f 



96 Lord Rayleigh on the Pressure developed 



Writing fi 3 = z, we have 



which may be expressed by means of V functions. Thus 



T=R oV / (^)- r:i r^ i}= ' 91468:Ro v / WP)- • (6) 



According to (4) U increases without limit as R diminishes. 

 This indefinite increase may be obviated if we introduce, 

 instead of an internal pressure zero or constant, one which 

 increases with sufficient rapidity. We may suppose such a 

 pressure due to a permanent gas obedient to Boyle's law. 

 Then, i£ the initial pressure be Q, the work of compression 

 is 47rQR 3 log (R fR), which is to be subtracted from (3). 

 Hence 



U 2 = 

 and U = when 



2P/IV 



3 P VR 3 



■)- 



2Qiy 



R 



log^; • 



(7) 



P(l-s) + Qlogs=0, 



...-.- (8) 



z denoting (as before) the ratio of volumes R 3 /R 3 . Whatever 

 be the (positive) value of Q, U comes again to zero before 

 complete collapse, and if Q>P the first movement of the 

 boundary is outwards. The boundary oscillates between two 

 positions, of which one is the initial. 



The following values of P/Q are calculated from (8) : 



z. 



P/Q. 



z. 



P/Q. 



1 



1000 



6-9147 



1 



arbitrary 



1 



100 



4-6517 



o 



0-6931 



1 



10 



2-5584 



4 



0-4621 



] 



4 



1-8484 



10 



0-2558 



1 



2 



1-3863 



100 



0-0465 



1 1 



arbitrary 



1000 



0-0069 



Reverting to the case where the pressure inside the cavity 

 is zero, or at any rate constant, we may proceed to calculate 



