324 George A. Sacher 



is a non-negative quadratic form, as is the correlation matrix 



iPn • • • Pin 



R 



(23) 



IPnl 



Pnn 



Given the covariance matrix V, the frequency distribution of the displacements 

 in n dimensions is determined. In the case that V is positive-definite, so that 

 the rank is equal to the order n, the distribution is (9) 



p(xj_, • • • , x„) 



1 



T^exp 



2K tk ^''"'''"''' 



(24) 



wherQ V is the determinant of V and K,j. is the cofactor of v,^. in V. The 

 coefficients, K,JF in the exponent of equation 24 are terms in the inverse of the 

 covariance matrix, 



l7{U=Y-i=A={A,,} 



V 



(25) 



If V is positive semi-definite, the rank r is less than the order n, and the 

 frequency distribution is an /--dimensional distribution in /• independent linear 

 functions of the x^ (9) 



The case that the rank of the matrix of covariances is less than the order is 

 frequently encountered in the initial description of biological systems in terms 

 of the variables of direct observation. 



VI. FLUCTUATION, ENTROPY, AND INFORMATION 



The entropy of a system at equilibrium has, in classical thermodynamics, a 

 precise value, i'oCvjg, • • • , x„q) where the .y,o are the values of the state variables 

 at equilibrium. However, if thermal agitation or other disturbance causes small 

 displacements of the state variables, the entropy decreases by an amount, AS. 

 Expanding AS in a Taylor series in terms of the displacements, .y, (10), we 

 obtain 





(30) 



