118 



It la obvious that the maximum value of (8) v/111 

 occur when a la less than the radius at which -SS. is a 

 naxlmuEi, but it is not Immediately obvious that (G) Is a 

 n.exiimin when a Is a minimum, since In this region increas- 

 inn a decreases p(a) but increases ~ . Ilov/ever, if «?e 

 assTome p(a) to follow an adiabntlc law pa •* constant and 

 neglect the p term in equation (1) of Appendix 1, we have 



W = G ^%ln) - "3^ Pl?MaL-|l^ (10) 



If (9) and (10) are assi^Jiiad it can be verified that (8) 

 increases v;ith decreasing a all the way dovvri to a^j^^j^, 

 whenever 'f> 1, as is always the case for perfect gases o 



If r is largo enough .-lo that the last terrri of (3) 

 can be noglocted v/e have 



(p - pj dt ^^ 



-^§ (11) 



The maximtim of (11) Viflll come when t and t are two tines 



for which a'' ^^ is a maxlmvm, in the expanding and 

 d^ I 



contracting stages respectively. From equation (1) of 

 Appendix 1 we have, neglecting G, 



>/da^ ^ Wa 



^'(^) = ^^ » ^ :2?a^ f.'.2) 



2'Tp 'S -p 



This is a maximum when 



