22/9 THE EFFECTS OF CONSTANCY 



For at the first field, a proportion p will be terminal, and 

 q (= 1 — p) will not. Of the latter, at the second field, the pro- 

 portion p will be terminal and q not ; so the total proportion stable 

 at the second field will be pq, and the number still unstable q 2 . 

 Similarly the proportion becoming terminal at the u-th field will 

 be pq*- 1 . So the average number of trials made will be 



p + 2pq + 3pq 2 + • • ■ + upq"- 1 + • . . _ 1 

 p + pq + pq 2 + . . . + pq"- 1 -f . . . p 



Temporary independence 



22/8. The relation of variable to variable has been treated by 

 observing the behaviour of the whole system. But what of their 

 effects on one another ? Thus, if a variable changes in value, can 

 we distribute the cause of this change among the other variables ? 

 In general it is not possible to divide the effect into parts, 

 with so much caused by this variable and so much caused by that. 

 Only when there are special simplicities is such a division possible. 

 In general, the change of a variable results from the activity of 

 the whole system, and cannot be subdivided quantitatively. 

 Thus, if dx/dt = sin x + xe y , and x = \ and y = 2, then in the 

 next 0-01 unit of time x will increase by 0-042, but this quantity 

 cannot be divided into two parts, one due to x and one to y. 

 Only when some special simplicity exists can the whole effect be 

 represented meaningfully as the sum of two effects, one from each. 

 Though not uncommon in theoretical physics, such simplicities are 

 rare in biological systems. 



22/9. Given a state-determined system, its field, a line of 

 behaviour in it, and a particular portion P of the line; given also 

 that x p is a part-function, then the following are equivalent, in 

 that the truth (or falsity) of any one implies the truth (or falsity) 

 of all the others: 



(1) x p is constant (inactive); 



(2) dxp/dt = 0; 



(S)fp{. . ., x 0t . . .) =0; 



(4) Xp = #jj independently of t; 



(5) Fp(x°; t) == x°, with such values of t as do not take the line 



out of P; 



277 



