ME. HOPKINS ON THE THEOET OF THE MOTION OF GLACIEES. 
733 
Also, since the same quantity of the fluid must pass through each transverse section 
of the tube in the same time, we must have 
and therefore 
Vu—ViUi, 
Let us interpret the above equation in different cases. We may flrst suppose the 
absence of all friction as a retarding force. We shall then have 
and if the section (a) of the tube do not contract so rapidly as to impede the motion of 
the water, the velocity acquired at the depth z will equal that of a body projected with 
velocity Vj, and falling freely by the action of gravity through the space z—z^. We 
shall therefore have 
W—vl)=9{z-z,), 
and therefore the atmospheric pressure, as if the tube were occupied by air. 
The least value of for any value of z in this case will be given by 
•"l 2 
= ^ . ul. 
2g[z-zi) 
If u satisfy this condition for all values of 2, the column of fluid will just All the tube 
without its motion being impeded, while the pressure on the tube, as just shown, will 
be the atmospheric pressure jpi. But suppose a to decrease, below a given section (Q), 
more rapidly as a function of z than is implied in the above equation. The motion of 
the fluid above the section Q would manifestly be retarded, and the retardation could 
only be due to the upward fluid pressure at Q being greater than the atmospheric pres- 
sure at the top of the tube. In like manner the increased downward fluid pressure 
at Q would accelerate the motion of the fluid immediately below that section. Any 
number of similar variations of with consequent variations of velocity and fluid pres- 
sure, might take place, subject to the above condition that the tube shall always be full. 
The fluid pressure at any point cannot be less, but may be greater than 
In this reasoning the tube is supposed to be independent of any other; but now 
suppose it to be confluent with a second and similar tube. At the point of junction we 
must have the condition that the pressures in the two tubes must be equal. This con- 
dition will be always satisfied, at whatever points in the surface of the mass the tubes 
may originate, provided each tube be such that the fluid pressure in it (as in the first 
case above explained) shall be equal at every point to the atmospheric pressure In 
such case water might pass from the upper surface through the mass, along any number 
of smooth tubes like those above described and communicating freely with each other. 
MDCCCLXII. 5 H 
