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/T 2 ], 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/t] and [l/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] = [ML 2 T~ 2 ] 

 will be found to be the dimensional equation for energy, and [ML 2 T^ 2 ] 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, 



Q = 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 x , M x , T 1} we have to find the value of 

 L x /L, M x /M , 1\/T, 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 x =Ll, M x = Mm, T 1 = Tt, and if Qi be the new quantity number, 



Q x =CLfM b Tf, 



= CL a l a M b m b T c t c = Ql a m h t c , 



or the conversion factor is [l a m b t c ], a quantity precisely of the same form as 

 the dimension formula [L a M b T c ]. 



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 s = v t + ^at 2 . The corresponding dimensional 

 equation is [L] = [{L/T)T] + [(L/T 2 )T 2 ], each term reducing to [L]. 



Dimensional considerations may often give insight into the laws regulating 

 physical phenomena. 2 For instance, Lord Rayleigh, in discussing the intensity 

 of light scattered from small particles, in so far as it depends upon the wave- 

 length, reasons as follows : 3 



The object is to compare the intensities of the incident and scattered ray; for these will 

 clearly be proportional. The number (i) expressing the ratio of the two amplitudes is a 

 function of the following quantities :—V, the volume of the disturbing particle; r, the 

 distance of the point under consideration from it ; X, the wavelength ; c, the velocity of 

 propagation of light ; D and D', the original and altered densities : of which the first three 

 depend only on space, the fourth on space and time, while the fifth and sixth introduce the 

 consideration of mass. Other elements of the problem there are none, except mere numbers 

 and angles, which do not depend upon the fundamental measurements of space, time, and 

 mass. Since the ratio i, whose expression we seek, is of no dimensions in mass, it follows 

 at once that D and D' occur only under the form D : D' , which is a simple number and may 

 therefore be omitted. It remains to find how i varies with V , r, \, c. 



Now, of these quantities, c is the only one depending on time ; and therefore, as i is of no 

 dimensions in time, c cannot occur in its expression. We are left, then, with V , r, and X ; and 

 from what we know of the dynamics of the question, we may be sure that i varies directly as 

 V and inversely as r, and must therefore be proportional to V -j- XV, V being of three di- 



2 Buckingham, E., Phys. Rev., vol. 4, p. 345, 1914; also Philos. Mag., vol. 42, p. 696, 1921. 



3 Philos. Mag., ser. 4, vol. 41, p. 107, 1871. See also Robertson, Dimensional analysis, 

 Gen. Electr. Rev., vol. 33, p. 207, 1930. 



SMITHSONIAN PHYSICAL TABLES 



