42 YOS, BADE AND JEHLE 



total mean square deviation ^s ((Ws — (Ws)av)^)av is the sum of the mean 

 square deviations of the abscissa and of the ordinate and should therefore be 

 > 2 • 2 = 4 or »2 -2 = 4. As regards more general manifolds, one may call 

 a distribution "effectively" a d dimensional one if the manifold average 

 y^ g ((Ws — (Ws)av)^)av is at least d times the discrimination limit 2 (cf. (15a)), 

 i.e. 2d, or better, large compared with 2d (condition 2). 



The caption of Fig. 4 illustrates the procedure of chopping up the interval 

 < I 5 I < 00 into a finite number of dimensions ("abscissa", "ordinate", etc.) 

 by forming combinations of mean square deviations (Ws — (Ws)av)^ so that 

 the distribution over those combinations is not too strongly correlated and that 

 to each combination (dimension) corresponds at least the effective mean square 

 deviation 2. This procedure obviously implies that each dimension comprises 

 at least one unit of the | 5 | scale (condition 3). In a hypothetical classical 

 oscillator limit case only the term 5 = (cf. (14) or (14a)) would be nonvanish- 

 ing and one would just get a one dimensional distribution. 



As adjacent Ws and Ws+i are correlated, it is evident that only manifolds 

 whose oscillator frequencies are distributed over wide spectral regions, and 

 whose polarizabilities in these diverse spectral regions are sufficiently large, 

 can be truly many dimensional. 



Such limits on the number of effective dimensions which a manifold may 

 be said to have do not provide for an unequivocal definition of the dimensional- 

 ity of the manifold. Fortunately that is not needed, and one can unambiguously 

 define the "degree of specificity", using Fig. 4 as an illustration. 



A many dimensional distribution implies many more or less independent re- 

 arrangement inequalities of which (10c) is made up as a sum over regions of the 

 variable s. In order to discuss the implications of such a many parametric dis- 

 tribution one can compare it with a hypothetical one dimensional distribution. 

 In Fig. 4, the two dimensional distribution of Fig. 3 is compared with a one 



~-^ 



V2 



Fig. 4 

 Abscissae = |(Wo - {Wo)av)1 

 Ordinates = {2(^1 - <Wi)av)- + 2(^2 - <W2)av)- + ■ ■ ■]' 



