionless products is greater than or 
 at most equal to the number of physical variables. In the general 
 case, therefore, the II theorem gives no new information. Hence it 
 is of the utmost importance to keep down to the minimum the num- 
 ber of dimensional constants used in the equation. 
 
 When, therefore, shall we expect dimensional constants, and in 
 any particular problem how shall we find what they are, and what 
 are their dimensional formulas? The answer to this question is 
 closely related to the answer to the question of how we shall choose 
 the list of physical quantities between which we are to search for a 
 relation. We have seen that it does not do to merely ask ourselves 
 ' l Does the result depend on this or that physical quantity ? ' ' for we 
 
50 DIMENSIONAL ANALYSIS 
 
 have seen in one problem that although the result certainly does 
 "depend" on the action of the atomic forces, yet we do not have 
 to consider the atomic forces in our analysis, and they do not enter 
 the functional relation. 
 
 To answer the question of what variables to include demands a 
 background of a great deal of physical experience. If we are to treat 
 a certain problem by the methods of mechanics we must have enough 
 background to be assured that the problem is a problem in mechan- 
 ics, and involves essentially no elements that are not treatable by the 
 ordinary equations of mechanics. We must know that certain 
 aspects of the problem can be neglected, and that certain others 
 alone are essential as far as certain features of behavior go. No one 
 would say that in any problem of mechanics the atomic forces are 
 not essential, but our experience shows that they combine into 
 certain complexes, which may be sufficiently characterized by an 
 analysis which does not go down to the ultimate component parts, 
 and that the results of our analysis, which disregards many even 
 essential aspects of the situation, have validity under certain condi- 
 tions whose restrictions are not irksome. The experience involved in 
 judgments of this sort reaches so far back that we know almost by 
 instinct whether a problem is suitable for mechanical treatment or 
 not. And if the problem is capable of mechanical treatment, we 
 know, by the very definition of what we mean by a mechanical sys- 
 tem, what the equations are which the motion of the component 
 parts of the system conform to, and what the form of the equations 
 is. In the same way, we know by instinct whether a system is a 
 thermodynamic system, or an electrical system, or a chemical sys- 
 tem, and in each case, because we know what we mean when we say 
 that a phenomenon is of such or such a nature, we know what are 
 the laws which govern the variations of the system, and the ele- 
 ments which must be considered in formulating the relations be- 
 tween the parts. But a very wide background of experience, extend- 
 ing over many generations, was necessary before we could say that 
 this particular group of phenomena is mechanical or electrical, or, 
 in general, that the phenomenon is physical. 
 
 Now my point of view is essentially that precisely the same ex- 
 perience which is demanded to enable us to say whether a system 
 is mechanical or electrical is the experience which is demanded in 
 order to enable us to make a dimensional analysis. This experience 
 will in the first place inform us what physical variables to include 
 
DIMENSIONAL CONSTANTS 51 
 
 in our list, and will in the second place tell us what dimensional 
 constants are demanded in any particular problem. 
 
 Let us for the present forget what we know of dimensional analysis 
 and imagine ourselves approaching a new problem. In the first place 
 we decide in the light of the experience of all the ages what the 
 nature of the problem is. Suppose that we decide that it is mechani- 
 cal. Then we know that the motion of the system is governed by the 
 laws of mechanics, and we know what these laws are. We write down 
 certain equations of motion of the system. We are careful to include 
 all the equations of motion, so that the system of equations by which 
 we have described the relations between the parts of the system has 
 a unique solution. Then we are convinced, because of our past 
 experience, that we have essentially represented all the elements of 
 the situation, that our equations correspond to the reality at least as 
 far as certain aspects of the phenomenon go, and the solution of the 
 equations will correctly represent the behavior of the system which 
 we have thus analyzed. We are not disappointed. The fact that our 
 predictions turn out to be verified means merely that we have 
 become masters of a certain group of natural phenomena. 
 
 Now the astute observer (Fourier 1 was the first astute observer) 
 notices that the equations by which the relation of the component 
 parts of the system is analyzed are expressed in such a general form 
 that they remain true when the size of the fundamental units is 
 changed. For instance, the equation stating that the force acting on 
 a particular part of our mechanical system is equal to the mass of 
 that part times its acceleration remains true however the size of the 
 fundamental units is changed, because in every system of units 
 which we use for mechanical purposes, the unit of force is defined 
 so that force has this relation to mass and acceleration. Every one 
 of the fundamental equations of motion is in the same way a com- 
 plete equation. The final solution is obtained from the equations of 
 motion by a purely mathematical process, which has no relation to 
 the size of the fundamental units. It follows, therefore, in general, 
 that the final result will also be complete, in the sense that the equa- 
 tion expressing the final result is a complete equation. 
 
 Dimensional analysis may, therefore, be applied to the results 
 which we obtain by solving the equations of motion. (We use equa- 
 tions of motion in a general sense, applying to thermodynamic and 
 electrical as well as mechanical systems. ) Now the arguments of the 
 function which we finally obtain by solving the equations of motion 
 
52 DIMENSIONAL ANALYSIS 
 
 can obviously be only those quantities which we put into the original 
 equations of motion, for the mathematical operations can introduce 
 no new arguments. 
 
 In particular, the dimensional constants which enter the final relation 
 are those, and those only, which we had to use in writing down the equa- 
 tions of motion. This is the entire essence of the question of dimensional 
 constants. 
 
 With regard to the dimensional formulas of dimensional con- 
 stants, we may merely appeal to experience with the observation 
 that all such constants are of the form of products of powers of the 
 fundamental quantities. But it is evident on reflection, that any law 
 of nature can be expressed in a form in which the dimensional 
 formulas of the constants are of this type, by the device, already 
 adopted, of introducing dimensional constants as factors with the 
 measured quantities in such a way as to make the equation com- 
 plete. We will hereafter assume that the equations of motion (which 
 are merely expressions of the laws of nature governing phenomena) 
 are thrown into such a form that the dimensional constants are of 
 this type ; this is seen to involve no real restriction. 
 
 It appears, therefore, that dimensional analysis is essentially of 
 the nature of an analysis of an analysis. We must know enough 
 about the situation to know what the general nature of the problem 
 is, and what the elements are which would be introduced in writing 
 down the equations determining the motion (in the general sense) 
 of the system. Then, knowing the nature of the elements, we can 
 obtain certain information about the necessary properties of any 
 relations which can be deduced by mathematical manipulations 
 with the elements. In so far as our knowledge of the underlying 
 laws of nature is adequate we may have confidence in the result, 
 but the result can have no validity not pertaining to the equations 
 of motion, and is in no way different from all our other knowledge. 
 The result is approximate, as the laws of motion are approximate, a 
 restriction which is imposed by the very nature of knowledge itself. 
 
 The man applying dimensional analysis is not to ask himself ' ' On 
 what quantities does the result depend?" for this question gets 
 nowhere, and is not pertinent. Instead we are to imagine ourselves 
 as writing out the equations of motion at least in sufficient detail to 
 be able to enumerate the elements which enter them. It is not neces- 
 sary to actually write down the equations, still less to solve them. 
 
DIMENSIONAL CONSTANTS 53 
 
 Dimensional analysis then gives certain information about the 
 necessary character of the results. It is here of course that the 
 advantage of the method lies, for the results are applicable to sys- 
 tems so complicated that it would not be possible to write the equa- 
 tions of motion in detail. 
 
 It is to be especially noticed that the results of dimensional 
 analysis cannot be applied to any system whose fundamental laws 
 have not yet been formulated in a form independent of the size of 
 the fundamental units. For instance, dimensional analysis would 
 certainly not apply to most of the results of biological measure- 
 ments, although such results may perfectly well have entire physical 
 validity as descriptions of the phenomena. It would seem that at 
 present biological phenomena can be described in complete equa- 
 tions only with the aid of as many dimensional constants as there 
 are physical variables. In this case, we have seen, dimensional analy- 
 sis has no information to give. In a certain sense, the mastery of a 
 certain group of natural phenomena and their formulation into 
 laws may be said to be coextensive with the discovery of a restricted 
 group of dimensional constants adequate to coordinate all the 
 phenomena. 
 
 Let us apply this view of the nature of dimensional constants to 
 the problem which we have already considered of the electro- 
 magnetic mass of a spherical distribution of electricity. This is evi- 
 dently a problem in electrodynamics, and must be solved by the use 
 of the field equations. These field equations consist of certain mathe- 
 matical operators operating on certain combinations of the electric 
 and magnetic forces and the velocity of light. In this particular 
 problem we want to solve the equations in such a form as to get the 
 electromagnetic mass; this is the integral throughout space of a 
 constant times the energy density, which in turn is given by the 
 distribution of the forces, which are determined by the distribution 
 of the charge. Hence if there is a relation of the form which we 
 suspect, the forces will eliminate from the final result. There is, 
 however, no reason to think that the characteristic constant "c" 
 of the equations will also eliminate from the result, and we must 
 therefore seek for a relation between the total charge, the mass, the 
 radius, and the constant of the field equations, which is the velocity 
 of light. This relation we have already found, and checked against 
 the results of a detailed solution of the field equations applied to 
 this particular problem. 
 
54 DIMENSIONAL ANALYSIS 
 
 We have seen that dimensional constants are going to enter the 
 final result only in so far as they enter the equations of motion. Now 
 a dimensional constant in an equation of motion is an expression of 
 a physical relation which is so universal as to be characteristic of all 
 the phenomena embraced in the particular group which we are 
 considering. Such a universal physical relation may be treated in 
 two ways. We may leave the dimensional constant in the equations 
 as an explicit statement of the relation, as i^ done in the field equa- 
 tions of electrodynamics, or we may define our fundamental units 
 with this relation in view, thus obtaining a system of units in which 
 the dimensional constant has disappeared but in which the number 
 of units which may be regarded as fundamental has been restricted 
 in such a way that all units belonging to the system automatically 
 bear the experimental relation to each other. The system of units so 
 obtained is of value only in treating that group of phenomena to 
 which the law in question applies. Thus it is a result of experience 
 that the mass times the acceleration of a body is proportional to the 
 force acting upon it. In this statement of the experimental facts 
 there is no restriction whatever upon the units of mass, or length, 
 or time, or force. The factor of proportionality will change in 
 numerical magnitude whenever any one of the four fundamental 
 units is changed in size. But now, instead of being bothered by a 
 continually changing factor of proportionality, we may arbitrarily 
 say that this factor shall be unity in all systems which we will con- 
 sider, and we will bring this result about by defining the unit of 
 force in our new system to be such that the force acting on a body 
 is equal to the mass times the acceleration. We have in this way 
 obtained a system of units adapted to dealing with all those physical 
 systems in which the laws of motion involve a statement of the 
 physical relation between force, mass, and acceleration, but if the 
 physical system should be such that this relation is not involved 
 in the motion of the system, then we would be unduly restricting 
 ourselves by using the mechanical system of units. 
 
 These considerations^ as to the possible systems of units answer 
 the question previously raised, as in the fourth problem of the 
 introduction, for example, as to the number of kinds of .units which 
 we shall take as fundamental. The answer depends entirely upon the 
 particular problem, and will involve the physical relations which 
 are necessary to a complete expression of the motion of the parts. 
 In any ordinary problem of dynamics, for example, the relation 
 
DIMENSIONAL CONSTANTS 55 
 
 between force and mass and acceleration is essentially involved in 
 the equations of motion. This relation may be brought into the 
 equations either by the use of four fundamental kinds of unit, 
 force, mass, length, and time, with the corresponding dimensional 
 constant of proportionality, or by using the ordinary mechanical 
 units of mass, length, and time, in which force is defined so that the 
 experimental relationship is always satisfied, and the dimensional 
 constant has disappeared. In either case the results of the dimen- 
 sional analysis are the same. For the difference between the number 
 of fundamental units and the number of variables, which deter- 
 mines the number of arguments of the unknown function, is the 
 same in either case, because when the number of units is augmented 
 by one by including the force, the number of variables is also aug- 
 mented by one by including the dimensional constant, and the differ- 
 ence remains constant. If, however, the problem were such that the 
 experimental relation between force, mass, and acceleration is not 
 involved in the equations of motion of the system, then the ordinary 
 mechanical units would be inappropriate, because we would obtain 
 less information when using them. For we could in this case use 
 four fundamental units without introducing a corresponding dimen- 
 sional constant into the list of variables, so that the difference be- 
 tween the number of variables and the units would be less by one 
 when using four than when using three fundamental units, and the 
 arguments of the function would be fewer in number, which is de- 
 sirable. We shall meet an example illustrating this point later. 
 
 EEFEEENCES 
 
 (1) Fourier, Theorie de Chaleur, 160. 
 
 As dealing with the general question of the proper number of 
 fundamental units may be mentioned 
 
 E. Buckingham, Nat. 96, 208, and 396, 1915. 
 
CflAPTER VI 
 
 EXAMPLES ILLUSTRATIVE OF DIMENSIONAL 
 ANALYSIS 
 
 LET us in the first place recapitulate the results of the preceding 
 chapter. Before undertaking a dimensional analysis we are to im- 
 agine ourselves as making an analysis to the extent of deciding the 
 nature of the problem, and enumerating the physical variables which 
 would enter the equations of motion (in the general sense) and 
 also the dimensional coefficients required in writing down the equa- 
 tions of motion. The dimensions of all these variables are then to be 
 written in terms of the fundamental units. These fundamental units 
 are to be chosen for each particular problem in such a way that 
 their number is as large as possible without involving the introduc- 
 tion of compensating dimensional constants into the equations of 
 motion. The dimensionless products of the variables are then to be 
 formed in accordance with the II theorem, choosing the products in 
 such a way from the great variety possible that the variables in 
 which we are particularly interested may stand conspicuously by 
 themselves. Having formed the products, the II theorem gives im- 
 mediately the functional relation. 
 
 In the following illustrative examples we have particularly to 
 consider the proper number of fundamental units, and the most 
 convenient way of choosing the dimensionless products. The matter 
 of dimensional constants we regard as clear. 
 
 As the first example we will take the first treated by Lord Eay- 
 leigh in Nature* Consider a wave advancing on deep water under 
 the action of gravity. This is evidently, a problem in hydrodynamics, 
 which is merely mechanics applied to liquids. The equations of 
 mechanics will therefore apply. Now the liquid when displaced from 
 equilibrium is restored by the force of gravity. This will involve the 
 density of the liquid and the intensity of gravity. Evidently these 
 quantities will enter the equations of motion. No other properties 
 of the liquid, such as the compressibility, will enter, because we 
 know from a discussion of the equations of hydrodynamics that such 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 57 
 
 properties are unimportant for phenomena of this scale of magni- 
 tude. Physically, of course, the compressibility affects the result 
 to a certain extent, so that the result of our analysis will not be 
 exact, but will be a valid approximation only to the extent that the 
 equations of hydrodynamics are valid approximations. There are 
 no dimensional constants entering the equations of hydrodynamics, 
 provided that we use ordinary mechanical units, in which mass, 
 length, and time are fundamental, for the laws of motion have 
 entered this^ system of units through the definition of force. The 
 equations, of course, are equations between the displacements and 
 the other elements. Now it is conceivable that we might eliminate 
 the various displacements from the equations, and come out at the 
 end with a relation between the velocity of propagation, the density, 
 and the intensity of gravity only, analogously to the pendulum 
 problem. 
 
 Let us try this. Write down the variables and their dimensional 
 formulas, as before. 
 
