604 



= 16 7T^ jj (T) - e w 



- 10 



4. (x) -J (T - x) dx 



>!d.. 



I = 11 a- J- (T) - 9 77 



3 



4. (x) -i (T - x) dx 



(31) 



here x = cos 6 



and </; (x) • 1. 6 - -^ sin 20 t^ sin u 5 



Final forms of equation of motion . 



Substituting from (31) in (1?) we have for the equation of motion 



y'hj dT' pj 6 dT 



,1 



<^ (x) -i (T- x) dx 



(32) 



(33) 



fl 



For alternative form we have 



<t,(x) J (T - x) dx = [<^j(x) J (T - x) * (^^(x) i- (T - x) + (^3 -J- (T - x)]J 



^^(x) i" (T - x) dx 



= !? J- fT) - ^ J- (T) 



6 315 



* I (f^M r (T - X) dx 



(3U) 



SO that the equation of rxjtion may also to written as 



6h 

 y' dr' 



J^ i;^ + r{-i) = I2 0"^^ ^ k j <^(x) i" (t - x) dx 



± 



J_ _ 6U K 



y^ 315 



(35) 



(36) 



For™ (33) is more suitable in the early stages when T is smJll while form (35) is more 

 suitable when T is large. The initial conditions ire 



-J (T) = -J' (T) = T ^ 



Since i* (T) is discontinuous it T = ind ■/' (t) is th^Teforc- infinite, the form (33) 

 should be used for -^ T <; 1. 



6. Incompressibl c ap-proximation . 



If the compressibility term, i.e. the integral term in form (35) Is small then a first 

 approximation is given by nuglactin^ it, i.e. 



? ^ 



F(J) = -2 e 



„-NT 



which gives 



1 d^;! , _ ^"0 „,-NT 



y' dT^ 



= - _" Nl-"' _ F'(J) ^ 



(37) 



(36) 



^ , 



