(1)
or, as A and B are any two points,
v2/2g + p/G + z = constant = H.
(2)
Now v2/2g is the head due to the velocity v, p/G is the head equivalent to the pressure, and z is the elevation above the datum (see § 16). Hence the terms on the left are the total head due to velocity, pressure, and elevation at a given cross section of the filament, z is easily seen to be the work in foot-pounds which would be done by 1 ℔ of fluid falling to the datum line, and similarly p/G and v2/2g are the quantities of work which would be done by 1 ℔ of fluid due to the pressure p and velocity v. The expression on the left of the equation is, therefore, the total energy of the stream at the section considered, per ℔ of fluid, estimated with reference to the datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total energy per ℔ of fluid is uniformly distributed along the stream line. If the free surface of the fluid OO is taken as the datum, and −h, −h1 are the depths of A and B measured down from the free surface, the equation takes the form
v2/2g + p/G − h = v12/2g + p1/G − h1;
(3)
or generally
v2/2g + p/G − h = constant.
(3a)