 Name of Quantity. Symbol. Dimensional Formula.* 
 Velocity of wave, v LT" 1 
 
 Density of liquid, d ML~ 3 
 
 Accleration of gravity, g LT~ 2 
 
 We now apply the n theorem. We have three variables, and three 
 fundamental kinds of unit. The difference between these numbers 
 is zero, and therefore, according to the theorem, there are zero 
 dimensionless products. That is, we have made some mistake^ and 
 no relation exists, unless this should be one of those exceptional 
 cases in which a product may be formed of fewer than the normal 
 number of factors. But an examination shows that this is no excep- 
 tion, and there is in fact no dimensionless product. This shows that 
 the suggested elimination was not possible, but that some other ele- 
 ments or combination of elements must enter the final result.. Of 
 course the detailed analysis wilFgive as the final result a detailed 
 description of the motion of the water, from which we must pick out 
 the wave motion and find its velocity. That is, along with the veloc- 
 ity in the final result there will be something characteristic of the 
 particular wave. The velocity of all the waves need not be the same, 
 but may depend on the wave length, for example. Physically, of 
 course, we knew this in the beginning, and we were stupid only for 
 
58 DIMENSIONAL ANALYSIS 
 
 purposes of instruction. Our experience with problems of this 
 nature would have led us to search for a relation between the varia- 
 bles which we put into the analysis, the velocity, and the wave 
 length. Let us introduce, therefore, into our list of quantities the 
 wave length. 
 
 Wave length A L 
 
 We have now four variables, and the dimensional formulas are 
 expressed in terms of three fundamental kinds of unit, so that the 
 II theorem leads us to expect the existence of one dimensionless 
 product, and the result will be that the dimensionless product is 
 equal to a constant. 
 
 The proof of the n theorem also showed that one exponent in a 
 dimensionless product can be assigned arbitrarily. Since we are 
 particularly interested in v, let us choose its exponent as unity, and 
 write the dimensionless product in the form v d- tt g-0 A-* . Putting 
 this equal to a constant and solving for v, gives for the result 
 
 v = Const d a g^ \y. 
 
 The dimensions of the factors on the right-hand side must 
 together be the same as that of the velocity, which stands alone on 
 the left-hand side. We solve for the unknown exponents of the fac- 
 tors of the right-hand side. Substitute the dimensional formulas for 
 the variables 
 
 LT- 1 = (ML- 8 )' 
 
 Now write down in succession the condition that the exponents of 
 M, of L, and of T be the same on the two sides of the equation. This 
 gives 
 
 a = condition on M 
 
 3a + /? + y= 1 condition on L 
 
 2 (3 = 1 condition on T. 
 
 Whence 
 
 a = 
 
 7 72 J j 
 
 and the final result is of the form 
 
 v == Const 
 
 w- f 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 59 
 
 The velocity of a gravity wave on deep water (the reason the 
 depth did not enter the final result was because we postulated that 
 the water was to be deep) is therefore proportional to the square 
 root of the wave length and the intensity of gravity, or is propor- 
 tional to the velocity acquired by a body falling freely under 
 gravity through a distance equal to the wave length. 
 
 It is to be noticed that the density of the liquid has disappeared 
 from the final result. This might have been anticipated; if the 
 density is doubled the gravitational force is also doubled on every 
 element, and the accelerations and therefore all the velocities are 
 unaltered, because the doubled force is compensated by a doubled 
 mass of every element. 
 
 Since the density disappears from the final result we have here 
 a dimensionless product of v, 1, and g only. This is, therefore, a 
 dimensionless product of three variables, expressed in three funda- 
 mental units. This is in general not possible, but demands some 
 special relation between the dimensional formulas of the variables. 
 
 We can see in a moment by writing out the product in terms of 
 unknown exponents, and then writing the algebraic equations which 
 the exponents must satisfy, that the condition that a dimensionlegs 
 product exist in a number of jterms just equal to the number of 
 fundamental units ia_that the detenfl 1TlftT it of the exponents in the 
 mrnensionarlbrnmla^of the factors vanish. This is obviously not 
 restricted to the case of three fundamental units, but applies to any 
 number. Conversely the condition that a particular element shall 
 enter as a factor into a dimensionless product with a number of 
 other factors equal in number to the fundamental units, is that the 
 determinant of the exponents of the other factors shall not be zero ; 
 otherwise the other factors by themselves form a dimensionless 
 product into which the factor in which we are interested does not 
 enter. 
 
 Consider now a second problem. An elastic pendulum is made 
 by attaching to a weightless spring of elastic constant k a box of 
 volume V which is filled with a liquid of density d. The mass of the 
 liquid in the box is acted upon by gravity, and we are required to 
 find an expression for the time of oscillation. As before we make 
 a list of the quantities and their dimensions. 
 
 
60 DIMENSIONAL ANALYSIS 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 Elastic constant (force per unit 
 
 displacement), k- MT~ 2 
 
 Time of oscillation, t - T 
 
 Volume of box, v L 3 
 
 Density of liquid, d ML~ 3 
 
 Acceleration of gravity, g LT~ 2 
 
 The problem is obviously one in ordinary mechanics, so that we 
 are justified in using the mechanical system of units, and there will 
 be no dimensional constant. The variables which we have listed 
 above are, therefore, the only ones, and are those in terms of which 
 the problem is formulated. Here there are five quantities and three 
 fundamental kinds of unit. There are therefore two dimensionless 
 products. In the analysis of the last chapter we saw that in finding 
 the dimensionless products we had to solve a system of algebraic 
 equations. Certain of the solutions could be assigned at pleasure, 
 and the others determined in terms of them. In this particular 
 problem we are interested especially in t, and let us say k. Then let 
 us choose the exponents of t and k in the dimensionless products 
 as those which are to be assigned at pleasure and in terms of which 
 the others are to be computed. Now the algebraic theorem showed 
 that there were two linearly independent sets of exponents which 
 we might assign to t and k, and that it is possible to choose these 
 two sets in an infinite number of ways. We will try to select the two 
 simplest. For the present purpose we will accomplish this by assign- 
 ing the value 1 to the exponent of t and to that of k for the one 
 set, and to the exponent of t and 1 to the exponent of k for the 
 other set. This is certainly a simple couple of pairs, and has the 
 effect of making both t and k appear in only one dimensionless 
 product. "We therefore have to find the two dimensionless products 
 
 t v tti d^ 1 g^i ? k v a a d^ 2 g? 2 . 
 
 We have now two sets of algebraic equations for the two sets of 
 unknown exponents a 1? ft, y x and a 2 , ft, y 2 . These equations are 
 
 3a 2 -3ft+y 2 + 
 Oa 2 + 0ft-y 2 -2 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 61 
 
 The solutions are 
 
 Hence the dimensionless products are 
 
 and( 
 
 and the solution is 
 
 - -,. 
 
 where the function f is undetermined. 
 
 Now the result so obtained is undoubtedly correct as far as it 
 goes, but an examination will show that we can do better, and 
 obtain a form in which there is no undetermined function. This 
 improvement can be effected by increasing the number of funda- 
 mental units. We were correct in using the ordinary mechanical 
 units, for the equations of motion involve the dynamical relation 
 between force, mass, and acceleration. The change is to be made in 
 a direction at first not obvious because we are so accustomed to using 
 the units written down. It is evident on reflection, however, that in 
 the equations of motion governing the system no use is made of the 
 fact that the numerical measure of the volume of the box is equal 
 to the cube of the length of one of its linear dimensions. It is quite 
 possible to measure volumes physically in terms of a particular 
 volume chosen as unity by cutting up the larger volume into smaller 
 volumes congruent with the unit, and counting the number of 
 times that the unit is contained in the larger volume. It may then 
 be proved that the number so obtained is proportional to the cube 
 of the number measuring one of the linear dimensions. In fact, this 
 is the method of proof originally adopted by Euclid in dealing with 
 both areas and volumes. After the geometrical fact has been proved, 
 it becomes natural to define the unit volume as that volume which 
 is equal to a cube whose sides are unity, but this definition and 
 restriction are of value only in those problems in which the relation 
 between volume and length enter essentially into the result. Such 
 is not the case here, because the volume of the box is of importance 
 only as determining, in conjunction with the density of the liquid, 
 the mass filling the box. We might perfectly well measure length 
 
62 DIMENSIONAL ANALYSIS 
 
 for this problem in inches, and the volume in quarts, provided, of. 
 course, that we measure density as mass per quart. 
 
 Let us then attempt the problem again, now taking volume as 
 an independent unit of its own kind. Then we shall have : 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 Elastic constant, k / ^ MT~ 2 
 
 Time of oscillation, t T 
 
 Volume of box, v >/ ^ 
 
 Density of liquid, d ' $ MV" 1 
 
 Acceleration of gravity, g & .. LT~ 2 
 
 We have now five variables, but four fundamental kinds of quan- 
 tity, so that there is only one dimensionless product. We are par- 
 ticularly interested in t, so we choose the exponent of t equal to 
 unity, and are required to find the other exponents so that 
 
 t k a v d^ g 6 is dimensionless. 
 
 This problem is so simple that we can solve for the unknowns by 
 inspection, or if we prefer, write out the equations, which are : 
 
 8 = 
 
 -2a-28 + lrz:0 
 -7 = 
 
 The solution of this set of equations is 
 
 =%, j s = -&y = -%,8 = o. 
 
 The dimensionless product is 
 
 tk*v-*d-*, 
 
 and the solution is 
 
 t = Const /? 
 
 The information embodied in this solution is evidently much 
 greater than in the more noncommittal one obtained with three 
 units. It is seen from the new solution, for example, that the time 
 of oscillation does not depend on the intensity of gravity. Physically, 
 of course, this means that gravity is effective only in changing the 
 mean position of equilibrium; as gravity increases the weight is 
 pulled down and oscillates about a position nearer the center of 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 63 
 
 attraction, but the period of oscillation is not changed thereby. It 
 was not at all obvious or necessary from the first form of the solu- 
 tion that the time would be independent of gravity, but that the 
 previous solution is not inconsistent with this one is seen by putting 
 the f of the previous solution equal to a constant times the inverse 
 square root of the argument, when the two solutions become 
 identical. 
 
 Instead of increasing the number of fundamental units from 
 three to four, we might have obtained the same result by observing 
 that the equations of motion are concerned only with the total mass 
 on the end of the string, and hence the volume and the density can 
 affect the result only in so far as they enter through their product, 
 the mass. According to this method of treatment we would have put 
 v and d together as one quantity, so that we would have been con- 
 cerned with only four quantities and three fundamental kinds of 
 unit. The result would have been the same as by the method which 
 we adopted. In fact, it will often be found possible by using special 
 knowledge of the problem to obtain in this way more detailed infor- 
 mation than would have been possible by the general analysis. 
 
 If we use the mass as one of the variables, the result assumes the 
 
 form t = Const V f , and again we have a dimensionless product of 
 fewer than the normal number of terms. 
 
 Now let us consider a problem illustrating how it is that the 
 result is unaffected by increasing the number of units if at the same 
 time the number of dimensional constants is increased. We .take the 
 same problem as above, except that we now give only the mass on 
 the end of the spring, and do not attempt to analyze the mass into 
 volume times density. The variables will be mass (m), time of oscil- 
 lation (t), and stiffness of spring (k). "We can omit the intensity of 
 gravity, because we have already seen it to be without effect. In 
 discussing this problem we propose to use five fundamental kinds of 
 unit, which we will choose as mass, length, time, as usual, and in 
 addition force, and velocity. This problem is evidently one in 
 mechanics involving in the statement of the relations between the 
 parts the experimental fact that force is proportional to mass times 
 accleration. Hence in formulating the equations of motion we will 
 have to introduce this proportionality factor, which will appear in 
 the analysis as a new dimensional constant. This factor is to connect 
 force, mass, and acceleration. But now acceleration must be rede- 
 
64 DIMENSIONAL ANALYSIS 
 
 fined if we are using velocity as a unit of its own kind. Acceleration 
 will now be denned as time rate of change of velocity, and will have 
 the dimensions VT 1 . The equation of motion thus written will 
 express a relation between the force and the velocity and the time. 
 But the force is connected with the displacement through the 
 elastic constant, so that to solve the equations a relation is needed 
 between displacement, velocity, and time. The experimental fact, of 
 course, is that velocity is proportional to the quotient of distance 
 by time. The factor of proportionality will appear in the final result 
 as a dimensional constant. We now have our list of quantities com- 
 plete. They compose three physical variables, and two dimensional 
 constants. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Time of oscillation, t T 
 
 Mass at end of spring, m M 
 
 Stiffness of spring, k FL" 1 
 
 The force dimensional constant, f FM" 1 TV" 1 
 The velocity dimensional con- 
 
 stant, v L- 1 TV 
 
 Here F is the dimensional symbol of force measured in units of 
 force, and V the dimensional symbol of velocity. The dimensional 
 formulas were obtained by the regular methods, noting only that 
 the stiffness of the spring is defined as the force exerted by the 
 spring per unit displacement of the end. 
 
 We have now to find the dimensionless products involving these 
 five variables. We note in the first place that there are five variables, 
 and five fundamental kinds of quantity, so that in general there 
 would be no dimensionless product. But it may be seen on writing 
 it out that the determinant of the exponents in the dimensional 
 formulas vanishes, so that in this special case there is a dimension- 
 less product with fewer than the normal number of factors. Of 
 course we knew that this must be the case from our previous discus- 
 sion. Now, as before, we select t as the quantity in which we are 
 particularly interested, write the dimensionless product in the form 
 
 and write down the condition that the product is dimensionless. This 
 gives 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 65 
 
 a 7 = condition on M 
 
 ft 8 = condition on L 
 7 + 8 + 1 = condition on T 
 
 (3 + y = condition on F 
 
 y + 8 = condition on V. 
 
 The solution is 
 
 a = -y 2 , p = y 2 , y = ~y 2 , < = -y 2 . 
 
 The dimensionless product is 
 
 tm-i k*f-* v-*, 
 and the final solution 
 
 t = Const V^-' 
 
 This is exactly the same as the solution already obtained, on put- 
 ting the dimensional constants f and v equal to unity, which of 
 course was their value in the ordinary mechanical system of units. 
 
 Although this example gives no new results, it is instructive in 
 showing that any system of fundamental units whatever is allow- 
 able, provided only that the dimensional constants required by the 
 special problem are also introduced. 
 
 We now consider a problem in which it is an advantage to treat 
 force as a unit of its own kind. This is the problem of Stokes of a 
 small sphere falling under gravity in a viscous liquid. The sphere 
 is so small that the motion is everywhere slow, so that there is 
 nowhere turbulence in the fluid. The elements with which we have 
 to deal in this problem are the velocity of fall; the density of the 
 sphere, the diameter of the sphere, the density of the liquid, the 
 viscosity of the liquid, and the intensity of gravity. The problem is 
 evidently one in mechanics, so that if we use the ordinary mechani- 
 cal units there will be no dimensional constants to introduce. But 
 we notice that the problem is of a very special kind for a mechanical 
 problem. The motion is slow, and the velocity is steady, the forces 
 acting on the sphere and the liquid being everywhere held in equi- 
 librium by the forces called out by the viscosity of the liquid. 
 That is, although this is a problem involving motion, it is a problem 
 involving unaccelerated motion, and the forces are in equilibrium 
 everywhere. The problem is essentially, therefore, one in statics, and 
 in solving the problem we need to make no use of the fact that in 
 those cases where there happens to be an acceleration the force is 
 
66 DIMENSIONAL ANALYSIS 
 
 proportional to mass times acceleration. In this problem, therefore, 
 we treat force as its own kind of quantity, and do not have to intro- 
 duce a compensating dimensional constant. Our analysis of the 
 problem is now as follows : 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Velocity of fall, v LT- 1 
 
 Diameter of sphere, D L 
 
 Density of sphere, d t ML~ 3 
 
 Density of liquid, d 2 ML~ 3 
 
 Viscosity of liquid, /* FL~ 2 T 
 
 Intensity of gravity, g FM" 1 
 
 The dimensional formula of viscosity is obtained directly from 
 its definition as force per unit area per unit velocity gradient. The 
 intensity of gravity is taken with the dimensions shown, because 
 obviously the equations of motion will not mention the accelerational 
 aspect of gravitational action, but only the intensity of the force 
 exerted by gravity upon unit mass. 
 
