520 Mr. A. J. Robertson on the Positive Wave of Translation. 



which is also the length of the body of water which has been 

 compressed in time t> the whole length in time T being 



V 

 AD = L-h~. 



a 



Also the force causing each column to rise varies as the 

 distance from the line QN ; 



cos -1 — j-^ versin -1 7- 



. *_ k k 



,\t— -— _ — j 



V jju 



and 



V fJU 



y=zJc versin V fjb*t. 



Let B\ represent the thickness ad of an ele- 

 mentary portion of the water. After time St it 

 becomes compressed into a smaller width Bx^ec, 

 or 



ab x ad—ec x (ab + ac), 

 i. e. 



oXx a^BX^a + k versin */fj,.t); 



a, 



6\j a 



SX a + k versing/ jul. t dX 



=- = -j~ ultimately ; 



d\f __ dX l dX_ J_/ L V\ 

 dt dX dt T V a J a 



a 



-\-k versin */ [x.t 

 Integrating, X x being when t = 0, 



L V 



X 1= ^ tl.JL.y/ » ta n .i{^£±«*tan^"l(e.) 



*/ .. V a-\-2k IV a 2 J 



V 



/* 



When 

 Also 



*=T, y = 2£; /.T-= 



7T 



V 7 /^ 



V fju.t= versin 1 t 



V fjb.t 



tan -iF=\/ 



Therefore, substituting in equation (6.), 



3/ 

 2&— / 



# = 2 



7T 



V a + 2£ V a 



2& ?/ 



2&— 3/ 

 When y — 2k, 



tan" 1 */ -f— =tan~ 1 oo = -, 



V a 2k— y 2 



("•) 



