54 Royal Society : — 



but, according to equation (1), A*«r — *; which value being substi- 



J 

 tuted in the above equation, gives 



2a 2 u 



and therefore 



=2**,, 



— =A, and a= ^gk ; ' ' ' ' ^ 



so that the velocity of advance of a wave (denned as in article 2) is 

 equal to that acquired by a body in falling through half the virtual 

 depth ; and this is true for all possible waves in which the upper sur- 

 face is a surface of uniform pressure. 



(In article 6 of the paper, the speed of advance of a wave of trans- 

 lation is expressed by combining the speed of a rolling wave, V yk, 

 with that of a supposed current, as stated in article 2. 



In articles 7, 8, and 9 the law which connects the speed of advance 

 of a wave with the virtual depth is compared with the already known 

 laws of the transmission of rolling waves in water of limited or unli- 

 mited depth. The principal results may be summed up as follows. 



Let T be the periodic time of a wave, in seconds ; h = ^— , the equi- 

 valent pendulum — that is, the height of the pendulum whose period 



is the same ; c= — ^— the rolling radius, being the radius of a 



2tz 

 circle whose circumference is equal to a wave-length ; v 1 the greatest 

 horizontal velocity, and w } the greatest vertical velocity of a surface- 

 particle ; a the velocity of advance ; then 



2it 



and 



, w, w, j v 



k — — ; * . c= — \ . h.) 



u \ 



(10) Oblique Advance of Forced Waves. — Let s be the velocity 

 with which a floating solid body is driven horizontally ; the wave 

 which that solid body pushes or drags along with it is forced to 

 advance at the velocity s also ; while the virtual depth of disturb- 

 ance, ky bears some relation to the depth of immersion and figure of 

 the solid body. If the speed of advance corresponding to that depth, 

 a= s/gk, is less than s, a pair of wave-ridges diverge obliquely from 

 the path of the floating body towards opposite sides ; and the sine of 



the angle which each of those ridges makes with that path is - . Such 



is the mode of formation of the obliquely spreading waves which travel 

 along with ships*. 



* See Watts, Rankine, Napier, and Barnes, ' On Ship-building,' Division I. 

 Article 156, p. 79. 