 We now have six variables, and four fundamental kinds of unit. 
 There are, therefore, two dimensionless products. One of them is 
 evident on inspection, and is d 2 /d . Now of the remaining quantities 
 we are especially interested in v. We need combine this with only 
 four other quantities to obtain a dimensionless product. We choose 
 D, di, JM, and g, and seek a dimensionless product of the form 
 
 The exponents are at once found to be 
 
 =-2,/T:= -1/8== -1,^ = 1 
 
 Hence the dimensionless products are 
 
 v D- 2 dr 1 n g- 1 and 
 and the final solution is 
 
 The function f is arbitrary, so that we cannot tell how the result 
 depends on the densities of the sphere and the liquid, but we do see 
 that the velocity of fall varies as the square of the diameter of the 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 67 
 
 sphere, and the intensity of gravity, and inversely as the viscosity 
 of the fluid. 
 
 This problem has of course long been solved by the methods of 
 hydrodynamics, and the solution is 
 
 See, for example, Millikan, Phys. Rev., 2, 110, 1913. The exact solu- 
 tion is obtained from the more general one above by giving 
 
 ) as the special value of the function. 
 
 If we had solved this problem with the ordinary mechanical units, 
 in which force is defined as mass times acceleration, we should have 
 had 'three instead of two dimensionless products, and the final result 
 would have been of the form 
 
 d.D 
 
 In this form we evidently can say nothing about the effect on the 
 velocity of any of the elements taken by themselves, since they all 
 occur under the arbitrary functional symbol. 
 
 There are many problems in which some specific information 
 about the nature of the physical system enables the information 
 given by dimensional analysis to be supplemented so that a more 
 restricted form of the solution can be obtained than would be pos- 
 sible by dimensional analysis alone. There is, of course, no law 
 against combining dimensional analysis with any information at our 
 command. 
 
 Let us take as a simple example the discussion of the problem of 
 the bending of a beam. This is a problem in elasticity. Let us en- 
 deavor to find how the stiffness of the beam depends on the dimen- 
 sions of the beam, and any other quantities that may be involved. 
 Now the equations of elasticity are equations of ordinary mechanics. 
 The mechanical system of three fundamental units is indicated. 
 The equations of elasticity from which the solution is to be obtained 
 will involve the elastic constants of the material. If the material is 
 isotropic, there will be two elastic constants, which may be chosen as 
 Young's modulus, and the shear modulus. Out analysis may now 
 run as follows: 
 
68 DIMENSIONAL ANALYSIS 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Stiffness (Force/deflection), S MT~ 2 
 
 Length, 1 L 
 
 Breadth, b L 
 
 Depth, d L 
 
 Young's modulus, E ML" 1 T~ 2 
 
 Shear modulus, ju ML- 1 T~ 2 
 
 There are six variables, and three fundamental kinds of unit. 
 Hence according to the general rule there should be three dimen- 
 sionless products. Three such products can obviously be written 
 down by inspection, and are 
 
 b/1, d/1, and /x/E. 
 
 Now none of these dimensionless products contains the quantity S 
 in which we are particularly interested, and it is evident that there 
 is something peculiar about this problem. It will in fact be found, 
 on going back to the system of algebraic equations on which the 
 solution depends, and writing down the matrix of the coefficients 
 obtained from the exponents in the dimensional formulas, that each 
 of the three rowed determinants formed out of the matrix is zero. 
 This is evident on inspection of the matrix. 
 
 100011 
 1 1 l_l_l 
 -2 0-2-2 
 
 This means that in this particular case there are more dimensionless 
 products than are given by the general rule. That such is the case 
 should have been evident beforehand. In the first place, an inspec- 
 tion of the dimensional formulas shows that M and T always enter 
 in the combination MT~ 2 , so that this combination together might 
 have been treated as a fundamental unit itself, so that there would 
 have been only two fundamental units instead of three, and four 
 instead of three dimensionless products. In the second place, this is 
 a problem in statics, in which mass and time do not enter into the 
 results. The dimensions of all the quantities could have been given 
 in terms of force and length as the fundamental units. This remark 
 is the physical equivalent of the analytical observation that M and 
 T always occur in the combination MT~ 2 (force is MT~ 2 multiplied 
 byL). 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 69 
 
 With the knowledge that there is still another dimensionless 
 product, we can see by inspection that it is 
 
 S/El, 
 so that the final solution is 
 
 b d /A 
 
 7 ? i; 
 
 = Blf 
 
 This solution gives no information about the variation of stiffness 
 with the dimensions of the beam. Now it is obvious from elementary 
 considerations of elasticity, that for slender beams, the stiffness 
 must be approximately proportional to the breadth, other things 
 being equal, for the boundary conditions are such that the solution 
 for a beam of twice the breadth may be obtained approximately by 
 simply placing beside each other two of the original beams. Hence 
 
 f must be of such a form that If (-,-, ) reduces to b < ( -, ) 
 
 \1 1 E/ \1 B/ 
 
 and the value of f must obviously be - <[-, ). The restricted 
 
 1 \l E/ 
 
 solution is therefore 
 
 I'E 
 
 The solution now shows that a beam of double the length can be 
 kept of the same stiffness by doubling the depth. The detailed solu- 
 tion of elasticity shows that the ratio of d to 1 enters as the cube, as a 
 factor of proportionality, so that the stiffness is proportional directly 
 to the cube of the depth, inversely to the cube of the length, directly 
 as the breadth, and to some unknown function of the elastic 
 constants. 
 
 This method of supplementing the results of dimensional analysis 
 by other information will often be found of value. There are numer- 
 ous examples in Lord Rayleigh's treatments. Rayleigh does not 
 always separate the analysis into a dimensional and another part, 
 but states that a result can be proved by dimensional analysis, 
 although it may require supplementing in some such way as above. 
 A good example will be found in his treatment of the scattering of 
 light by the sky. 2 The result that the scattering varies inversely as 
 the fourth power of the wave length of the incident light is obtained 
 by using in addition to dimensional knowledge the fact that "From 
 
W 
 
 70 DIMENSIONAL ANALYSIS 
 
 what we know of the dynamics of the situation i (ratio of ampli- 
 tude of incident and scattered light) varies directly as T (volume ' 
 of scattering particle) and inversely as r (distance of point of 
 observation from scattering particle)." 
 
 Thus far we have considered only problems in mechanics, but of 
 course the method is not restricted to such problems, but can be 
 applied to any system whose laws can be formulated in a form 
 independent of the size of the fundamental units of measurement. 
 
 Let us consider, for example, a problem from the kinetic theory 
 of gases, and find the pressure exerted by a perfect gas. The atoms 
 of the gas in kinetic theory are considered as perfect spheres, com- 
 pletely elastic, and of negligible dimensions compared with their 
 distance apart. The only constant with dimensions required in 
 determining the behavior of each atom is therefore its mass. The 
 behavior of the aggregate of atoms is also evidently characterized 
 by the density of the gas or the number of atoms per unit volume. 
 The problem is evidently one of mechanics, and the pressure exerted 
 by the gas is to be found by computing the change of momentum 
 per unit time and per unit area of the atoms striking the walls of 
 the enclosure. The mechanical system of units is therefore indicated. 
 But in addition to the ordinary mechanical features there is the 
 element of temperature to be considered. How does temperature 
 enter in writing down the equations of motion of the system? 
 Obviously through the gas constant, which gives the average kinetic 
 energy of each atom as a function of the temperature. Our analysis 
 of the problem therefore runs as follows : 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Pressure exerted by gas, p ML" 1 T~ 2 
 
 Mass of the atom, m M 
 
 Number of atoms per unit of 
 
 volume, N L~ 8 
 
 Absolute temperature, 6 
 
 Gas constant per atom, k ML 2 T~ 2 0" 1 
 
 "We have here five variables, and four kinds of units. There is, 
 therefore, one dimensionless product. Since p is the quantity in 
 which we are interested, we choose the exponent of this as unity. 
 We have to find 
 
 p m 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 71 
 
 The work of solution is as before. The values of the exponents are 
 
 a = 0, =-1, y = -l, 8 = -l. 
 The dimensionless product is 
 
 p N- 1 0- 1 k- 1 , 
 and the final solution is 
 
 p = Const N k 0. 
 
 That is, the pressure is proportional to the gas constant, to the 
 density of the gas, and to the absolute temperature, and does not 
 depend on the mass of the individual atoms. The formula for pres- 
 sure is, of course, one of the first obtained in any discussion of 
 kinetic theory, and differs from the above only in that the numerical 
 value of the constant of proportionality is determined. 
 
 In this problem, or in other problems of the same type, we could, 
 if we preferred, eliminate temperature as an independent kind of 
 variable and define it as equal to the energy of the atom. This 
 amounts merely to changing the size of the degree, but does not 
 change the ratio of any two temperatures, and is the sort of change 
 of unit which is required according to the fundamental assumptions. 
 If we define temperature in this way, the gas constant is of course 
 to be put equal to unity. We would now have three fundamental 
 units, and four variables. There is of course again only one dimen- 
 sionless product, and the same result would be obtained as before. 
 Let us go through the work ; it is instructive. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Pressure exerted by gas, p ML" 1 T~ 2 
 
 Mass of atom, m M 
 
 Number of atoms per unit vol- 
 ume, N L~ 3 
 Absolute temperature, ML 2 T~ 2 
 
 We now have to find our dimensionless product in the form 
 
 p m a N Ov. 
 The exponents are at once found to be 
 
 a = 0,=-]L, r = --l, 
 and the solution of the problem is 
 
 p Const N 6. 
 
72 DIMENSIONAL ANALYSIS 
 
 This solution is like the one obtained previously except for the 
 presence of the gas constant, but since the gas constant in the new 
 system of units is unity, the two solutions are identical, as they 
 should be. 
 
 This procedure can obviously be followed in any problem whose 
 solution involves the gas constant. Temperature may be either 
 chosen as an independent unit, in which case the gas constant 
 appears explicitly as a variable, or temperature may be so denned 
 that the gas constant is always unity, and temperature has the 
 dimensions of energy. The same procedure is not incorrect in prob- 
 lems not involving the gas constant in the solution. But if in this 
 class of problem temperature is denned as equal to the kinetic 
 energy of an atom (or more generally equal to the energy of a 
 degree of freedom) and the gas constant is made equal to unity, 
 the fundamental units are restricted with no compensating advan- 
 tage, so that although the results are correct as far as temperature 
 is proportional to the energy of a degree of freedom, they will not 
 give so much information as might have been obtained by leaving 
 the units less restricted. 
 
 It is obvious that these remarks apply immediately to the heat 
 transfer problem of Eayleigh treated in the introductory chapter. 
 
 Many persons feel an intuitive uncertainty with regard to the 
 dimensions to be assigned to temperature. This is perhaps because 
 of the feeling that a dimensional formula is a statement of the 
 physical nature of the quantity as contained in the definition. Now 
 the absolute temperature, as we have used it above, is the thermo- 
 dynamic absolute temperature, defined with relation to the second 
 law of thermodynamics. It is difficult to see how such a complex 
 of physical operations as is involved in the use of the second law 
 (such as Kelvin first gave in his definition of absolute temperature) 
 can be reproduced in a simple dimensional formula. It is, however, 
 evident that measurements of energy, for example, are involved in 
 an application of the second law, so that perhaps in some way the 
 ordinary mechanical units ought to be involved in the dimensional 
 formula. But we have seen that the dimensional formula is con- 
 cerned only with an exceedingly restricted aspect of the way in 
 which the various physical operations enter the definition, namely 
 with the way in which the numerical measure of a quantity changes 
 when the fundamental units change in magnitude. Now a little 
 reflection shows that any such procedure as that of Lord Kelvin 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 73 
 
 applied to the definition of absolute temperature through the use 
 of the second law imposes on the number measuring a given con- 
 crete temperature no restriction whatever in terms of the units in 
 which heat or energy, for example, are measured. The size of a 
 degree of thermodynamic temperature may be fixed entirely arbi- 
 trarily so that there are any number of degrees between the freez- 
 ing and the boiling points of water, for example, absolutely without 
 reference to the size of any other unit. We are concerned in the 
 dimensional formula with the definition in terms of the second law 
 only in so far as this definition satisfies the principle of the absolute 
 significance of relative magnitude, that is, the principle that the 
 ratio of the measures of two concrete examples shall be independent 
 of the size of the units. Now it is evident that the thermodynamic 
 definition of absolute temperature does leave the ratio of any two 
 concrete temperatures independent of the size of the units. The 
 dimensional formula of temperature, therefore, need contain no 
 other element* and temperature may be treated as having its own 
 dimensions. 
 
 There is no necessity in using the absolute thermodynamic tem- 
 perature. We might, for instance, define the number of degrees in a 
 given temperature interval as the number of units of length which 
 the kerosene in a certain capillary projecting from a certain bottle 
 of kerosene moves when the bottle is brought from one temperature 
 to another. The temperature so defined evidently satisfies the prin- 
 ciple of the absolute significance of relative magnitude, for if the 
 size of the unit of length measured along the capillary is cut in half, 
 the number of degrees in every temperature interval is doubled. 
 The advantage of the thermodynamic scale is one of simplicity; in 
 the kerosene scale the behavior of a perfect gas could not be charac- 
 terized in terms of a single constant, and the Fourier equations of 
 heat conduction could not be written, except over a very narrow 
 range, in terms of a single coefficient of thermal conductivity. 
 
 Besides the question of the dimensions of temperature, there is 
 one other question connected with the application of dimensional 
 analysis to problems in thermodynamics which is apt to be puzzling ; 
 this is the matter of the so-called logarithmic constants. In books on 
 thermodynamics equations are very common which on first sight 
 do not appear to be complete equations or to be dimensionally 
 homogeneous. These equations often involve constants which cannot 
 change in numerical magnitude by some factor when the size of the 
 
74 DIMENSIONAL ANALYSIS 
 
 fundamental units is changed, but must change by the addition of 
 some term. An example will be found on page 6 of Nernst's Yale 
 lectures on the Applications of Thermodynamics to Chemistry. This 
 equation is : 
 
 logC = - 
 
 RT R R 2R 
 
 In this equation C is a concentration of a given gaseous sub- 
 stance, A is a heat, a, b, and c are dimensional coefficients in the 
 usual sense into which we need not inquire further, except that a/R 
 is dimensionless, and i is a constant of integration. It is obvious that 
 this formula as it stands does not allow the size of the fundamental 
 units to be changed by making the usual sort of changes in the 
 various quantities. But a rearrangement of terms is possible which 
 throws the formula into the conventional form. If we group together 
 the terms log C, a/R log T, and i into the single term 
 
 C 
 log 
 
 where i' is a new constant, we evidently have a complete equation in 
 
 a 
 
 the usual sense of the word, and i' has the dimensions of C T R . 
 
 This sort of rearrangement of terms is always possible if the 
 formula has had a theoretical derivation, as have all the formulas 
 of these treatises, and the logarithmic constant appears only as a 
 formal exception. 
 
 The logarithmic constant is met with so often in thermodynamic 
 formulas because in most thermodynamic expressions there is an 
 undetermined constant of integration arising from the fact that 
 energy, or work, or entropy, or thermodynamic potential has no 
 absolute significance, but is only the difference between two values, 
 and the coordinates of the initial point which fix the origin of 
 entropy, for example, may be chosen at pleasure. 
 
