59 
obtaining the density at the orifice from the thermo-dynamic 
relation between density and pressure we have the weight 
discharged per second by multiplying the product of velocity 
with density by the effective area of the orifice. This is 
Thomson and Joule’s equation for the flow through an 
orifice. And so far the logic is perfect and there are no 
assumptions but those involved in the general theories of 
thermo-dynamics and of fluid motion. 
But in order to apply this equation it is necessary to 
know the pressure at the orifice, and here comes the assump- 
tion that has been tacitly made : that the pressure at the 
orifice is the 'pressure in the receiving vessel at a distance 
from the orifice. 
3. The origin of this assumption is that it holds when a 
denser liquid like water flows into a light fluid like air and 
approximately when water flows into water. 
Taking no account of friction the equations of hydro- 
dynamics show that this is the only condition under which 
the ideal liquid can flow steadily from a drowned orifice. 
But they have not been hitherto integrated so far as to show 
whether or not this would be the case with an elastic fluid. 
In the case of an elastic fluid the difficulty of integration 
is enhanced. But on examination it appears that there is 
an important circumstance connected with the steady motion 
of gases which does not exist in the case of liquid. This 
circumstance which may be inferred from integrations 
already effected determines the pressure at the orifice irre- 
spective of the pressure in the receiving vessel when this is 
below a certain point. 
4. To understand this circumstance it is necessary to con- 
sider a steady narrow stream of fluid in which the pressure 
alls and the velocity increases continuously in one direction. 
Since the stream is steady, equal weights of the fluid 
nust pass each section in the same time, or if u be the 
Velocity, p the density, and A the area of the stream, the 
