Here, to lessen confusion, w is written in place of u but the lengthy values 

 of a and b have not been inserted into Equation [ S'+b ] . 



For moderate damping, e > ; in any case b < and the formula holds 

 if e > - 'L/{2^). Very likely, numerically greater values of negative e 

 need not be admitted. If they are, the continuous values of the square 

 root can be extended by choosing the negative sign in Formula [32a] and, 

 hence, using the right-hand member of [3^a] with signs reversed. 



q, and q may now be found as (q., ) and (q^) . For clarity, 



replace the symbols a, b, and.u in Equations [32a,b] by a, B, and r, 



respectively. Substitute from Equation [3^a] for the square root in 



Equations [31a,b] and then, in the brace in each of these equations, 



collect the real terms as a and the imaginary terms as i6. Then Equations 



[32a,b] give: 



qi = q (r + ^) ; a = Ce + w ; g = _ g [^ + ^ (l+ 2^e)] [35a,b,c] 

 1 2r 2w 



— • 1 2 



a = q (r + il) ; a = - Ce + w ; 6 = - g [-^ + — (l + 2? e)] [35d,e,f] 

 £^ 2r 2w 



where in either case 



=VH-vZ77) 



and w is given by Equation [ 3^b ] . Here in both q and q , a > provided 



1 2 



e > K^g . For then in [ S^+b ] , a = e + 5^ (e^ - g^ ) > S^e^ ^ so that 



2 2 2 II 



w > a >C e and w > |?e| . Otherwise it may be necessaiy again to extend 



the range of continuous values by reversing signs. 



33 



