332 



as 



V,'e shall employ the Laplace transform LC«w^ of x^Ct) , defined 



lH= J *, c<-;«.-'^*dtr . 



(8) 



After taking the Laplace transform of both sides of Eq, (6) and integrating 

 by parts, one obtains 



f*« C«-»'/»)FCo) ) 



P^-^«-'-^>^r^.-^*5^0-eOe---^, 



(9) 



C-»t«e> 



<it*> 



Tcipa.e, y . C«*'-».i/e)FCu>) 



' C"*- CO 



where C is the imaginary unit \J"-i , and C is any real constant greater than 

 the least value of u> for which the transform LCt«') exists. If one deter- 

 mines the zeros u?^ , with vanishing or n^ative real parts, of FCw^, 



Fio>^)^o 



(10) 



one may evaluate the integral of Eq, (9) by the theory of residues to obtain 



c). ^&a;. 



Ut)^ ZL -^ 



<^^^ 



»-«-U)h» 



c-*/^ 



e.F'C-'H) e,©»FC-'/0) 



(11) 



P'(.0,}. 3u>^"^+ =«4=: + -!_ - ii^A e-'*'*®'^ 



<». 



The permanent deflection of the plate, with possible neglect of a small 



23 



