80 



Mr. C. Chree on the Theory of 



The integral appearing here is of the well-known type 



appearing in the theory of probability. 



To determine the mode in which v commences to vary 

 from its initial value (28), expand the several exponentials 

 and retain only lowest powers of t. In this way we find 



_ ft? A 2 — lik— a _ c 2 a x A + c 2 b 2 A. 2 + b 2 a — \a l b x ^ 2 ,_ 

 2b 2 K + a l 26 2 A-f a x ' \ ) 



Thus the initial departure of v from its steady value depends 

 on t 2 , and so is extremely slow compared to that of V. 



If we neglected the terms a + a ± v -\-bxV , we should get in 

 place of (29), 



^2 



C 2 



v =£ r{ A-Bt) + i±B, 



b.,B 



2b 



2b, 



(<-$y 



r- 



-A, 



A\2 



('-!) 



dt. 



(31) 



This shows that v is always in excess of the value 



given it on the ordinary or, as we shall term it, equilibrium 

 theory. 



On the same simplified hypothesis we replace (30) by 



•=^A(1-Vtf). 



(32) 



In obtaining (29) and (30) we have tacitly assumed that 

 (B/A)t is small as well as t. If the rate of change of V is 

 very rapid, then even when t is very small (32) had better 

 be replaced by 



^wA'-^i'-H 1 )}- • • (33) 



This result may be applied in this case up to the time 



t=A/B 



at which V vanishes. For the value of v when V vanishes, 

 it gives 



1 - • (34) 



^wA^l^ A ' 



BJ* 



a 2 not zero, but V constant. 

 §13. For shortness write (14) in the form 



^ + rt 2 (^-^)0 , + ^) = 0, 



(35) 



