the Method of Dimensions. Ill 



Still another step may be taken by using an independent 

 unit of viscosity as well as one of force. The dimensional 

 equation for viscosity has then to be written 



[>] = [O'fi-Hi 



iii which C is a new derivation constant with the dimensions 



[C] = ^f-HH-^. . ... . (57) 



The initial equation is now to be written 



F(R, D, S, />,/., 0, C) = 0, . . . (58) 



with 7 quantities, to be measured in terms of 5 independent 

 units ; aud the solution by the II theorem is 



__ = C^(_-^J, .... (59) 



which reduces to (50) as soon as C and C are fixed. 



The foregoing illustrates the truth of Bridgman's remark* 

 that " the number of fundamental units is largely a matter 

 of convenience/' So far as concerns the application of the 

 principle of dimensional homogeneity,' we may, if we like, 

 use an arbitrary independent unit for every different kind of 

 quantity involved in the problem under investigation, or even 

 for each separate quantity, if there are several of one kind. 

 But if we do so, we must recognize the facts in some other 

 way, — the facts being that some of the units are derivable 

 from others in accordance with laws, which are operative 

 and without which the phenomenon would be different from 

 what it really is. We may pretend, for a time, that there 

 are no such dimensional relations, but if we take the pretence 

 seriously, we get into difficulty. 



It is now obvious that there is no definite answer to the 

 question : " How many fundamental units are required in 

 treating a given problem ? " The number is not determined 

 by the phenomenon under investigation but by the initial 

 equation used to describe it. The physical nature of the 

 phenomenon does not fix either the number n of quantities 

 to be used in the description, or the number k of fundamental 

 units required for expressing their values. But it does fix 

 their difference (n — k), i. e. the number of the independent 

 variables II which must appear in the final equation 



/(n l5 n 2 , ...n„_ A .) = o, .... (60) 



to which the principle of dimensional homogeneity leads. 

 * Pliys. Rev. vol. viii. p. 428 (Oct. 1916). 



