HYDRAULIC INVESTIGATIONS. 



pressed by y, now occupies the length y + v; and putting 



av 



a *> x — z, z 21 — ■■ . The direction of the surface of the 



y "t" 



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 fprce, acting on any 

 thin transverse vertical lamina, may be equal to its weight ; 

 then the velocity y must be such, that while the inclined 

 margin of the waye 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 to that velocity: and this time is 



expressed by — ; but the space described by the margin of 



the wave is not exactly z, because the lamina in question 



has moved horizontally during its acceleration, through a 



space which must be equal to b ; the distance actually de? 



z + b Qb 

 scribed will therefore be z -f b> and we have — =r=_ = — , z 

 -*- y v 



+ 6=i^'a W +6y — B»=il?!l+2fty,2f*+ ivy 



— v — * v 



av* v*, , — N » av % t?* v \. 



= Tb—Tw *m mm*- t& hni > m bein s tbe prq * 



per coefficient, v 5g m V b> and v* = m % b, ~^r+ y^ — »»* 



(i" + le)' y ' ~ m v (I + ie) ± * Vt and y + v ~ m v 



( — + «)+t^« But wnen f i s small, we may take y + 



, n -i ma \/ b . j ; v \ 



v nearly m V ■£ , and z = — -7-77-7 ~ *S C 2 a $)» and ¥ 



25 a 4- -/ (2 « b)t while the height of a fluid, in which the 



velocity would bey, is nearly a -f- £ >J (2 a b) : consequently, 



when the velocity v is at all considerable, y must be some- 

 what greater than the velocity of a wave moving on the sur- 

 face of the elevated fluid ; and probably the surface of the 



elevated 



