the Method of Dimensions. 713 



12. Remarks on the Second Method of treating 

 Dimensional Constants. 



This second mode of treating ordinary dimensional 

 constants may be followed in any other similar case ; and 

 while it is usually of no practical value because the more 

 familiar procedure is also more natural and is at least as 

 easy, the second method is, nevertheless, rather instructive. 



When we set [p] = [m/ -3 ] = [1] and derive mass from 

 length by setting [m]=[Z 3 ], we reduce the number of 

 fundamental units by one ; but we do not reduce the 

 number of arbitrarily chosen standards. For what we 

 have done is to adopt as the unit of mass, not the mass 

 of a particular body such as the international kilogram, 

 but some fixed multiple of the mass of unit volume of 

 the liquid under consideration. The number of standards 

 is the same as before, but we have changed the nature 

 of the arbitrary choice and shifted the responsibility of 

 preserving a unit from a certain piece of metal to a certain 

 liquid and a certain length or volume standard. 



This procedure may be compared with that for universal 

 constants. In treating problems which involve the operation 

 of the law of gravitation, it is necessary to recognize the 

 existence of this law, and we usually do it, as in the first 

 part of section 8, by putting the gravitation constant into 

 the initial equation and taking its dimensions from the law 

 of gravitation. When we pursue the second plan and 

 eliminate one fundamental unit, we dispense with one 

 concrete standard, to be sure, but only by throwing the 

 responsibility for furnishing a substitute on the universe, — 

 the properties of our universe provide the required standard 

 In the sense that we are actually confined to operations in a 

 particular universe which has fixed properties, \ve> have 

 dispensed with one arbitrary selection of a standard ; but 

 it is only because the arbitrary choice has already been 

 made for us by the Creator. If it be admitted that the 

 universe might have been created differently, the choice 

 must still be regarded as arbitrary, even though we did 

 not make it. 



Similar remarks might be made in other instances in 

 which some property, not of a particular body but of 

 the universe, is made to provide a standard for fixing a 

 unit. The number of concrete standards to be preserved 

 by human agency is diminished, but not the number of 

 separate standards required, except in the souse that the 

 one universe may furnish standards for various kinds ot 



Phil. Mag. S. 6. Vol. 12. No. 251. Tor. 1921. 3 B 



