Dr. Young’s Hydraulic Investigations. 
17S 
will be equal to the velocity of a body which has fallen 
through the height ~ b. Supposing x infinite, the height 
capable of producing the necessary pressure becomes ~ c - 
which may be called g, and for every other value of x this 
height is 1 1 — - ) g> or g — or, since h becomes g 
2 b, so that b is always equal to half the difference between 
g and the actual height of the column above the given point, 
or to half the height of the point above the base of the 
column. 
If two values of x, with their corresponding heights, are 
given, as b and x, corresponding to c and d, and it is required 
to find a ; we have 
dbx—chx 
x — a 
x 
dx — cb 
: d, dbx — dax = cbx — 
or - = Thus if the height 
cba, and a — — , 
dx — cb a dx—cx 
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 4~ ?* ) and d = c { 1 -f- mn ), - = — ( ^ + d jYt nvt) — 1 ) 
v ^ ; v ^ ‘ a be ((!+»).(! + OT»-(I + n) 
__ m n + h s j nce S q Uare Q f n i s evanescent, and - = - + 1 
mn 1 1 a m • 
For example, if m =. 4, - = - , and if m — 2, b : a : : 3 : 2. 
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 