 The formulas of thermodynamics also often present a strange 
 appearance in the way that concrete quantities (that is, quantities 
 with dimensions) appear as the arguments of transcendental func- 
 tions. Thus on page 5 of the same book of Nernst's, we find the 
 formula. 
 
 dT 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 75 
 
 This comes from an application of Clapeyron's equation to a sub- 
 stance whose vapor obeys the perfect gas law, and the volume of 
 whose vapor is large compared with that of the liquid. In spite of 
 the appearance of a pressure under the logarithm sign, this equa- 
 tion is seen on examination to be a complete equation, and holds 
 valid for all sizes of the fundamental units. This may at once be 
 
 seen on expanding - , which is equal to -- , and is therefore 
 dT pdT 
 
 of zero dimensions in p. Expressions of this sort in which the loga- 
 rithm is taken of a quantity with dimensions are particularly com- 
 mon in thermodynamics, and often arise from the equations of the 
 perfect gas. The occurrence of such logarithmic terms should, it 
 seems to me, be difficult for those to interpret who like to regard a 
 dimensional formula as expressing a concrete physical operation on 
 a concrete physical thing. 
 
 That the occurrence of such expressions is not contrary to the n 
 
 theorem is seen from the expanded form, -- . The slope of the 
 
 p dT 
 
 curve, , would be one of the variables in which the dimensionless 
 
 products are to be expressed, and there is evidently no exception. 
 Our theorem has merely stated that the results are expressible in 
 terms of dimensionless products; we have no reason to think that 
 the man who derived the formula was accommodating enough to 
 write the formula so that this would appear without some rearrange- 
 ment of the terms. 
 
 Let us close this chapter of special examples with several electri- 
 cal examples. 
 
 As the first example consider an electric circuit possessing capa- 
 city and inductance. An oscillatory discharge is excited in it. How 
 does the period of the discharge depend on the constants of the 
 circuit? The solution of this problem is to be obtained from the 
 detailed equations of the electric circuit, written in the usual form, 
 in electromagnetic units. None of the electrostatic effects of the cur- 
 rent, or the interactions with a magnet, have to be considered in 
 the equations, which are of the form 
 
 d t 
 
76 DIMENSIONAL ANALYSIS 
 
 Hence in establishing the units fundamental to this equation it is 
 evidently sufficient to consider only three fundamental kinds of 
 quantity, namely, quantity of electricity, time, and energy. Then 
 current is to be defined as quantity per unit time, coefficient of self- 
 induction is such a quantity that when multiplied by half the square 
 of the current it gives an energy, and similarly the capacity is such 
 a quantity that it gives an energy when divided into the square of a 
 quantity. We may now make our usual analysis of the problem. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 Quantity of electricity, q Q 
 
 Current, i QT" 1 
 
 Coefficient of self-induction, L Q~ 2 T 2 E 
 
 Capacity, c Q^" 1 
 
 Periodic time, t T 
 
 Now the time of oscillation might conceivably involve the con- 
 stants of the circuit, and the initial charge of the condenser. That 
 is, we are to look for a relation between q, L, c, and t. Since we are 
 especially interested in t, we try to find a dimensionless product of 
 the form 
 
 t L* c^ i?. 
 
 The exponents are at once found to be 
 
 giving as the solution of the problem 
 
 t = Const ^ c. 
 
 This of course is the solution which would be found by actually 
 solving the equations of the circuit, except for the value of the con- 
 stant coefficient. It is to be noticed that the initial charge does not 
 enter. This problem is evidently the electrical analogue of the 
 mechanical problem of the simple pendulum. 
 
 It is perhaps worth noticing again that some knowledge of the 
 nature of the solution is necessary before dimensional analysis can 
 be used to advantage. The Australian bushman, when attacking this 
 problem for the first time, might be tempted to look for a relation 
 of the dimensions of a time between the constants of the circuit, 
 and the instantaneous current, and the instantaneous charge in the 
 condenser. If he had included i in his list of variables, he would have 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 77 
 
 found that q/i also has the dimensions of a time, and his solution 
 would have been of the form 
 
 L 
 
 t = /L c f ( 
 
 \ 
 
 which is not incorrect, since it reduces to the form already found on 
 putting f equal to a constant, but it gives less information than the 
 previous form. 
 
 "We now consider a problem in electrostatics. The conception of 
 the medium introduced by Faraday tells us that it is possible to 
 regard the medium as the seat of the essential phenomena in the 
 electrostatic field, and that the condition of the medium at any 
 instant is uniquely determined by the electric vector at that point. 
 Let us seek for the connection between the space density of energy 
 in the electrostatic field and the intensity of the field. Since this is a 
 problem in statics, the phenomena can be adequately described in 
 terms of two fundamental units, those of force and length. Further- 
 more the field equations of electrostatics contain no dimensional con- 
 stants, so that the velocity of light does not enter the results, as it 
 did the problem of the mass of the spherical distribution of charge. 
 In terms of the two fundamental units of force and length we may 
 make our fundamental definitions as follows. Unit electrostatic 
 charge is that charge which at distance of unity from an equal 
 charge in empty space exerts on it a force of unity. The electric 
 vector is that vector which when multiplied into the charge gives the 
 force on the charge. The dielectric constant is the ratio of the force 
 between two charges when in empty space, and when surrounded 
 by the medium in question. The dimensions of dielectric constant 
 are obviously zero. The dimensions of energy with this system of 
 units are obviously force multiplied by distance. 
 
 We now formulate the problem. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Charge, e 
 
 Field strength, E 
 
 Energy density, u FL~ 2 
 
 We are to seek for a relation between E and u. Generally there 
 would not be a relation between these quantities, because there are 
 two fundamental quantities and two variables. But under the spe- 
 
78 DIMENSIONAL ANALYSIS 
 
 cial conditions of this problem a relation exists, and the result is 
 obvious on inspection to be 
 
 u = Const E 2 . 
 
 In treatises on electrostatics the constant is found to be %. 
 
 If instead of the energy density in empty space we had tried to 
 find the energy density inside a ponderable body with dielectric 
 constant e, the above result would have been modified by the appear- 
 ance of an arbitrary function of the dielectric constant as a factor. 
 Dimensional analysis can give no information as to the form of the 
 function. As a matter of fact, the function is equal to the dielectric 
 constant itself. 
 
 This problem is instructive in showing the variety of ways in 
 which it is possible to choose the fundamental units. Since the prob- 
 lem is one which may be reduced to formulation in mechanical 
 terms (the definitions of electrical quantities are given immediately 
 in terms of mechanical quantities) we might have used the ordinary 
 three units of mechanics as fundamental, and written the dimen- 
 sional formulas in terms of mass, length, and time. We would have 
 obtained the following formulation. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Charge, e 
 
 Field strength, E 
 
 Energy density, u 
 
 Again we are to seek for a relation between the energy density 
 and the field strength. Now here we have two variables and three 
 kinds of fundamental units, so that again the general rule would 
 allow no dimensionless products, and no relation, but the relation 
 between the exponents is such that the dimensionless product does 
 exist, and in fact is found to be exactly the same as before. The 
 new formulation in terms of different fundamental units does not 
 change the result, as it should not. 
 
 Many persons will object to the dimensional formulas given for 
 these electrostatic quantities on the ground that we arbitrarily put 
 the dielectric constant of empty space equal to unity, whereas we 
 know nothing about its nature, and therefore have suppressed cer- 
 tain dimensions which are essential to a complete statement of the 
 problem. This point of view will of course not be disturbing to the 
 reader of this exposition, who has come to see that there is nothing 
 
EXAMPLES OF DIMENSIONAL ANALYSIS 79 
 
 absolute about dimensions, but -that they may be anything consistent 
 with a set of definitions which agree with the experimental facts. 
 However, let us by actual example carry through this problem, 
 including the dielectric constant of empty space explicitly as a new 
 kind of fundamental quantity which cannot be expressed in terms 
 of mass, length, and time. Call the dielectric constant of empty 
 space k, and use the same letter to stand for the quantity itself, and 
 its dimension. Then the unit of electrostatic charge is now defined 
 by the relation, force = e 2 /k r 2 . Field strength is to be defined as 
 before as eE = Force. If we formulate the problem in terms of 
 these fundamentals, the electrostatic field equations will now con- 
 tain k explicitly, so that the dimensional constant k appears in the 
 list of variables. The formulation of the problem is now as follows : 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Charge, e 
 
 Field strength, E 
 
 Energy density, u 
 
 Dielectric constant of empty 
 
 space, k k 
 
 We again look for a dimensionless product in which the terms are 
 E, u, and k, and find the result to be 
 
 u = Const k E 2 . 
 
 This reduces to that previously found on putting k equal to unity, 
 which was the value of k in the previous formulation of the problem. 
 The form above appears somewhat more general than the form 
 previously obtained in virtue of the factor k, but this factor does 
 not tell us anything more about nature, but merely shows how the 
 formal expression of the result will change when we change the for- 
 mulation of the definitions at the basis of our system of equations. 
 The inclusion of the factor k in the result and in the definitions is 
 therefore of no advantage to us, and never can be of advantage, if 
 our considerations are correct. 
 
 There has been much written on the ' ' true ' ' dimensions of k, and 
 much speculation about the various physical pictures of the me- 
 chanical structure of the ether which follow from one or another 
 assumption as to the ' ' true ' ' dimensions, but so far as I am aware, 
 no result has been ever suggested by this method which has led to 
 the discovery of new facts, although it cannot be denied that a num- 
 
80 DIMENSIONAL ANALYSIS 
 
 ber of experiments have been suggested by these considerations, as 
 for example those of Lodge on the mechanical properties of the 
 ether. 
 
 KEFEBENCES 
 
 (1) Rayleigh, Nat. 95, 66, 1915. 
 
 (2) Rayleigh, Phil. Mag. 41, 107 and 274, 1871. 
 
 A number of miscellaneous examples will be found treated in a 
 paper by C. Runge, Phys. ZS. 17, 202, 1916. 
 
 Speculations as to the " correct" dimensions to be given to elec- 
 trical quantities, and deductions as to the properties of the ether 
 are found in the following papers : 
 
 A. W. Riicker, Phil. Mag. 27, 104, 1889. 
 W. Williams, Phil. Mag. 34, 234, 1892. 
 O. Lodge, Phil. Mag. Nov., 1882. 
 
 Nat. July 19, 1888. 
 
 "Modern Views of Electricity," Appendix. 
 R. A. Fessenden, Phys. Rev. 10, 1 and 83, 1900. 
 K. R. Johnson, Phys. ZS. 5, 635, 1904. 
 A. C. Crehore, Phys. Rev. 14, 440, 1919. 
 
 G. F. Fitzgerald, Phil. Mag. 27, 323, 1889'. This article is quoted 
 entire. 
 
 ' ' Some attention has lately been called to the question of the 
 dimensions of the electromagnetic units, but the following sug- 
 gestion seems to have escaped notice. 
 
 "The electrostatic system of units may be defined as one in 
 which the electric inductive capacity is assumed to have zero 
 dimensions, and the electromagnetic system is one in which the 
 magnetic inductive capacity is assumed to have zero dimen- 
 sions. Now if we take a system in which the dimensions of both 
 these quantities are the same, and of the dimensions of a slow- 
 ness, i.e., the inverse of a velocity (T/L), the two systems be- 
 come identical, as regards dimensions, and differ only by a 
 numerical coefficient, just as centimeters and kilometers do. 
 There seems a naturalness in this result which justifies the 
 assumption that these inductive capacities are really of the 
 nature of a slowness. It seems possible that they are related 
 to the reciprocal of the square root of the mean energy of 
 turbulence of the ether. ' ' 
 
CHAPTER VII 
 
 APPLICATIONS OF DIMENSIONAL ANALYSIS TO 
 MODEL EXPERIMENTS. OTHER ENGINEER- 
 ING APPLICATIONS 
 
 HITHERTO we have applied dimensional analysis to problems which 
 could be solved in other ways, and therefore have been able to check 
 our results. There are, however, in engineering practise a large 
 number of problems so complicated that the exact solution is not 
 obtainable. Under these conditions dimensional analysis enables us 
 to obtain certain information about the form of the result which 
 could be obtained in practise only by experiments with an impos- 
 sibly wide variation of the arguments of the unknown function. In 
 order to apply dimensional analysis we merely have to know what 
 kind of a physical system it is that we are dealing with, and what 
 the variables are which enter the equation ; we do not even have to 
 write the equations down explicitly, much less solve them. In many 
 cases of this sort, the partial information given by dimensional 
 analysis may be combined with measurements on only a part of the 
 totality of physical systems covered by the analysis, so that together 
 all the information needed is obtained with much less trouble and 
 expense than would otherwise be possible. This method is coming 
 to be of more and more importance in technical studies, and has 
 recently received a considerable impetus from the necessities of 
 airplane design. The method has received wide use at the National 
 Physical Laboratory in England, and at the Bureau of Standards 
 in this country, and has been described in numerous papers. At the 
 Bureau of Standards Dr. Edgar Buckingham has been largely in- 
 strumental in putting the results of dimensional analysis into such 
 a form that they may be easily applied, and in making a number 
 of important applications. 
 
 The nature of the results obtainable by this method may be illus- 
 trated by a very simple example. Suppose that it is desired to con- 
 struct a very large and expensive pendulum of accurately prede- 
 termined time of swing. Dimensional analysis shows that the time 
 
82 DIMENSIONAL ANALYSIS 
 
 of oscillation of the entire system of all pendulums is given by the 
 formula t = Const VVg- Hence to determine the time of any 
 pendulum whatever, it is sufficient to determine by experiment only 
 the value of the constant in the equation. The constant may evi- 
 dently be found by a single experiment on a pendulum of any 
 length whatever. The experimental pendulum may be made as 
 simple as we please, and by measuring the time of swing of it, the 
 time of swing of the projected large pendulum may be obtained. 
 
 The case of the pendulum is especially simple in that no arbitrary 
 function appeared in the result. Now let us consider the more gen- 
 eral case which may be complicated by the appearance of an arbi- 
 trary function. Suppose that the variables of the problem are 
 denoted by Q 1? Q 2 , etc., and that the dimensionless products are 
 found, and that the result is thrown into the form 
 
 Q, = Q:-Q?> *(Q?Qf- -,Q?Qf- -) 
 
 where the arguments of the function and the factor outside embrace 
 all the dimensionless products, so that the result as shown is general. 
 Now in passing from one physical system to another the arbitrary 
 function will in general change in an unknown way, so that little 
 if any useful information could be obtained by indiscriminate 
 model experiments. But if the models are chosen in such a restricted 
 way that all the arguments of the unknown function have the same 
 value for the model as for the full scale example, then the only 
 variable in passing from model to full scale is in the factors outside 
 the functional sign, and the manner of variation of these factors 
 is known from the dimensional analysis. 
 
 Two systems which are so related to each other that the argu- 
 ments inside the unknown functional sign are equal numerically 
 are said to be physically similar systems. 
 
