Hydraulic Investigations. 187 



may remain nearly horizontal : and this proposition will be 

 the more accurately true, the smaller the velocity of the 

 moveable end may be. For, calling this velocity v, the 

 original depth a, the increased depth x, and the velocity of 

 the anterior part of the wave y, we have, on the supposition 

 that the extent of the wave is already become considerable, 



x = ■■ S -, taking the negative or positive sign according 



to the direction of the motion of the end ; since the quantity 

 of fluid, which before occupied a length expressed by y, now 



— av 



occupies the length y-\-v ; and putting a sx = 2, z = -— -, 



The direction of the surface of the margin of the wave is 



indifferent to the calculation, and it is most convenient to 



suppose its inclination equal to half a right angle, so that 



the accelerating; force, acting on any thin transverse vertical 



lamina, mav be equal to its weight : then the velocity y 



must be such, that while the inclined margi-n of the wave 



passes by each lamina, the lamina may acquire the velocity 



v by a force equal to its own weight : consequently the time 



of its passage must be equal to that in which a body acquires 



the velocity v in falling through a height b, corresponding 



lb 

 to that velocity : and this time is expressed by — 3 but the 



space described by the. margin of the wave is not exactly z, 

 because the lamina in question has moved horizontally du- 

 ring its acceleration, through a space which must be equal 

 to /' ; the distance actually described will therefore be % ± b, 



% ± b 'lb . 2by . . 2lyy 



and we have -~— = , % + b at- ' av 4- Ini — bv = - 



y v — v ' — J v 



av 9 v* __ < av* v* 



+ wjt* y 1 + * v y = 2h~ 2> { y + * ^ 2 = a? + & hut m 



being the proper coefficient, v = m ^/ b, and v 1 = m r b 9 

 av 1 ■ v* _ ,-a b \ /a b \ 



■tb+T6= m ( 2 fie)'** m i w)m ■'&* v ' an - 



y + v = 771 */ (— 4- — i\ + I v. But when v is small, we 



.» — 1 a j ma ^ b *.. ,. 



may take y + v nearly m ,/-, and z = Jj^Tj^v* <S{<>ab) % 



and x m a ± J {?ab), while the height of a fluid, in which 



th$ 



