May, 1948.] Criterion for Scaling Model Experiments. 157 



where the J,* are constants determined by 



a/6.^ = (2.11) 



and where r is the rank of the matrix ||oy'H. 



This result is, of course, the well-known 'V-theorem" as described 

 by Buckingham (2) or Bridgman (3). The virtue of having used the 

 transformation theory lies chiefly in the following two points : 



1. The "alias-alibi" duality of any transformation immediately al- 

 lows the conclusion that although the deductions were based on the 

 assumption that the "absolute" magnitude of the physical quantities 

 were unchanged during the transformation, the invariants, y'', will 

 certainly satisfy the equation 



$(3,*) = 



throughout a transformation in which the fundamental magnitudes are 

 fixed and the magnitudes of the x^ are "actually" changed but in such a 

 way as to keep the y'' constant. In other words, empirical physical 

 information satisfying 



F{x^) = 



on a given scale may be used to predict a physical relation on any other 

 scale, provided only that the x" are varied in such a way as to maintain 

 the constancy of the y''. 



2. The simplification of a physical problem brought about by the 

 reduction of the degrees of freedom, n, by the number r, is not mathe- 

 matically different in kind from the complete solution of the problem 

 (the determination of the form of the unknown function, F) which is 

 obtained when (n — 1) independent transformation parameters are 

 obtained. 



3. A FORMAL SOLUTION FOR *,'. 



Equations 2.11 are equivalent to the matrix equation 



AB = 0, (3.1) 



where A = l|oy'||, B = l|i.*||. The rank of A is r; by mere rearrangement 

 of rows and columns, the non-vanishing determinant can be made to 

 appear in the upper right-hand comer of A (under the last r columns 

 and the first r rows; it will be supposed that the subscript is the row 

 index, the superscript the column index). Call this portion of the 

 matrix Ai, call the first (n — r) columns and first r rows A 2; let the last 

 {m — r) rows be called Ao (ii r = m, Ao does not exist). 



Make a corresponding partition of B, calling the first (m — r) rows 



