the Method of Dimensions. 707 



eliminate one unit and so relegate one of our three primary 

 standards to the position of a secondary or working standard. 

 Thus we shall have reduced the number of fundamental 

 units from 3 to 2, and the number of quantities to be 

 measured from 5 to 4, the difference remaining unchanged 

 so that there will be the same number of TI's as before. 



If we elect to retain the metre and kilogram as primary 

 standards and eliminate time as an independent unit, we 

 have by (28) [t~] = [m-W]. The dimensions of the 

 other quantities are therefore 



[>] = [n^l-^l M = [ m Z~3], 1 



[D] = [/] . [>] = [wOT- 6 *]. J ' ' ^ J) 



And if, with these dimensions, we apply the II theorem to 

 equation (27), the result is 



whence 



-= Vpf^), (31) 



or, if viscosity plays no appreciable part, 



a = N v>. (32) 



Upon comparing the result contained in equations (31) 

 and (32) with the former result contained in (25) and (26), 

 it is seen that the two results are identical as soon as it is 

 recognized that 7 is a constant. For practical purposes 

 and so long as we are restricted to observations in our 

 present universe, the two results are the same ; but the 

 earlier and more general result contains some further 

 information which may be or may not be of interest. For 

 it shows how the frequency would be changed if we could 

 pass into another universe where 7 had a different value 

 characteristic of that universe. It also shows how a would 

 vary if the one universe were transformed into the other 

 by a continuous variation of 7. If, in passing from one 

 point of space or time to another, the value of 7 varied 

 continuously, equations (25) and (26) would remain 

 applicable, 7 being tl.en not a permanent constant of 

 any one universe but a variable point function in space 

 and time. 



