158 J- C. Decius. 



Bi, the remaining rows Bi. Then Eq. 3.1 is equivalent to 



^2^2 + AlB, 



[J. F. I. 



Ao 



B, 



B, 



= 0. 



(3.2) 



But since the rows of Ao are Unearly dependent upon the rows of 

 |/l2i.(4i||, any solution of 



AiBi = - AiBi (3.3) 



automatically makes A o 



5i" 



vanish. Since | ^ 1 1 ?^ 0, ^ r' exists and 



Bi = - Ar'AiB^ 



(3.4) 



The elements of Bi are now completely arbitrary, and it is convenient, 

 as will be shown below, to let B2 equal the negative unit matrix, whence 



5, = Ar'A, 



(3.5) 



Combination with Eqs. 2.10 and 2.11 gives the result that the 

 invariants are : 



y" = n — -— - 



(3.6) 



that is, since the y* are to be held constant, the first (n — r) variables 

 must be individually proportional to a set of products involving only 

 the last r variables which may be changed arbitrarily. 



4. THE SCALING CRITERION. 



In the application of these results to the design of model experiments 

 certain difficulties arise which have not previously been considered in 

 the general case. It frequently occurs that some of the variables essen- 

 tial to a given problem may not be readily changed with scale. As 

 examples, the acceleration of gravity, or, in certain cases, even the 

 properties of liquid and solid media cannot be readily altered in general 

 so as to satisfy Eqs. 3.6 and must therefore be regarded as fixed. In 

 order to set up a general criterion for determining whether scaling is 

 possible, it will be useful to classify the variables in three types: those 

 which are fixed, those which are to be arbitrarily (and independently) 

 scaled, and those which are unrestricted. We shall use the subscripts/ 

 and 5 to designate the first two types respectively; r„ r/, r^s will stand 

 for the rank of the sub-matrix of A corresponding to all the variables of 

 types s, /, and both j and /, respectively: ts, t/, tsf will stand for the 

 number of 31* which involve variables of type 5,/, 5 and /or/, respectively; 

 and M„ «/ are the numbers of variables of type 5 and of type/. A neces- 