 It is evident. that a model experiment can give valuable informa- 
 tion if the model is constructed in such a way that it is physically 
 similar to the full scale example. The condition of physical simi- 
 larity involves in general not only conditions on the dimensions of 
 the model but on all the other physical variables as well. 
 'As an example let us consider the resistance experienced by a 
 body of some definable shape in moving through an infinite mass of 
 fluid. Special cases of this problem are the resistance encountered 
 by a projectile, by an airplane, by a submarine in deep water, or by 
 a falling raindrop. The problem is evidently one of mechanics, and 
 
APPLICATIONS TO MODEL EXPERIMENTS 83 
 
 involves the equations of hydrodynamics. The conditions are ex- 
 ceedingly complicated, and would be difficult to formulate in pre- 
 cise mathematical terms, but we perhaps may imagine it done by 
 some sort of a super-being. The important thing to notice is that no 
 dimensional constants appear in the equation of hydrodynamics if 
 the ordinary mechanical units in terms of mass, length, and time 
 are used, so that the result will involve only the measurable physical 
 variables. The variables are the resistance to the motion, the velocity 
 of motion, the shape of the body, which we may suppose specified 
 by some absolute dimension and the ratio to it of certain other 
 lengths (as, for instance, the shape of an ellipsoid may be specified 
 by the length of the longest axis and the ratio to this axis of the 
 other axes) and the constants of the fluid, which are its density, 
 viscosity, and compressibility, the latter of which we may specify 
 by giving the velocity of sound in the fluid. We suppose that gravity 
 does not enter the results, that is, the body is in uniform motion at 
 a constant level, so that no work is done by the gravitational forces. 
 The formulation of the problem is now as follows. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Resistance, R MLT~ 2 
 
 Velocity, v LT^ 1 
 
 Absolute dimension, 1 L 
 
 Density of fluid, d ML~ 3 
 
 Viscosity of fluid, p ML^T- 1 
 
 Velocity of sound in the fluid, v' LT" 1 
 Shape factors, fixing shape of 
 
 body, r , r 2 , etc. . 
 
 We have here six variables, not counting the shape factors, which 
 may have any number depending on the geometrical complexity of 
 the body, so that there are three dimensionless products exclusive 
 of the shape factors, which are already dimensionless. One of these 
 three dimensionless products is obvious on inspection, and is v'/v. 
 We have to find the other dimensionless products in the way best 
 adapted to this particular problem. Since we are interested in the 
 resistance to the motion, we choose this as the term with unit 
 exponent in one of the products, so that we may write the result 
 with R standing alone on the left-hand side of the equation. We find 
 by the methods that we have used so many times that there are two 
 
84 DIMENSIONAL ANALYSIS 
 
 dimensionless products of the forms R v~ 2 1~ 2 d" 1 and /u, v" 1 1" 1 d" 1 , 
 so that the final solution takes the form 
 
 R = v 2 1 2 d f (^/v 1 d, v'/v, r , r 2 , - -) . 
 
 This formula is so broad as to cover a wide range of experimental 
 conditions. If the velocity is low, the problem reduces to one of 
 equilibrium in which the forces on the solid body immersed in the 
 fluid are held in equilibrium by the forces due to the viscosity in the 
 fluid. The resistance does not depend on the density of the fluid, nor 
 on the velocity of sound in it. Evidently if the density is to disap- 
 pear from the above result, the argument /u/vld must come outside 
 the functional sign as a factor, and for slow motion the law of 
 resistance takes the form 
 
 Rr=vl/xf (r ,r 2 , ---- ). 
 
 The resistance at low velocities is therefore proportional to the vis- 
 cosity, to the velocity, and to the linear dimensions, and besides this 
 depends only on the geometrical shape of the body. We have already 
 met a special case in the Stoke 's problem of the sphere, in the intro- 
 ductory chapter. 
 
 For a domain of still higher velocities the density of the fluid 
 plays an important part, since some of the force acting on the body 
 is due to the momentum carried away from the surface of the body 
 by the fluid in the form of eddies (and the momentum carried away 
 obviously depends on the density of the fluid), but the velocity of 
 sound has not yet begun to affect the result, which means that the 
 fluid acts sensibly like an incompressible liquid. This is the realm 
 of velocities of interest in airplane work. Under these conditions the 
 argument v'/v drops out of the function, therefore, and the result 
 becomes 
 
 Let us stop to inquire how the information given by this equation 
 can be used in devising model experiments. What we desire to do is 
 to make a measurement of the resistance encountered by the model 
 under certain conditions, and to infer from this what would be the 
 resistance encountered by the full size example. It is in the first 
 place obvious that the unknown function must have the same value 
 for the model and the original. This means, since the function is 
 entirely unknown, that all the arguments must have the same value 
 
APPLICATIONS TO MODEL EXPERIMENTS 85 
 
 for the model and the original. r x , r 2 , etc., must therefore be the 
 same for both, or in other words, the model and the original must be 
 geometrically of the same shape. Furthermore, ju/vld must have the 
 same numerical value for both. If the model experiment is to be per- 
 formed in air, as it usually is, /A and d are the same for the model 
 and original, so that vl must be the same for model and original. 
 That is, if the model is one-tenth the linear dimensions of the orig- 
 inal, then its velocity must be ten times as great as that of the 
 original. Under these conditions the formula shows that the resist- 
 ance encountered by the model is exactly the same as that encoun- 
 tered by the original. Now this requirement imposes such difficult 
 conditions to meet in practise, demanding velocities in the model 
 of the order of thousands of miles per hour, that it would seem at 
 first sight that we had proved the impossibility of model experi- 
 ments of this sort. But in practise the function of //./v 1 d turns 
 out to have such special properties that much important informa- 
 tion can nevertheless be obtained from the model experiment. If 
 measurements are made on the resistance of the model at various 
 speeds, and the corresponding values of the function calculated 
 (that is, if the measured resistances are divided by v 2 1 2 d), it will be 
 found that at high values of the velocity the function f approaches 
 asymptotically a constant value. This means that at high velocities 
 the resistance approaches proportionality to the square of the veloc- 
 ity. It is sufficient to carry the experiment on .the model only to 
 such velocities that the asymptotical value of the function may be 
 found, in order to obtain all the information necessary about the 
 behavior of the full scale example, for obviously we now know that 
 the resistance is proportional to the square of the velocity, and the 
 model experiment has given the factor of proportionality. The only 
 doubtful point in this proposed procedure is the question as to 
 whether it is possible to reach with the model speeds high enough 
 to give the asymptotic value, and this question is answered by the 
 actual experiment in the affirmative. 
 
 The fact that at the velocities of actual airplane work the- resist- 
 ance has become proportional to the square of the velocity means, 
 according to the analysis, that the viscosity no longer plays a 
 dominant part. This means that skin friction has dropped out as an 
 important part of the retarding force, and that all the work of 
 driving the airplane is used in creating eddies in the air. Freedom 
 from viscosity and complete turbulence of motion are seen by the 
 
86 DIMENSIONAL ANALYSIS 
 
 analysis to be the same thing. This view of the phenomena is abun- 
 dantly verified by experiment. 
 
 Let us consider the possibility of making model experiments in 
 some other medium than air. If we choose water as the medium for 
 the model we must so choose the dimensions and the velocity of the 
 model that /x/v d 1 for the model is equal to p/v d 1 for the origi- 
 nal. Now for water /x is 10~ 2 and d is 1, whereas for air //, is 170 X 
 10~ 6 and d is 1.29 X 10~ 3 . Substituting these values shows that vl 
 for the model must be about one-thirteenth of the value for the 
 original. As a factor one-thirteenth is itself about the reduction in 
 size that would be convenient for the model ; this would mean that 
 the model in water must travel at about the same rate as the original 
 in air. Such high velocities in water are difficult, and there seems 
 no advantage in using water over the actual procedure that is pos- 
 sible in air. 
 
 Consider now still higher velocities, such as those of a projectile, 
 which may be higher than the velocity of sound in the medium, so 
 that the medium has difficulty in getting out of the way of the body, 
 and we have a still different order of effects. At these velocities the 
 viscosity has entirely disappeared from the result, which now takes 
 the form 
 
 R = v 2 l 2 df (v'/v, ri ,r 2 , ). 
 
 If we are now to make model experiments, it is evident that the 
 model projectile must be of the same shape as the original, and 
 furthermore that v'/v must have the same value for the model and 
 the original. If the model experiment is ,made in air, v' for the 
 model will be the same as for the original, so that v must be the same 
 also. That is, the original and the model must travel at the same 
 speed. Under these conditions the formula shows that the resistance 
 varies as the area of cross section of the projectile. The requirement 
 that the model must travel at the same speed as the original imposes 
 such severe restrictions in practise that model experiments on pro- 
 jectiles are most difficult, and apparently have not yet been success- 
 fully made, but all the information has been derived from experi- 
 ence with actual projectiles. 
 
 We may try to avoid the difficulty by making the experiment in 
 another medium, such as water. But the velocity of sound in water 
 is of the order of five times that in air, so that the conditions would 
 require that the velocity of the model projectile in water should be 
 
APPLICATIONS TO MODEL EXPERIMENTS 87 
 
 five times that of the actual projectile in air, an impossible require- 
 ment. 
 
 Besides these applications to model experiments, the results of 
 dimensional analysis may be applied in other branches of engineer- 
 ing. At the Bureau of Standards extensive applications have been 
 made in discussing the performance of various kinds of technical 
 instruments. A class of instruments for the same purpose have 
 certain characteristics in common so that it is often possible to write 
 down a detailed analysis applicable to all instruments of the par- 
 ticular type. Dimensional analysis gives certain information about 
 what the result of such an analysis must be, so that it is possible to 
 make inferences from the behavior of one instrument concerning the 
 behavior of other instruments of somewhat different construction. 
 This subject is treated at considerable length and a number of 
 examples are given in Aeronautic Instruments Circular No. 30 of 
 the Bureau of Standards, written by Mr. M. D. Hersey. 
 
 BEFEKENCES 
 
 Applications of dimensional analysis to subjects of a technical 
 character will be found in the following papers : 
 0. Reynolds, Phil. Trans. Roy. Soc. 174, 935, 1883. 
 Rayleigh, Phil. Mag. 34, 59, 1892, and 8, 66, 1904. 
 E. Buckingham, Phys. Rev. 4, 345, 1914. 
 
 Trans. Amer. Soc. Mech. Eng. 37, 263, 1915. 
 
 Engineering, March 13, 1914. 
 
 E. Buckingham and J. D. Edwards, Sci. Pap. Bur. Stds. No. 359, 
 1920. 
 
 M. D. Hersey, Jour. Wash. Acad. 6, 569, 1916. 
 Sci. Pap. Bur. Stds. No. 331, 1919. 
 
 Aeronautical Instruments Circulars of the Bureau of Stand- 
 ards, No. 30, 1919, and No. 32, 1918. 
 
 W. L. Cowley and H. Levy, Aeronautics in Theory and Experi- 
 ment, Longmans, Green and Co., 1918, Chap. IV. 
 
 E. B. Wilson, Aeronautics, John Wiley and Sons, 1920, Chap. XI. 
 
CHAPTER VIII 
 APPLICATIONS TO THEORETICAL PHYSICS 
 
 THE methods of dimensional analysis are worthy of playing a much 
 more important part as a tool in theoretical investigation than has 
 hitherto been realized. No investigator should allow himself to pro- 
 ceed to the detailed solution of a problem until he has made a 
 dimensional analysis of the nature of the solution which will be 
 obtained, and convinced himself by appeal to experiment that the 
 points of view embodied in the underlying equations are sound. 
 
 Probably one difficulty that has been particularly troublesome in 
 theoretical applications has been the matter of dimensional con- 
 stants ; it is in just such theoretical investigations that dimensional 
 constants are most likely to appear, and with no clear conception 
 of the nature of a dimensional constant or when to expect its ap- 
 pearance, hesitancy is natural in applying the method. But after 
 the discussion of the preceding pages, the matter of dimensional 
 constants should now be readily handled in any special problem. 
 
 The indeterminateness of the numerical factors of proportionality 
 is often also felt to be a disadvantage of the dimensional method, 
 but in many theoretical investigations it is often possible to obtain 
 approximate information about the numerical order of magnitude 
 of the results. Our considerations with regard to dimensional 
 analysis show that any numerical coefficients in the final result are 
 the result of mathematical operations performed on the original 
 equations of motion (in the general sense) of the system. Now it is 
 a result of general observation that such mathematical operations 
 usually do not introduce any very large numerical factors, or any 
 very small ones. Any very large or small numbers in our equations 
 almost always are the result of the substitution of the numerical 
 value of some physical quantity, such as the number of atoms in a 
 cubic centimeter, or the electrostatic charge on the electron, or the 
 velocity of light. Accordingly, if the analysis is carried through 
 with all the physical quantities kept in literal form, we may expect 
 that the numerical coefficients will not be large or small. 
 
 This observation may be used conversely. Suppose that we suspect 
 
APPLICATIONS TO THEORETICAL PHYSICS 89 
 
 a connection between certain quantities, but as yet do not know 
 enough of the nature of the physical system to be able to write down 
 the equations of connection, or even to be sure what would be the 
 elements which would enter an equation of connection. We assume 
 that there is a relation between certain quantities, and then by a 
 dimensional analysis find what the form of the relation must be. We 
 then substitute into the relation the numerical values of the physi- 
 cal quantities, and thus get the numerical value of the unknown 
 coefficient. If this coefficient is of the order of unity (which may 
 mean not of the order of 10 10 according to our sanguinity) the sus- 
 pected relation appears as not intrinsically improbable, and we 
 continue to think about the matter to discover what the precise 
 relation between the elements may be. If, on the other hand, the 
 coefficient turns out to be large or small, we discard the idea as 
 improbable. 
 
 An exposition of this method, and an interesting example were 
 given by Einstein 1 in the early days of the study of the specific 
 heats of solids and their connection with quantum phenomena. The 
 question was whether the same forces between the atoms which 
 determine the ordinary elastic behavior of a solid might not also be 
 the forces concerned in the infra red characteristic optical fre- 
 quencies. This view evidently had important bearings on our whole 
 conception of the nature of the forces in a solid, and the nature of 
 optical and thermal oscillations. 
 
 For the rough analysis of the problem in these terms we may 
 regard the solid as an array of atoms regularly spaced at the corners 
 of cubes. In our analysis we shall evidently want to know the mass 
 of the atoms, and their distance apart (or the nunjber per cm 3 ). 
 Furthermore, if our view of the nature of the forces is correct the 
 nature of the forces between the atoms is sufficiently characterized 
 by an elastic constant, which we will take as the compressibility. 
 These elements should now be sufficient to determine the infra red 
 characteristic frequency. We make our usual analysis of the 
 problem. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 Characteristic frequency, v T" 1 
 
 Compressibility, k M -1 LT 2 
 
 Number of atoms per cm 8 , N L~ 3 
 
 Mass of the atom, m M 
 
90 DIMENSIONAL ANALYSIS 
 
 There should be one dimensionless product in these quantities, 
 and it is at once found to be k v 2 N* m. The final result is therefore 
 
 k = Const v~ 2 N-S m- 1 . 
 
 "We now take the numerical values pertaining to some actual sub- 
 stance and substitute in the equation to find the numerical value of 
 the coefficient. For copper, k = 7 X 10~ 13 , v 7.5 X 10 12 , m = 1.06 
 X 10~ 22 , and N = 7.5 X 10 22 . Substituting these values gives for 
 the constant 0.18. This is of the order of unity, and the point o'f 
 view is thus far justified. It is of course now a matter of history 
 that this point of view is the basis of Debye 's analysis of the specific 
 heat phenomena in a solid, and that it is brilliantly justified by 
 experiment. 
 
 Another example of this sort of argument concerning the magni- 
 tude of the constants is given by Jeans. 2 The question was whether 
 the earth has at any time in its past history passed through a stage 
 of gravitational instability, and whether this instability has had any 
 actual relation to the course of evolution. A preliminary examina- 
 tion by the method of dimensions showed Jeans what must be the 
 form of the relation between the variables such as mean density, 
 radius, elastic constants, etc., at the moment of gravitational insta- 
 bility, and then a substitution of the numerical values for the earth 
 gave a coefficient of the order of unity. This preliminary examina- 
 tion showed, therefore, that it was quite conceivable that gravita- 
 tional instability might be a factor at some time past or future in 
 the earth 's history, and a more detailed examination of the problem 
 was accordingly undertaken. 
 
