184 Hydraulic Investigations* 



of the height, is -- . t - , which will become - -. -.— . Now 

 fc ' Ox I— a: 2x b — a 



in a tube filled with an elastic fluid, the height being h, the 



force in similar circumstances would be rk, and if we make 



h= --. j _ ■> the velocity of the propagation of an impulse 



will be the same in both cases, and will be equal to the ve- 

 locity of a body which has fallen through the height \ h. 

 Supposing x infinite, the height capable of producing the 



necessary pressure becomes i— , which may be called g, 



and for every other value of x this height is Q 1 — Jg 3 ot 



g _ ' -•?, or, since h becomes ~, g — 2 h, so that h is al- 

 ways equal to half the difference between g and the actual 

 height of the column above the given point, or to half the 

 Jieight of the point above the base of the column. 



If two values pf x, with their corresponding heights, are 

 given, as b and ^corresponding to c and d, and it is required 



to find a; we have — ,— : c : : '— — : d, dhx — dax = cbx— 

 dlx—cbx h dx—ch . c . . . , 



cla > and a ~ -dx^cT' or 7 == ^-7^ Thusifthebei S ht 



equivalent to the tension vary in the ratio of any power m 

 of the diameter, so that, n being a small quantity, x = b 



(1 +n) and d=ci\ + mn) 9 -= WWm^W^^W^ 



mn + n . . c . {■ b m + 1 



~ — ■ — -, since the square or n is evanescent, and = m • 



I 5 

 For example, if m = 4, - = — , and if tk = 2, 1 : a : : 3 : 2. 

 1 \ a 4 



IV. Of the Magnitude of a diverging Pulsation at different 



Points. 



The demonstrations of Euler, Lagrange, and Bernoulli, 



respecting the propagation of sound, have determined that 



the velocity of the actual motion of the individual particles 



of an., elastic fluid, when an impulse is transmitted through 



a conical 



