522 Mr. A. J. Robertson on the Positive Wave of Translation, 



and L+ V 



d* s a a . _ ■ "'. 



~ a r fi~~^ — — — / , xg -ftsin v/^.^ = force which has 



produced the motion. 



Again, force producing vertical motion is 



(k— y)jUL = kfjL COS \SjjL.t', 



8 'i = J S + J Vcosv^.*. . (12.) 



x=h t=T t=T 



Therefore, remembering that each side of the equation =0 

 when t = 9 



ttF fa + k 1 ,-. 1 



g 7 V\ IlT" versin VA6.^ + ^(cos2 VfJL.t-l)> 



V ±1 ,- £(l-cos VfA.t) ,- . /- 



= . • /*.— ^= — : 7=^-. +k Vfi sin V>.*. 



When VfA.t — TT, or 2 = T, 



L - 



V . 

 Whence, substituting for L+ — its value obtained from (8.), 

 ° a v 



7T 2 



^=_ 2 .(a + %, (13.) 



and 



7r L . / a-\-k 



T: 





^ « + & v g 

 Now \/ is the time which a body falling freely by 



o * 



gravity takes to fall through the space — - — . 



Therefore the period of the half-wave is to the time of fall- 

 ing through — - — as L is to a + k. 



<£ < 



Since in uniform motion, if the times be proportional 

 to the distances the velocities are equal, the velocity of the 

 crest of the wave would be that acquired by a body falling 



