177 
Dr. Young's Hydraulic Investigations. 
falling body ; and in the same manner if the pipe be inclined, 
the motion of the impulse may be compared with that of 
a body descending or ascending freely along an inclined 
plane. 
These propositions may be thus demonstrated : let a be the 
diameter of the pipe in its most natural state, and let this dia- 
meter be increased to b by the pressure of the column c, the 
tube being so constituted that the tension may vary as the 
force. Then the relative force of the column c is represented 
by be, since its efficacy increases, according to the laws of 
hydrostatics, in the ratio of the diameter of the tube ; and this 
force must be equal, in a state of equilibrium, to the tension 
arising from the change from a to 6, that is, to b — a ; conse- 
quently the height c varies as • and if the tube be en- 
larged to any diameter x, the corresponding pressure required 
to distend it will be expressed by a height of the column equal 
DC 
r — , since 
b — a 
b — a 
~T~ 
Now 
b — a 
if the diameter be enlarged in such a degree, that the length 
of a certain portion of its contents may be contracted in the 
ratio 1 : l — r, r being very small, then the enlargement will 
be in the ratio l : l + - , that is, x' will be ^ ; but the in ere- 
1 2 2 
ment of the force, or 
of the height, is — 
be 
b — a 
which will 
become — - lc Now in a tube filled with an elastic fluid, 
zx b — a 
the height being h, the force in similar circumstances would 
be rh , and if we make h = A ^ , the velocity of the pro- 
pagation of an impulse will be the same in both cases, and 
A a 
JMDCCCVIII. 
