156 J- C. Decius. [J- F. I. 



An acceptable solution of this system is 



3- = 11 , (2.5) 





where Ot = l -zri ) is a constant, so that, y being actually an invariant, 

 V ok" / 



which can be assumed to have the numerical value of unity, the ac- 

 ceptable form for the combination of magnitudes is 



m 



x = n (x*)°t (2.6) 



with the transformation law: 



X = exp [oa:X*]x. (2.7) 



It is immediately perceived that the numbers au characterize the kind 

 of the derived quantity and that they may be regarded as the com- 

 ponents of an w-dimensional vector in the basis in which the /th com- 

 ponent of the ^th fundamental quantity is the Kronecker symbol, Si*. 

 In other words, the general symbol, o,', is the ordinary exponent oc- 

 curring in the conventional dimensional formula for the ith kind of 

 variable relative to the jth fundamental kind of quantity; for example, 

 let i = \ designate energy; j = 1, 2, 3 correspond to mass, length, and 

 time, respectively, then Ci' = 1, Og' = 2, 03' = — 2. 



Now let x^ be a set of derived magnitudes expressible in terms of the 

 basic set, x'. Then the transformation laws are : 



x< = exp \_ajn^'~\x\ ^2 g) 



i = \,2, ■ ■• n; j = 1,2, ■ • ■ m. 

 Now let 



F{x') = 



be the expression of an unknown physical relation in terms of the magni- 

 tudes, x\ Equations 1.2, 1.3, and 1.5 require that the relation be ex- 

 pressible in terms of 3'*, the solutions of 



. . .dy'' . 

 ' dx' 

 or 



dy'' 



ai 



' d In -r' 

 for J = 1, 2, ■ • • w 

 Solution of Eq. 2.9 gives 



= (2.9) 



y" = 11 {xY', (2.10) 



k = 1,2, ■ ■ ■ n — r, 



