INTRODUCTION XXX111 



CONVERSION FACTORS AND DIMENSIONAL FORMULAE 



For the ratios of length, 

 mass, time, temperature, dielectric constant and permeability units the small 

 bracketed letters, [f], [w], [f\, [0~], M, and \jjl] will be adopted. These symbols 

 will always represent simple numbers, but the magnitude of the number will 

 depend on re relative magnitudes of the units the ratios of which they repre- 

 sent. When the values of the numbers represented by these small bracketed 

 tetters as well as the powers of them involved in any particular unit are known, 

 the factor for the transformation is at once obtained. Thus, in the above ex- 

 ample, the value of / was 1/3, and the power involved in the expression for volume 

 was 3; hence the factor for transforming from cubic feet to cubic yards was P 

 or 1/3 3 or 1/27. These factors will be called conversion factors. 



To find the symbolic expression for the conversion factor for any physical 

 quantity, it is sufficient to determine the degree to which the quantities length, 

 mass, time, etc., are involved. Thus a velocity is expressed by the ratio of the 

 number representing a length to that representing an interval of time, or \_L/T~], 

 and acceleration by a velocity number divided by an interval-of-time number, 

 or [L/r 2 J, and so on, and the corresponding ratios of units must therefore enter 

 in precisely the same degree. The factors would thus be for the just stated cases, 

 \_l/f\ and {J/t 2 ^. Equations of the form above given for velocity and acceleration 

 which show the dimensions of the quantity in terms of the fundamental units 

 are called dimensional equations. Thus [_E] = \_M L 2 T~ 2 ~\ will be found to 

 be the dimensional equation for energy, and \_MLrT~-~} the dimensional formula 

 for it. These expressions will be distinguished from the conversion factors by 

 the use of bracketed capital letters. 



In general, if we have an equation for a physical quantity, 



<2 = CL a M b T c , 

 where C is a constant and L, M, T represent length, mass, and time in terms 

 of one set of units, and it is desired to transform to another set of units in terms 

 of which the length, mass, and time are L n M n T n we have to find the value of 

 L,/L, MJM, TJT, which, in accordance with the convention adopted above, 

 will be /, m, t, or the ratios of the magnitudes of the old to those of the new units. 



Thus L, = LI, M, = Mm, T , = Tt, and if Q t be the new quantity number, 



Q, = CLfM*T t % 



= CL a l a M b m b TH c = Ql a mH r , 



or the conversion factor is {l a mH c ~\, a quantity precisely of the same form as the 

 dimension formula [_L a M b T <r \. 



Dimensional equations are useful for checking the validity of physical equa- 

 tions. Since physical equations must be homogeneous, each term appearing in 

 them must be dimensionally equivalent. For example, the distance moved by 

 a uniformly accelerated body is 5 = v t + \at 2 . The corresponding dimensional 

 equation is [L] = \_{L/T)T] + [(L/r 2 )r 2 ], each term reducing to [Z-]. 



Dimensional considerations may often give insight into the laws regulating 

 physical phenomena. 1 For instance Lord Rayleigh, in discussing the intensity 



1 See "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations." 

 E. Buckingham, Physical Review, (2) 4, 345, 1014; also Phil. Mag. 42, 696, 1021. 

 2 



