454 Dr. H. Jeffreys on Periodic 



e -mz term separately, as had to be done in the previous 

 problem. 

 Then 



w = _ (HPi emz + Wb #» _ £*& #.# _a*& #.* 



2 too 2lq> ly ty 



V 



( X v ^-^)-y*X*} ' ^ 



2©X + 2a>\ 7 2 -4ft> 2 7 2 -4o) 2 



tyXgoibv , rtrk >. 



^v-4»')-7* g ~" (29) 



afib e ^ + ^> 2 *** + 2ft)X ^ ,,3, + 2<bX *»* 



2o)X . 2©X + 7 2_4o) 2 7 2 -4ftr 



2(o\godjv 



(30) 



v 2 ( 7 2 -4w'0-7 2 X 2 



The boundary conditions are that there is no motion 

 at the ground, so that A = B = W = when z—0, and that 

 p = gp \wdt when z — li. Now 8 is insignificant at the 

 free surface, and hence the last relation is equivalent to 



tyU = C/W (31) 



The first three lead to 



fJ'iPi _ _ ^2 2 ^2 _ aXyctOv (o*' 2 (]h+p 4 ) ,^) 



7 + 2co 7 — 2&) v 2 (y' J — 4 ft) 2 ) - 7 2 A, 2 7 2 — 4&) 2 ' ^ w ' 



/7+2ft) 7- 2ft)\/ ffafry 7>a+/> 4 \ X 2 



\ 2/*i ^2 A»' 2 (7 2 -4ft> 2 j-7 2 /V 2 7^_4ftr7' 



Now g^/y 2 is large even for bodies of continental 

 dimensions ; gv/y 2 also is large. Hence the equation (31) 

 is found to lead to 



*fa-p*) = (^ - rt^ ipt +p>) - ^ff^.^r ( 34 ) 



Substituting in (33) we have, remembering that \jv 

 is small, 



{-7, /7 + 2ft> 7-2ft)\ X 2 ] f^_^ . -,^7 



