Entropic Contributions to Mortality and Aging 323 



Then taking logarithms and collecting the constant terms into lumped constants 

 the relation was obtained 



G(0 — log <7(/) - « -f /> AA (17) 



where G(t) is called the Gompertzian. 



The linear approximation to A^ is satisfactory for a first-order description 

 of the relation of injury to age and to dosage of agents that cause permanent 

 injury, such as x-rays (7). 



The fact that /i also tends to decrease with age does not alter the generaliza- 

 tion made previously (7) that the Gompertzian is a linear measure of mean 

 physiologic state. The entire exponent 



2^2 ~ 2D 

 can be expanded, yielding an expression of the form 



^=^a„ + a,A?. + a.,Aft (18) 



Furthemiore if the mechanism depends on several variables, the same expansion 

 procedure again yields an exponent term that is a linear function of the dis- 

 placements of all of the parameters. Thus, within the range of parameter values 

 that occur in the course of natural aging, the Gompertzian is an approximate 

 linear measure of the mean physiologic state. 



V. DESCRIPTION OF THE «-DIMENSIONAL 

 FLUCTUATION PROCESS 



The consideration of the general 77-dimensional case will take as its starting 

 point the empirical description of the w-variate process in tenns of its moments. 

 The observational data consist of a large number w of sets of observations 

 on one individual or on w indistinguishable individuals, where each set is a 

 measurement of each of n variables at a given time. The first moments are the 

 n mean values, 



1 m 



x,o = - 2 X,, (19) 



m J = 1 



The second central moments are the covariances 



m 



Vi, = — (20) 



The covariances are related to the standard deviations and correlation coefficients 

 as 



Vik = f^i (^k Pik (21) 



where c, are the standard deviations and p^,^ is the total correlation coefficient 

 between the /th and ^th variables. The covariance matrix 



nhi • ' • I'm 



■ ... p 



III '^ II n 



(22) 