 Consider another application of the same argument, this time 
 with a negative result. Let us suppose that we are trying to con- 
 struct an electrodynamic theory of gravitation, and that we regard 
 the gravitational field as in some way, as yet undiscovered, con- 
 nected with the properties of the electron, to be deduced by an 
 application of the field equations of electrodynamics. Now in the 
 field equations there occurs a dimensional constant c, the ratio of 
 the electromagnetic and electrostatic units, which is known to be 
 of the dimensions of a velocity, and numerically to be the same as 
 the velocity of light. 
 
 In searching for a relation of the sort suspected, we therefore 
 consider as the variables the charge on the electron, the mass of the 
 
APPLICATIONS TO THEORETICAL PHYSICS 91 
 
 electron (for charge and mass together characterize the electron), 
 the velocity of light, and the gravitational constant. We make the 
 following analysis : 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Gravitational constant, G 
 
 Charge on electron, e 
 
 Mass of electron, m M 
 
 Velocity of light, c LT~* 
 
 We have four variables, and three fundamental units, so that we 
 expect one dimensionless product. This is at once found to be 
 Gm 2 e~ 2 , the velocity of light not entering the hypothetical relation, 
 and the final result taking the significantly simple form 
 
 G=r Const (e/m) 2 . 
 
 We now substitute numerical values to find the magnitude of the 
 constant. G = 6.658 X 10~ 8 , and e/m = 5.3 X 10 17 , so that Const 
 = 2.35 X 10~ 43 . 
 
 The constant is seen therefore to be impossibly small, and we give 
 up the attempt to think how there might be a relation between these 
 quantities, although the simplicity of the dimensional relation be- 
 tween G and e/m is arresting. 
 
 Identity of dimensional formulas must not be thought, therefore, 
 to indicate an a priori probability of any sort of physical relation. 
 When there are so many kinds of different physical quantities ex- 
 pressed in terms of a few fundamental units, there cannot help 
 being all sorts of accidental relations between them, and without 
 further examination we cannot say whether a dimensional relation 
 is real or accidental. Thus the mere fact that the dimensions of the 
 quantum are those of angular momentum does not justify us in 
 expecting that there is a mechanism to account for the quantum 
 consisting of something or other in rotational motion. 
 
 The converse of the theorem attempted above does hold, however. 
 If there is a true physical connection between certain quantities, 
 then there is also a dimensional relation. This result may be used 
 to advantage as a tool of exploration. 
 
 Consider now a problem showing that any true physical relation 
 must also involve a dimensional relation. Suppose that we are try- 
 
92 DIMENSIONAL ANALYSIS 
 
 ing to build up a theory of thermal conduction, and are searching 
 for a connection between the mechanism of thermal conduction and 
 the mechanism responsible for the ordinary thermodynamic be- 
 havior of substances. The thermodynamic behavior may be consid- 
 ered as specified by the compressibility, thermal expansion, specific 
 heat (all taken per unit volume), and the absolute temperature. If 
 only those aspects of the mechanism which are responsible for the 
 thermodynamic behavior are also effective in determining the 
 thermal conductivity, then it must be possible to find a dimensional 
 relation between the thermodynamic elements and the thermal con- 
 ductivity. We have the following formulation of the problem. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Thermal conductivity, /x 
 Compressibility per unit vol- 
 
 ume, k M~ 2 L 4 T 2 
 
 Thermal expansion per unit 
 
 volume, A Mr^L 3 ^ 1 
 
 Specific heat per unit volume, C L 2 ^ 2 ^ 1 
 
 Absolute temperature, 
 
 We are to seek for a dimensionless product in these variables. 
 There are five variables, and four fundamental kinds of quantity, 
 so that we would expect one dimensionless product. Since /* is the 
 quantity in which we are particularly interested, we choose it as the 
 member of the product with unity for the exponent, write the 
 product in the form 
 
 and attempt to solve for the exponents in the usual way. We soon 
 encounter difficulties, however, for it appears that the equations are 
 inconsistent with each other. This we verify by writing down the 
 determinant of the exponents in the dimensional formulas for k, A, 
 C, and 0. The determinant is found to vanish, which means that the 
 dimensionless product does not exist. Hence the hypothetical rela- 
 tion between thermal conductivity and thermodynamic data does 
 not exist, and the mechanism of the solid must have other properties 
 than those sufficient to account for the thermodynamic data alone. 
 We now give a simple discussion of the problem of radiation from 
 a black body. A much more elaborate discussion has been given by 
 
APPLICATIONS TO THEORETICAL PHYSICS 93 
 
 Jeans. 3 The paper of Jeans is also interesting because he uses a 
 system of electrical units in which the dielectric constant of empty 
 space is introduced explicitly. It is easy to see on a little examina- 
 tion that he would have obtained the same result with a system of 
 units in which the dielectric constant of empty space is defined as 
 unity. 
 
 Let us now consider a cavity with walls which have absolutely 
 no specific properties of their own, but are perfect reflectors of any 
 incident radiation. Inside the cavity is a rarefied gas composed of 
 electrons. If the gas is rarefied enough we know from such con- 
 siderations as those given by Richardson in considering thermionic 
 emission that the electrons function like a perfect gas, the effect of 
 the space distribution of electrostatic charge being negligible in 
 comparison with the forces due to collisions as gas particles. The 
 electron gas in the cavity is to be maintained at a temperature 0. 
 The electrons are acted on by two sets of forces; the collisional 
 forces with the other electrons, which are of the nature of the forces 
 between atoms in ordinary kinetic theory, and the radiational field 
 in the ether. Since the electrons are continually being accelerated, 
 they are continually radiating, and they are also continually absorb- 
 ing energy from the radiational field of the ether. The system must 
 eventually come to equilibrium with a certain energy density in the 
 ether, the electrons possessing at the same time the kinetic energy 
 appropriate to gas atoms at the temperature of the enclosure. The 
 detailed solution of the problem obviously involves a most com- 
 plicated piece of statistical analysis, but a dimensional analysis 
 gives much information about the form of the result. 
 
 In solving this problem we shall have to use the field equations 
 of electrodynamics, so that the velocity of light will be a dimensional 
 constant in the result. The charge and the mass of the electron must 
 be considered, the absolute temperature, and the gas constant, be- 
 s cause this determines the kinetic energy of motion of the electrons 
 as a function of temperature. The number of electrons per cm 3 does 
 not enter, because we know from kinetic theory that the mean 
 velocity of the electrons is independent of their number. The second 
 law of thermodynamics also shows that the energy density in the 
 enclosure is a function of the temperature, and not of the density 
 of the electron gas. 
 
 Our formulation of the problem is now as follows : 
 
94 DIMENSIONAL ANALYSIS 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Energy density, u ML" 1 1 1 " 2 
 
 Velocity of light, c 
 
 Mass of electron, m M 
 
 Charge of electron, e 
 
 Absolute temperature, 6 6 
 
 Gas constant, k 
 
 The ordinary electrostatic system of units is used. There are here 
 six variables and four fundamental kinds of unit, hence two dimen- 
 sionless products, unless there should be some special relation be- 
 tween the exponents. Since we are especially interested in u we 
 choose this as the member of one of the products with unit exponent. 
 We find in the usual way that two products are 
 
 u e 6 k~ 4 0~ 4 and k m- 1 c~ 2 . 
 The result therefore takes the form 
 
 u = k 4 e- 6 6* f (k e m- 1 c~ 2 ). 
 
 We as yet know nothing of the nature of the arbitrary function. 
 The argument of the function, however, is seen to have a definite 
 physical significance, kflrn" 1 is half the square of the velocity of the 
 electron (k0 being its kinetic energy), so that the argument is one- 
 half the square of the ratio of the velocity of the electron to the 
 velocity of light. Now this quantity remains exceedingly small in 
 the practical range of temperature, so that whatever the form of the 
 function, we know that we have a function of a quantity which is 
 always small. By an extension of the reasoning which we employed 
 for the numerical value of any coefficients to be met with in dimen- 
 sional analysis, we may say that the probability is that the numeri- 
 cal value of such a function is sensibly the same as its value for the 
 value zero of the argument, that is, the function may be replaced 
 by a constant for the range of values of the variable met with in 
 practise. Hence with much plausibility we may expect the result 
 to be of the form 
 
 u = Const k 4 e- 6 0*. 
 
 is the only physical variable on the right-hand side of this equa- 
 tion, so that as far as physical variables go the result may be written 
 in the form 
 
 u = a P. 
 
APPLICATIONS TO THEORETICAL PHYSICS 95 
 
 This, of course, is the well-known Stefan 's law, which checks with 
 experiment. The result therefore justifies to a certain extent the 
 views which led us to the result. 
 
 Our argument about the size of numerical coefficients would lead 
 us to expect that the constant in the first form of the result could 
 not be too large or too small. That is, if we put a = Const k* e" 6 , the 
 result should have a certain simplicity of form, such as might seem 
 to be a plausible result of a mathematical operation. Now Lewis and 
 Adams 4 have called attention to the fact that within the limits of 
 experimental error the constant of Stefan's law may be written in 
 the form 
 
 a k 4 /(47re) 6 . 
 
 Although (4 7r) 6 is not an especially small number in the sense of 
 the original formulation by Einstein of the probability criterion for 
 numerical coefficients, it is nevertheless to be regarded as small con- 
 sidering the size of the exponents of the quantities with which it is 
 associated, | and it is undeniable that the result is of such simplicity 
 that it seems probable that the coefficient may be the result of a 
 mathematical process, and is not merely due to a chance combina- 
 tion of elements in a dimensionally correct form. ! 
 
 At any rate, whatever our opinions as to the validity of the argu- 
 ment, the striking character of the result sticks in our minds, and 
 we reserve judgment until the final solution is forthcoming, in the 
 same way that the periodic classification of the elements had to be 
 carried along with suspended judgment until the final solution was 
 forthcoming. It may be mentioned that Lorentz and his pupils have 
 tried a detailed analysis on these terms, with unsuccessful results. 
 
 The above analysis gives other opportunities for thought. It is 
 significant that the quantum h does not enter the result, although it 
 appears to be inseparably connected with the radiation processes, 
 at least in ponderable matter. We know that h enters the formula 
 for the spectral distribution of energy, and we also know from 
 thermodynamics that the distribution of energy throughout the 
 spectrum in a cavity such as the above is the same as the distri- 
 bution in equilibrium with a black body composed of atoms. The 
 spectral distribution in the cavity which we have been considering 
 must therefore involve h. Does this mean that h can be determined 
 in terms of the electronic constants, the gas constant, and the con- 
 
96 DIMENSIONAL ANALYSIS 
 
 stants of the ether, so that no mechanism with which we are not 
 already familiar is needed to account for h? Of course Lewis has 
 used Planck's formula for h in terms of a, etc., in order to obtain a 
 numerical value for h in terms of other quantities. 
 
 As an additional example of the application of dimensional analy- 
 sis in theoretical investigations let us examine the possibility of 
 explaining the mechanical behavior of substances on the basis of a 
 particular form of the law of force between atoms. "We suppose that 
 the law of force can be written in the form 
 
 F = A r~ 2 + B r~ n . 
 
 A is to be intrinsically negative, and represents an attractive force, 
 and B is positive, and represents a force of repulsion which becomes 
 very intense on close approach of the atoms. The atoms of different 
 substances may differ in mass and in the numerical value of the 
 coefficients A and B, but the exponent n is to be the same for all 
 substances. We also suppose that the temperature is so high that the 
 quantum h plays no important part in the distribution of energy 
 among the various degrees of freedom, but that the gas constant 
 is sufficient in determining the distribution. The external variables 
 which may be imposed on the system are the pressure and the tem- 
 perature. When these are given the volume is also determined, and 
 all the other properties. We have, therefore, the following list of 
 quantities in terms of which any of the properties of the substance 
 are to be determined. 
 
 Name of Quantity. Symbol. Dimensional Formula. 
 
 Pressure, p ML -1 T- 2 
 
 Temperature, 6 
 
 Mass of the atom, m M 
 
 Gas constant, k MLT" 2 ^ 1 
 
 A (of the law of force), A ML 3 T~ 2 
 
 B (of the law of force), B ML n+1 T- 2 
 
 In addition to these we will have whatever particular property 
 of the substance is under discussion. In the above list there are six 
 quantities in terms of four fundamental units. Therefore from this 
 list of permanent variables there are two dimensionless products. 
 Let us find them. We will choose one involving p and not 0, and the 
 
APPLICATIONS TO THEORETICAL PHYSICS 97 
 
 other 6 and not p, since p and are the physical variables under 
 our control. The products are at once found to be 
 
 _n + 2 * 
 
 pA n - 2 B n - 2 
 and 
 
 n -1 1 
 
 A~^2 B5TT2 k 6. 
 
 The existence of these two products already gives us information 
 about the behavior of the body in those cases in which pressure and 
 temperature are not independently variable quantities, as they are 
 not on the vapor pressure curve, or on the melting curve, or on the 
 curve of equilibrium between two allotropic modifications of the 
 solid. Under these conditions we have 
 
 n-f 2 4 / n-1 1 \ 
 
 p A~^* B^ 7 * = f lA"^^ B irr 2 k 0/, 
 
 where f is the same function for all substances. A and B vary from 
 substance to substance. Hence this analysis shows that in terms of 
 a new variable p C x for the pressure, and a new variable B C 2 for 
 temperature, the equations for the equilibrium curves of all sub- 
 stances are the same. These new pressure and temperature variables 
 are obtained by multiplying the ordinary pressure and temperature 
 by constant factors, and may be called the reduced pressure and 
 temperature. Van der Waal's equation is a particular case of such 
 an equation, which becomes the same for all substances in terms of 
 the reduced variables. 
 
 Now consider any other physical property of the substance which 
 is to be accounted for in terms of the variables of the analysis above. 
 We have to form another dimensionless product in which it is 
 involved. This dimensionless product may most conveniently be 
 expressed in terms of the quantities m, k, A, and B, since these are 
 physically invariable for the particular substance. The expression 
 of any physical quantity is always dimensionally possible in terms 
 of these quantities, unless the determinant of the exponents of m, k, 
 A, and B vanishes, and this is seen to vanish only in the case n = 
 +2, which is the trivial case of the force reducing to an attraction 
 alone. Hence in the general case any physical property, which we 
 may call Q, may be expressed in the form 
 
 (n+2 * n-1 1 \ 
 
 p A~^~2 B^ A~^ B 5 ^ k 0), 
 
98 DIMENSIONAL ANALYSIS 
 
 where the Const may involve m, k, A, and B in any way, but does 
 not involve p or 6. Now if we define Q/Const as the ' ( reduced ' ' value 
 of Q, then we have the important result that for all substances of 
 this type the equation connecting the reduced value of a quantity 
 Q with the reduced pressure and temperature is the same. This 
 applies not only to thermodynamic properties, but to all properties 
 which are to be explained in terms of the same structure, such as 
 thermal conductivity or viscosity. 
 
 The values of the factors by means of which the measured values 
 of the physical variables are converted to "reduced" values will 
 enable us to compute A, B, and m for the substance in question, if n 
 can be otherwise determined. 
 
