James Clerk Maxwell

“ If the flow of the fluid were limited by a hole, there would be a minimum velocity instead of a maximum.
The differential equation which determines the nature of small disturbances is in general of the fourth order, but may be reduced to the third by a proper choice of the value of the mean overflow.
Theory of Differential Gearing.
In some contrivances the main shaft is connected with the governor by a wheel or system of wheels which are capable of rotation round an axis, which is itself also capable of rotation about the axis of the main shaft. ”

“ In the first kind, the centrifugal piece has a constant distance from the axis of motion, but its pressure on a surface on which it rubs varies when the velocity varies. In the moderator this friction is itself the retarding force. In the governor this surface is made moveable about the axis, and the friction tends to move it; and this motion is made to act on a break to retard the machine. A constant force acts on the moveable wheel in the opposite direction to that of the friction, which takes off the break when the friction is less than a given quantity. ”

“ I have pointed out that, under certain conditions, the sudden disturbances of the machine do not act through the differential system on the governor, or vice versa. When these conditions are fulfilled, the equations of motion are not only simple, but the motion itself is not liable to disturbances depending on the mutual action of the machine and the governor.
Distinction between Moderators and Governors.
In regulators of the first kind, let be the driving-power and the resistance, both estimated as if applied to a given axis of the machine. ”

“ Let be the normal velocity, estimated for the same axis, and the actual velocity, and let be the moment of inertia of the whole machine reduced to the given axis. Let the governor be so arranged as to increase the resistance or diminish the driving-power by a quantity , then the equation of motion will be
When the machine has obtained its final rate the first term vanishes, and
Hence, if is increased or diminished, the velocity will be permanently increased. Regulators of this kind, as Mr Siemens [2] , has observed, should be called moderators rather than governors. ”

“ In M. Foucault’s arrangement, the force acting on the centrifugal piece is the weight of the balls acting downward, and an upward force produced by weights acting on a combination of levers and tending to raise the balls. ”

“ The energy of motion of the fluid in the tube is
The energy of the fluid which escapes in unit of time is
The work done by the prime mover in turning the shaft in unit of time is
The work spent on the liquid in unit of time is
Equating this to the work done, we obtain the equations of motion
These equations apply to a tube of given section throughout. If the fluid is in open channels, the values of and will depend on the depth to which the channels are filled at each point, and that of will depend on the depth at the overflow.
In the governor described by Mr. ”