u 



605 



Now since p and therefore F(i) increases with deflection we have F' (-Z) > and up to 

 time of maximum deflection ^ > and therefore d^-i < up to time of maximum deflection. 



dT^ 

 A 



The compressibility term is K <^,(x) 4" (T - x) dx 



= K-^" (T - e) 



(^(x) 



dx 



since (^^(x) ^ 



= LIL-r {1 - e) 



6U 



where < € < 1 (39) 



Hence, if compressibility is snail we can solve on the assumption that it is negligible 



and then estimate the error by expression (39). Since 4" (T) < the error will involve a 



relief pressure and therefore the calculated deflection on the Incompressible approximation will 

 bo '\n over-estimation. 



7. Properties of the function (/;,(x) 



The following properties of <^,(x) defined by 



(LM = i y{l- 2y2)(0 cot e- 1 +4 y^ t-^y" + -^ y*) +-ii-i^ 

 ^3 3 15 105 315 



where x = cos 

 and y = sin 

 may be noted:- 



(to) 



(a) <jij(x) = </: (X) 



■^ sin 2^ + I sin 4 (9 



(b) <t(x) 5-0, ^ X ^ 1 



(c) (?^(x) - ^ as X - 



(d) <^(x) - ^ as X - 1, y - 



(32) 

 (41) 

 (42) 

 (»3) 



<^(x) dx = X 



(uu) 



