May, 1948.] Criterion for Scaling Model Experiments. 155 



generated by an integration of the differential operation signified 

 by (1.3). 



The important consequence for the present purposes is that if any 

 function of the variables x\ say 



F{x\ ■ ■ ■ X") = 0, (1.4) 



is known to be invariant under the group represented by (1.1), then it 

 is always possible to express (1.4) in terms of a complete set of funda- 

 mental invariants y\ >'^ ■ • • y"-', which are determined by 



Uy = (1.5) 



as functions of the x'. Thus 



Fix\ ■ ■ ■ X") = Hy\ ■ ■ ■ y"~') = (1.6) 



and the number of independent variables has been reduced by r, which 

 is equal to the rank of the matrix | 77- ) • 



2. DETERMINATION OF THE INVARIANTS OF A GENERAL PHYSICAL EQUATION 

 UNDER TRANSFORMATIONS OF THE UNITS. 



The nature of the number, x', resulting from a physical measurement 

 of the jth kind of physical quantity is such that it is determined only 

 within a transformation of the type : 



X' = exp [X']x', (2.1)' 



— CO < X' < + 00. 



If it is desired to deduce the possible kinds of functions of the magni- 

 tudes of a basic set of physical quantities, which give the magnitude of a 

 resultant physical quantity derivable from the basic ones, it is possible 

 to apply the principles of Section 1 in the following manner. The 

 resultant magnitude itself must obey a transformation law of the same 

 form as (2.1) 



X = exp Qf(XO> (2.2) 



in which the value of/ may be taken as zero when X' = 0, all/. Then 

 Eq. 1.3 requires that the relation between x and x' be expressible in 

 terms of a function y determined by 



The a'' being arbitrary constants, (2.3) is equivalent to the m relations 



-^^=-(^)-Jy- (2 4) 



alnx* VaXVoalnx' ^ ■ ' 



k = 1, 2, ■ • ■ m. 



' It is clear that the range of the \- indicated will contain values satisfying all the group 

 postulates of Section 1. 



