112 



W A^(^) 



-^^- 



If this Is nov; inoorted into (15) and the right of (15) ex- 



(o) 

 pressed in terns of R, , and of a and its derivatives, by 



(22) and (23), a set of second order differential equations 



R.^t: 



for the R'' ^(t) result. All the equations v/ill be horao- 



ceneous except that for A-O; since at t»0, R « ^^' =0, 



' at 



this means that 



r^) 



for A> 



/. 



For 



/-o 



we obtain the equation 



p da d R, 



^ -dT ^ 



(<>) 



da\ dR ^ 



^ 3(hf) -gi^ (3 ^ 



da d a 

 dt-^ 



-r a "^t^ 



,« 



„ ->- 1 - /daN , a da da 



It can be verified that (8) is an intecral of (29), so that 



the solution (9) obtained by the simpler method is checked. 



Another check on (29) is obtained by noting that the left 



/'') da 



side reduces to aero if we set R 



to a shift in the starting time. 



Tt ' 



corresponding 



The same procedure can bo used to c^t the dif- 



OS.) 

 ferontial equations for the R J^ < usinc (19), (27), (17), 



/>! tl 



(16), and (84). For />! the resulting equation is homo- 

 geneous, so 



for A<>1 



fi) 



(20 ) 



(29) 



n-= 



(30) 



For X~l v;e have on collecting the terms 



<#^-3^#^=tjj«Y^t 





(31) 



