May, 1948.] Criterion for Scaling Model Experiments. 159 



sary and sufficient condition that scaling be possible is that 



n,-[-r, = r„. (4.1) 



To prove the necessity of the condition, suppose n, + rj 9^ r.s- 

 Since r,/ cannot exceed n, + r/, we must suppose n, -\- rj > r,/ or 

 n, + Uf — r,f > rif — r/, but since 



t,j = n, + tif - r.f 

 and 



t, = n, - r, 

 we should have 



t„ > tf 



so that there would be either or both of the following types of y*; 



1 . those containing arbitrarily scaled variables only, 



2. those containing arbitrarily scaled and fixed, but not unrestricted 

 variables, 



neither of which types can be allowed. 



To prove that the condition is sufficient, distinguish the two possible 

 cases: 



1. r,f = r 



2. r,f < r. 



In case (1), that part of A,f with the r,/ X r,f non-vanishing de- 

 terminant can be taken as ^ i ; consequently (since such a determinant 

 can be found which contains columns from all the arbitrarily scaled 

 variables) all the arbitrarily scaled variables will appear in the set 

 *"-'+", the fixed variables (if any) not contained in A 1 will be expressible 

 in terms of fixed variables in Ai only, and there will be just enough 

 restricted variables to complete the set of (« — r)y'''s with one such 

 variable to each y^ involving arbitrarily scaled variables. 



If, on the other hand, case (2) obtains, the same proof will hold (with 

 the modification that some unrestricted variables will appear in the 

 set x"-^^) provided only that the r,/ X r,/ non-vanishing determinant 

 contained in A,f is contained in some r X r non-vanishing determi- 

 nant of A. 



That this latter requirement is always satisfied follows, for example, 

 from the definition of rank in terms of linear independence as shown by 

 Birkhoflf and MacLane (4). This completes the proof of the scaling 

 criterion. 



In concluding this section it should be remarked that the criterion 

 furnished by Eq. 4.1 is independent of the basis of fundamental units 

 adopted, since the only quantities appearing are numbers of variables 

 and the rank of various sub-matrices, taken by columns, which are, of 

 course, all invariant under the group of homogeneous linear transforma- 

 tions on the a/, A -^ A = TA, \T\9^ 0, which corresponds to all con- 

 ceivable choices of a basic set of fundamental units. 