 It is evident on consideration of the above work that the only 
 assumption which we have made about n is that it is dimensionless, 
 and that we have not used the assumption stated in the beginning 
 that n is the same for all substances. We may therefore drop this 
 assumption, and have the theorem that for all substances whose 
 behavior can be determined in terms of atoms which are character- 
 ized by a mass and a law of force of the form Ar~ 2 + Br~ n , with no 
 restriction on A, B, or n, there is a law of corresponding states for 
 all physical properties. \ 
 
 Evidently it would be possible to carry through an analysis like 
 the above in which the external variables p and are replaced by 
 any other two which might be convenient, such as certain of the 
 thermodynamic potentials, and the same result would have been 
 obtained, unless there should happen to be special relations between 
 the dimensional exponents. Whether there are such special relations 
 can be easily determined in any special case. 
 
 Before anyone starts on a detailed development of such a theory 
 of the structure of matter as this, he would make a preliminary 
 examination to see whether the properties of substances do actually 
 obey such a law of corresponding states, and govern his future 
 actions accordingly. The value of the advance information obtained 
 in this way is incontestable. 
 
 The analysis above reminds one in some particulars of that of 
 Meslin, 5 but is much more general, in that the analysis of Meslin 
 applied only to the equation of state, and had to assume the exist- 
 ence of critical, or other peculiar points. 
 
 As a final application of dimensional analysis to theoretical 
 
APPLICATIONS TO THEORETICAL PHYSICS 99 
 
 physics we consider the determination of the so-called absolute sys- 
 tems of units. 
 
 The units in ordinary use are ones whose absolute size is fixed 
 in various arbitrary ways, although the relations between the differ- 
 ent sorts of units may have a logical ring. Thus the unit of length, 
 the centimeter, was originally denned as bearing a certain relation 
 to a quadrant of the earth 's circumference, and the unit of mass is 
 the mass of a quantity of water occupying the unit volume. There is 
 something entirely arbitrary in selecting the earth and water as the 
 particular substances which are to fix the size of the units. 
 
 We have also met in the course of our many examples dimensional 
 constants. These constants usually are the expression of some pro- 
 portionality factor which enters into the expression of a law of 
 nature empirically discovered. Such dimensional constants are the 
 constant of gravitation, the velocity of light, the quantum, the con- 
 stant of Stefan's law, etc. Now the numerical magnitude of the 
 dimensional constants depends on the size of the fundamental units 
 in a way fixed by the dimensional formulas. By varying the size of 
 the fundamental units, we may vary in any way that we please the 
 numerical magnitude of the dimensional constant. In particular, by 
 assigning the proper magnitudes to the fundamental units we might 
 make the numerical magnitudes of certain dimensional constants 
 equal to unity. Now the dimensional constants are usually the 
 expression of some universal law of nature. If the fundamental units 
 are so chosen in size that the dimensional constants have the value 
 unity, then we have determined the size of the units by reference to 
 universal phenomena instead of by reference to such restricted 
 phenomena as the density of water at atmospheric pressure at some 
 fixed temperature, for instance, and the units to that extent are 
 more significant. 
 
 There is no reason why one should be restricted to dimensional 
 constants of universal occurrence in fixing the size of the units, but 
 any phenomenon of universal occurrence may be used. Thus the 
 units may be so chosen that the charge on the electron is unity. 
 
 Any system of units fixed in this way by reference to phenomena 
 or relationships of universal occurrence and significance may be 
 called an absolute system of units. The first system of absolute units 
 was given by Planck 6 in his book on heat radiation. He connected 
 the particular system which he gave with the quantum, and it might 
 appear from Planck's treatment that before the discovery of the 
 
100 DIMENSIONAL ANALYSIS 
 
 quantum there were not enough, dimensional constants of the proper 
 character known to make possible a universal system of units, but 
 such is not the case. Planck was the first to think of the possibility 
 of absolute units, and used the quantum in determining them, but 
 there is no necessary connection with the quantum, as may be seen 
 in the following discussion. 
 
 Let us now determine from the dimensional formulas the set of 
 absolute units given by Planck. To fix this set of units we choose 
 the constant of gravitation, the velocity of light, the quantum, and 
 the gas constant. "We require that the fundamental units be of such 
 a size that each of these dimensional constants has the value unity 
 in the new system. The discussion may be simplified for the present 
 by omitting the gas constant, for this is the only one which involves 
 the unit of temperature, and it is obvious that after the units of 
 mass, length, and time have been fixed, the gas constant may be 
 made unity by properly choosing the size of the degree. In deter- 
 mining the size of the new units we find it advantageous to choose 
 the form of notation used in the third chapter in changing units. 
 Consider, for example, the constant of gravitation. We write this as 
 
 Constant of gravitation = G = 6.658 X 10~ 8 gmr 1 cm 3 see" 2 . 
 
 The value in the new system of units is to be found by substituting 
 in the expression for G the value of the new units in terms of the 
 old. Thus if the new unit of mass is such that it is equal to x gm, 
 and the new unit of length is equal to y cm, and the new unit of 
 time to z sec, we shall have as the equation to determine x, y, and z, 
 since the numerical value of the gravitational constant is to be unity 
 in the new system 
 
 6.658 X 10~ 8 gm" 1 cm 3 see" 2 = 1 (x gm)- 1 (y cm) 3 (z sec)" 2 . 
 
 The other two dimensional constants give the two additional equa- 
 tions needed to determine x, y, and z. These other equations are 
 immediately written down as soon as the dimensional formulas and 
 the numerical values of the velocity of light and the quantum are 
 known. The equations are 
 
 3 X 10 10 cm sec" 1 = 1 (y cm) (z sec)" 1 
 6.55 X 10~ 27 gm cm 2 sec" 1 = 1 (x gm) (y cm) 2 (z sec)- 1 . 
 
 This set of three equations may be readily solved, and gives x = 
 
APPLICATIONS TO THEORETICAL PHYSICS 101 
 
 5.43 X 10~ 5 , y = 4.02 X 10~ 33 , and z = 1.34 X 10~ 43 . This means 
 
 that 
 
 the new unit of mass is 5.43 X 10~ 5 gm 
 the new unit of length 4.02 X 10~ 33 cm 
 the new unit of time 1.34 X 10~ 43 sec. 
 
 So far all is plain sailing, and there can be no question with 
 regard to what has been done. The attempt is sometimes made to go 
 farther and see some absolute significance in the size of the units 
 thus determined, looking on them as in some way characteristic of a 
 mechanism which is involved in the constants entering the defini- 
 tion. Thus Eddington 7 says: "There are three fundamental con- 
 stants of nature which stand out preeminently, the velocity of light, 
 the constant of gravitation, and the quantum. From these we can 
 construct a unit of length whose value is 4 X 10~ 33 cm. There are 
 other natural units of length, the radii of the positive and negative 
 charges, but these are of an altogether higher order of magnitude. 
 "With the possible exception of Osborne Reynold's theory of matter, 
 no theory has attempted to reach such fine grainedness. But it is 
 evident that this length must be the key to some essential structure. ' ' 
 
 Speculations such as these arouse no sympathetic vibration in the 
 convert to my somewhat materialistic exposition. The mere fact that 
 the dimensional formulas of the three constants used was such as to 
 allow a determination of the new units in the way proposed seems 
 to be the only fact of significance here, and this cannot be of much 
 significance, because the chances are that any combination of three 
 dimensional constants chosen at random would allow the same pro- 
 cedure. Until some essential connection is discovered between the 
 mechanisms which are accountable for the gravitational constant, 
 the velocity of light, and the quantum, it would seem that no signifi- 
 cance whatever should be attached to the particular size of the units 
 defined in this way, beyond the fact that the size of such units is 
 determined by phenomena of universal occurrence. 
 
 Let us now continue with our deduction of the absolute units, 
 and introduce the gas constant. For this we have the equation 
 
 Gas constant = k = 2.06 X 10~ 16 gm cm 2 see" 2 0" 1 
 
 1 (xgm) (ysec) 2 (zsec)~ 2 (wfl)- 1 . 
 
 x, y, and z are already determined, so that this is a single equa- 
 tion to determine w. The value found is 2.37 X 10 32 . This means 
 
102 DIMENSIONAL ANALYSIS 
 
 that the new degree must be equal to 2.37 X 10 32 ordinary Centi- 
 grade degrees. 
 
 In the wildest speculations of the astrophysicists no such tem- 
 perature has ever been suggested, yet would Professor Eddington 
 maintain that this temperature must be the key to some funda- 
 mental cosmic phenomenon? 
 
 It must now be evident that it is possible to get up systems of 
 absolute units in a great number of ways, depending on the univer- 
 sal constants or phenomena whose numerical values it is desired to 
 simplify. "With any particular selection of constants, the method 
 in general is the same as that in the particular case above. In general 
 there will be four fundamental kinds of unit, if we want to restrict 
 ourselves to the electrostatic system of measuring electrical charges, 
 and define the magnitude of the charge in such a way that the force 
 between two charges is equal to their product divided by the square 
 of the distance between them, or if we do not restrict ourselves to the 
 electrostatic system, there may be five fundamental kinds of quan- 
 tity. There seems to be nothing essential in the number five, which 
 merely arises because we usually find it convenient to use the me- 
 chanical system of units in which the constant of proportionality 
 between force and the product of mass and acceleration is always 
 kept fixed at unity. The convenience of this system is perhaps more 
 obvious in the case of mechanical phenomena, because of the univer- 
 sality of their occurrence. But if temperature effects were as univer- 
 sal and as familiar to us, we would also insist that we always deal 
 only with that system of units in which the gas constant has the 
 fixed value unity. 
 
 Having, therefore, fixed the number of fundamental units which 
 we deem convenient, and having chosen the numerical constants 
 whose values we wish to simplify, we proceed as above. It is evident 
 that it will in general be necessary to assign as many constants as 
 there are fundamental units, for otherwise there will not be enough 
 equations to give the unknowns. Thus above, we fixed four con- 
 stants, gravitational, velocity of light, quantum, and gas constant, 
 and we had four fundamental kinds of units. Now it is important 
 to notice that four algebraic equations in four unknowns do not 
 always have a solution, but the coefficients must satisfy a certain 
 condition. This condition is, when applied to the dimensional for- 
 mulas into which the unknowns enter, that the determinant of the 
 exponents must not vanish. In general, a four-rowed determinant 
 
APPLICATIONS TO THEORETICAL PHYSICS 103 
 
 selected at random would not be expected to vanish. In the case of 
 the determinants obtained from the dimensional formulas of the 
 constants of nature this is not the case, however, because the dimen- 
 sional formulas are nearly all of them of considerable simplicity, 
 and the exponents are nearly always small integers. It very often 
 happens that the determinant of the exponents of four constants 
 chosen at random vanishes, and the proposed scheme for determin- 
 ing the absolute units turns out to be impossible. The vanishing of 
 the determinant means that all the quantities are not dimensionally 
 independent, so that we really have not four but a smaller number 
 of independent quantities in terms of which to determine the un- 
 knowns. For instance, we have found that the gravitational constant 
 dimensionally has the same formula as the square of the ratio of 
 the charge to the mass of the electron. This means that we could 
 not set up a system of absolute units in which the gravitational 
 constant, the charge on the electron, and the mass of the electron 
 were all equal to unity. Now let us write down some of the impor- 
 tant constants of nature and see what are the possibilities in the 
 way of determining systems of absolute units. 
 
 Gravitation constant, G 6.658 X 10" 8 gm" 1 cm 3 sec~ 2 
 
 Velocity of light, c 3 X 10 10 cm sec" 1 
 
 Quantum, h 6.547 X 10" 27 gm-cm 2 sec" 1 
 
 Gas constant, k 2.058 X 10" 16 gm cm 2 see" 2 C- 1 
 
 Stefan constant, a 7.60 X 10~ 15 gm cm" 1 see" 2 C- 4 
 
 First spectral constant, C 0.353 X gm. cm 4 sec~ 3 
 
 Second spectral constant, a' 1.431 cm C 
 
 Rydberg constant, R 3.290 X 10 15 sec" 1 
 
 Charge of the electron, e 4.774 X 10" 10 gm' cm* sec- 1 
 
 Mass of the electron, m 8.8 X 10~ 28 gm 
 
 Avogadro number, N 6.06 X 10 23 gm" 1 
 
 Second Avogadro number, N' 7.29 X 10 15 gm" 1 cm~ 2 sec 2 C 
 
 Some of the quantities in the above list require comment. The 
 Stefan constant "a" is defined by the relation u = a0 4 , where u is 
 the energy density in the hohlraum in equilibrium with the walls 
 at temperature 0. The first and second spectral constants are the 
 constants in the formula 
 
 Cr*l -1-1 
 
 -vp-ij 
 
104 DIMENSIONAL ANALYSIS 
 
 for the distribution of energy in the spectrum. The Avogadro num- 
 ber N is denned as the number of molecules per gm molecule, and 
 its dimensions may be obtained from the formula for it ; N = (no. of 
 molecules per gm) X (mass of molecule / mass of hydrogen mole- 
 cule). Its dimensions are evidently the reciprocal of a mass, and the 
 numerical value is merely the reciprocal of the mass of the hydro- 
 gen molecule. The second Avogadro number N' is denned as the 
 number of molecules per cm 3 in a perfect gas at unit temperature 
 and at unit pressure. We know that this number is independent of 
 the particular gas, and is therefore suited to be a universal constant. 
 Its dimensions are evidently those of vol" 1 pressure" 1 temp, and the 
 numerical value may be found at once in terms of the other 
 constants. 
 
 We have now a list of twelve dimensional constants in terms of 
 which to define an absolute system of units. Since these constants 
 are defined in that system in which there are four fundamental 
 kinds of unit, in general any four of the twelve would sufiice for 
 determining the absolute system of units, but the relations are so 
 simple that there are a large number of cases in which the deter- 
 minant of the exponents vanishes, and the choice is not possible. For 
 instance, C has dimensionally the same formula as he 2 , so that no 
 set of four into which C, h, and c all enter is a possible set. k has the 
 dimensions of cha'" 1 , so that the set k, c, h, and a' is not possible. 
 N' has the dimensions of k" 1 , so that no set of four into which both 
 k and N' enter is possible. The examples might be continued further. 
 The moral is that it is not safe to try for a set of absolute units in 
 terms of any particular group of constants until one is assured that 
 the choice is possible. For instance, one set that might seem quite 
 fundamental turns out to be impossible. It is not possible so to 
 choose the magnitudes of the units that the velocity of light, the 
 quantum, the charge on the electron, and the gas constant all have 
 the value unity. 
 
 By way of contrast, certain sets which are possible may be men- 
 tioned. It will be found that the determinant of the exponents of 
 the following does not vanish; G, c, h, k; G, c, e, k; N, c, h, k; 
 N, c, e, k. 
 
 If one has certain criteria of taste which make certain of the above 
 list of quantities objectionable as universal constants, somewhat 
 startling results may be obtained. Let us decline to consider the 
 quantities R, m, N, and N'. There remain G, c, h, k, a, C, a', and e. 
 
APPLICATIONS TO THEORETICAL PHYSICS 105 
 
 Now it will be found that the last seven of these have the property 
 that it is not possible to choose any four of them whose exponential 
 determinant does not vanish. Hence any set of four quantities in 
 terms of which the absolute system of units is to be determined, if 
 selected from the above list of eight, must include the gravitational 
 constant. This fact is what has made possible Tolman's Principle of 
 Similitude. 8 It seems to me that it is not possible to ascribe any 
 significance to the fact that there exist these relations between the 
 various dimensional constants, but it must be regarded as an entirely 
 fortuitous result due to the limited number of elements of which the 
 dimensional formulas are composed, and their relative simplicity. 
 
 Another interesting speculation on the nature of the absolute 
 units requires comment. G. N. Lewis 4 has stated it to be his convic- 
 tion that any set of absolute units will be found to bear a simple 
 numerical relation to any other possible set of absolute units. The 
 justification of this point of view at present is not to be found in 
 any accurate results of measurement, but is rather quasi-mystical 
 in its character. This point of view led Lewis to notice the remark- 
 ably simple relation between the Stefan constant and the electronic 
 charge and the gas constant, but so far as I know it has not been 
 fruitful in other directions, and I have already indicated another 
 possible significance of the simplicity of the relation. 
 
 Now let us examine this hypothesis~of Lewis's with a numerical 
 example. We have already found the magnitude of the fundamental 
 units which would give the value unity to the gravitational con- 
 stant, the velocity of light, the quantum, and the gas constant. Let 
 us now find what size units would make the gravitational constant, 
 the velocity of light, the gas constant, and the charge on the electron 
 all equal to unity. The work is exactly the same in detail as before, 
 and it is not necessary to write out the equations again. It will be 
 found that the following units are required. 
 
 New unit of mass, 1.849 X 10~ 6 gm 
 
 New unit of length, 1.368 X 10~ 34 cm 
 
 New unit of time, 4.56 X 10~ 45 sec 
 
 New unit of temperature, 8.07 X 10 30 C 
 
 Now the ratio of all these units to the ones previously determined 
 will be found to be 1/29.36. On the face of it, 29.36 does not appear 
 to be a particularly simple number, but on examining the way in 
 which it came into the formulas, it will be found that 29.36 is the 
 
106 DIMENSIONAL ANALYSIS 
 
 approximate value of 4 TT ( - ) , and this somewhat complicated 
 
 \ 15 / 
 
 numerical expression came from Planck 's relation between Stefan 's 
 constant and the spectral radiation constants. In fact, using Lewis's 
 value for a, Planck's formula for h becomes 
 
 It would seem that there will be considerable hesitation in calling 
 a numerical coefficient of this form "simple." If this is simple, it 
 is hard to see what the criterion of numerical simplicity is, and 
 Lewis's principle, at least as a heuristic principle, becomes of ex- 
 ceedingly doubtful value. Lewis 's 9 own feeling is that the coefficient 
 in the above form cannot be regarded as simple, and the fact that it 
 cannot is presumptive evidence that the formula as given by Planck 
 can be regarded only as an approximation, and that sometime a 
 more rigorous theory will be possible in which the number which is 
 at present within the experimental error equal to 29.36 will be 
 expressed in a way which will appeal to everyone as simple as made 
 up of simple integers and TT'S. 
 
 The justification of such speculations is thus for the future. The 
 spirit of such speculations is evidently opposed to the spirit of this 
 exposition, and we are for the present secure in our point of view 
 which sees nothing mystical or esoteric in dimensional analysis. 
 
 BEFERENCES 
 
 (1) A. Einstein, Ann. Phys. 35, 686, 1911. 
 
 (2) J. H. Jeans, Trans. Roy. Soc. 201 (A), 157, 1903. 
 
 (3) J. H. Jeans, Proc. Roy. Soc. 76, 545, 1905. 
 
 (4) G. N. Lewis and E. Q. Adams, Phys. Rev. 3, 92, 1914. 
 
 (5) Meslin, C. R. 116, 135, 1893. 
 
 (6) M. Planck, Heat Radiation, English translation by Masius, 
 p. 175. 
 
 (7) A. S. Eddington, Report on Gravitation, Lon. Phys. Soc., 
 1918, p. 91. 
 
 (8) R. C. Tolman, Phys. Rev. 3, 244, 1914. 
 
 (9) Private communication from Lewis. 
 
PROBLEMS 
 
 1. THE gas constant in the equation pv = RT, has the value 0.08207 
 when the pressure p is expressed in atmospheres, the volume v is the 
 volume in liters of 1 gm mol, and T is absolute Centigrade degrees. 
 What is R when p is expressed in dynes/cm 2 and v is in cm 3 ? 
 
 2. The thermal conductivity of copper is 0.92 cal. per cm 2 per sec 
 per 1 per cm temperature gradient. What is it in B.T.U. per hour 
 per square foot for a temperature gradient of 1 Fahrenheit per 
 foot? (This last is the engineering unit.) 
 
 3. If the numerical value of e 2 /ch is 0.001161 in terms of the gm, 
 cm, and sec, what is its value in terms of the ton, mile, and hour? 
 e is the charge of the electron in E.S.U., c is the velocity of light in 
 empty space, and h is Planck's quantum of action. 
 
 4. The thrust exerted by an air propeller varies with the number 
 of revolutions per second and the speed of advance along the axis 
 of revolution. Show that the critical speed of advance at which the 
 thrust vanishes is proportional to the number of revolutions per 
 second. 
 
 5. Show that the acceleration toward the center of a particle 
 moving uniformly in a circle of radius r is Const v 2 /r. 
 
 6. Show that the time of transverse vibration of a heavy stretched 
 wire is Const X length X (linear density /tension )*. 
 
 7. The time of longitudinal vibration of a bar is Const X length 
 X (volume density /elastic constant)*. 
 
 8. The velocity of sound in a liquid is Const X (density/modulus 
 of compressibility)'. 
 
 9. Given that the twist per unit length of a cylinder varies in- 
 versely as the elastic constant, or as the moment of the applied 
 force, prove that it also varies inversely as the fourth power of the 
 diameter. 
 
 10. There is a certain critical speed of rotation at which a mass 
 of incompressible gravitating fluid becomes unstable. Prove that 
 the angular velocity at instability is independent of the diameter 
 and proportional to the square root of the density. 
 
 11. There is a certain size at which a solid non-rotating gravitat- 
 ing sphere becomes unstable under its own gravitation. Prove that 
 the radius of instability varies directly as the square root of the 
 elastic constant and inversely as the density. 
 
 12. Given that the velocity of advance of waves in shallow water 
 is independent of the wave length, show that it varies directly as the 
 square root of the depth. 
 
 13. The velocity of capillary waves varies directly as the square 
 
108 DIMENSIONAL ANALYSIS 
 
 root of the surface tension, and inversely as the square root of the 
 wave length and the density. 
 
 14. A mass attached to a massless spring experiences a damping 
 force proportional to its velocity. The mass is subjected to a periodic 
 force. Show that the amplitude of vibration in the steady state is 
 proportional to the force. 
 
 15. The time of contact of two equal spheres on impact is propor- 
 tional to their radius. Given further that the time varies inversely 
 as the fifth root of the relative velocity of approach, show that it 
 varies as the 2/5th power of the density, and inversely as the 2/5th 
 power of the elastic constant. 
 
 16. The specific heat of a perfect gas (whose atoms are character- 
 ized by their mass only) is independent of pressure and tempera- 
 ture. 
 
 17. Show that if a gas is considered as an assemblage of molecules 
 of finite size exerting no mutual forces on each other except when in 
 collision the viscosity is independent of the pressure and is propor- 
 tional to the square root of the absolute temperature. 
 
 18. Show that if the thermal conductivity of the gas of problem 
 17 is independent of the pressure it is also proportional to the 
 square root of the absolute temperature. 
 
 19. A periodic change of temperature is impressed on one face of 
 a half-infinite solid. Show that the velocity of propagation of the 
 disturbance into the solid is directly as the square root of the fre- 
 quency, and the wave length is inversely as the square root of the 
 frequency. The disturbance sinks to 1/eth of its initial value in a 
 number of wave lengths which is independent of the frequency and 
 the thermal constants of the material. 
 
 20. A long thin wire is immersed in a medium by which its ex- 
 ternal surface is maintained at a constant temperature. Heat is 
 supplied to the wire by an alternating current of telephonic fre- 
 quency at the rate Q coswt per unit volume. Show that the ampli- 
 tude of the periodic fluctuation of the average temperature of the 
 wire is of the form r= Q d 2 /k f ( w c d 2 /k), where d is the diameter 
 of the wire, k the thermal conductivity, and c the heat capacity per 
 unit volume. If the wire is thin, show by a consideration of the 
 numerical values of k and c for metals that is independent of o> 
 and c and assumes the approximate form 6 = Const Q d 2 /k. 
 
 21. The internal energy of a fixed quantity of a perfect gas, 
 reckoned from Abs. and pressure, is independent of the pres- 
 sure and proportional to the absolute temperature. Hence the inter- 
 nal energy reckoned from an arbitrary temperature and pressure 
 as the initial point is independent of pressure and proportional to 
 the excess of the absolute temperature over that of the initial point. 
 
 22. Why may not the argument of problem 21 be applied to the 
 entropy of a fixed amount of a perfect gas ? 
 
 23. T. W. Richards, Jour. Amer. Chem. Soc. 37, 1915, finds 
 empirically the following relation for different chemical elements 
 
PROBLEMS 109 
 
 ft = 0.00021 A/D 1 - 26 (T m - 50) 
 where 8 = -(] is the compressibility, A is the atomic weight, D 
 
 is the density, T m the melting temperature on the absolute Centi- 
 grade scale. What is the minimum number of dimensional constants 
 required to make this a complete equation, and what are their 
 dimensions ? 
 
 24. Show that the strength of the magnetic field about a magnetic 
 doublet varies inversely as the cube of the distance, and directly as 
 the moment of the doublet. 
 
 25. What are the dimensions of the dielectric constant of empty 
 space in the electromagnetic system of units ? What is its numerical 
 value ? 
 
 26. What are the dimensions of the magnetic permeability of 
 empty space in the electrostatic system of units ? What is its numeri- 
 cal value? 
 
 27. Given a half -infinite conducting medium in the plane surface 
 of which an alternating current sheet is induced. Show that the 
 velocity of propagation of the disturbance into the medium varies 
 as the square root of the specific resistance divided by the periodic 
 time, and the extinction distance varies as the square root of the 
 product of specific resistance and the periodic time. 
 
 28. Show that the self-induction of a linear circuit is proportional 
 to the linear dimensions. 
 
 29. A sinusoidal E.M.F. is applied to one end of an electrical line 
 with distributed resistance, capacity, and inductance. Show that the 
 velocity of propagation of the disturbance is inversely proportional, 
 and the attenuation constant is directly proportional to the square 
 root of the capacity per unit length. 
 
 30. An electron is projected with velocity v through a magnetic 
 field at right angles to its velocity. Given that the radius of curva- 
 ture of its path is directly proportional to its velocity, show that the 
 radius of curvature is also proportional to the mass of the electron, 
 and inversely proportional to the field and the charge. 
 
 31. In all electrodynamical problems into whose solution the 
 velocity of light enters, the unit of time may be so defined that the 
 velocity of light is unity, and two fundamental units, of mass and 
 time, suffice. Write the dimensions of the various electric and mag- 
 netic quantities in terms of these units. Obtain the formula for the 
 mass of an electron in terms of its mass and radius. . . . Problems 
 involving the gravitational constant may also be solved with only 
 the units of mass and length as fundamental. Discuss the formula 
 for the mass of the electron with gravitational units. 
 
 32. The Rydberg constant (of the dimensions of a frequency) de- 
 rived by Bohr 's argument for a hydrogen atom is of the form N = 
 Const m e*/h 3 , where e and m are the mass and the charge of the 
 electron, and h is Planck's quantum of action. 
 
INDEX 
 
 Absolute significance of relative mag- 
 nitude, 20, 21. 
 Absolute units, 99-106. 
 Adams, 95, 106. 
 Airplane, 84, 85. 
 Angle, dimensions of, 3. 
 Atomic forces, 4, 96. 
 Avogadro's number, 103. 
 
 Boussinesq, 9. 
 
 Buchholz, 35. 
 
 Buckingham, 24, 40, 46, 55, 81, 87. 
 
 Bureau of Standards, 81, 87. 
 
 Change ratio, 29. 
 Changing units, 28-35. 
 Complete equation, 37. 
 Corresponding states, 97-98. 
 Cowley, 87. 
 Crehore, 80. 
 
 Debye, 90. 
 
 Determinant of exponents, 34, 43, 59, 
 
 68, 103. 
 
 Dielectric constant, 78. 
 Dimension, 23. 
 Dimensional constant, 14, 15, 36, 52, 
 
 64. 
 
 Dimensional formulas, 23, 24. 
 Dimensional homogeneity, 41. 
 Dimensionless products, 40, 43. 
 Drop of liquid, 3. 
 
 Eddington, 101, 106. 
 Edwards, 87. 
 Einstein, 89, 95, 106. 
 Elastic pendulum, 59-65. 
 Electromagnetic mass, 12, 53. 
 Electronic charge, 103. 
 Electronic mass, 103. 
 Energy density, 77-79. 
 
 Equations of motion, 52. 
 Euler's theorem, 39. 
 
 Falling sphere, 65-67, 84. 
 
 Faraday, 77. 
 
 Fessenden, 27, 80. 
 
 Fitzgerald, 80. 
 
 Fourier, 51, 55. 
 
 Fractional exponents, 46, 47. 
 
 Gas constant, 71, 72, 100, 101, 103. 
 Gravitating bodies, 5. 
 Gravitation constant, 6, 90, 100, 103. 
 Gravitational instability, 90. 
 Gravity wave, 56-59. 
 
 Heat transfer, 9, 72. 
 Hersey, 87. 
 
 Infra red frequency, 89. 
 Instruments, 87. 
 
 Jeans, 40, 46, 90, 93, 106. 
 Johnson, 80. 
 
 Levy, 87. 
 
 Lewis, 95, 96, 105, 106. 
 
 Lodge, A., 35. 
 
 Lodge, O., 80. 
 
 Logarithmic constant, 36, 73-75. 
 
 Lorentz, 95. 
 
 Meslin, 98, 106. 
 Millikan, 67. 
 Models, 82, 85, 86. 
 
 National Physical Laboratory, 81. 
 
 Nernst, 74. 
 
 Number of fundamental units, 11, 24, 
 
 48, 54, 55, 61, 63, 66, 68. 
 Numerical coefficients, 88-90. 
 
112 
 
 INDEX 
 
 Pendulum, 1, 81. 
 Physical relations, 91, 92. 
 ,Pi theorem, 36-47, 40, 49, 57, 75. 
 Planck, 96, 99, 100, 106. 
 Pressure of gas, 70-71. 
 Primary quantities, 18-20, 23. 
 Principle of Similitude, Eayleigh's, 
 
 10. 
 
 Principle of Similitude, Tolman 's, 105. 
 Product of powers, 8, 22, 39. 
 Projectile, 86. 
 
 Quantum, 91, 95, 99, 100, 103. 
 
 Eadiation from black body, 93. 
 Eayleigh, 4, 9, 10, 11, 24, 56, 69, 72, 
 
 80, 87. 
 
 Eeduced variable, 97, 98. 
 Eesistance to submerged motion, 82- 
 
 86. 
 
 Eeynolds, 87, 101. 
 Eiabouchinsky, 10, 11, 24. 
 Eichardson, 93. 
 Eouth, 46. 
 Eiicker, 26, 80. 
 Eydberg constant, 103. 
 
 Scattering of sky light, 69. 
 Secondary quantities, 19-20, 22, 23. 
 Spectral constants, 103. 
 Stefan, 95, 103, 105, 106. 
 Stiffness of beam, 67-69. 
 Stokes, 65, 84. 
 
 Temperature, dimensions of, 10-11, 24, 
 
 71, 72, 73. 
 
 Thermal conductivity, 92. 
 Thomson, James, 29. 
 Thompson, S. P., 27, 46. 
 Time constant of electric circuit, 76. 
 Tolman, 24, 26, 27, 105, 106. 
 Transcendental functions, 42, 44, 45. 
 "True" dimensions, 24, 79. 
 
 Ultimate rational units, 105. 
 Uniqueness of dimensions, 25, 26. 
 
 Velocity of light, 12, 53, 100, 103. 
 
 Webster, 35. 
 Williams, 26, 46, 80. 
 Wilson, 87. 
 
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