Page:On Governors.pdf/12

12 the equations of motion in $$\scriptstyle\ \theta$$ and $$\scriptstyle\ \phi$$ will be

If $$\scriptstyle\ M'=0$$, then the motions in $$\scriptstyle\ \theta$$ and $$\scriptstyle\ \phi$$ will be independent of each other. If $$\scriptstyle\ M$$ is also 0, then we have the relation

and if this is fulfilled, the disturbances of the motion in $$\scriptstyle\ \theta$$ will have no effect on the motion in $$\scriptstyle\ \phi$$. The teeth of the differential system in gear with the main shaft and the governor respectively will then correspond to the centres of percussion and rotation of a simple body, and this relation will be mutual.

In such differential systems a constant force, $$\scriptstyle\ H$$, sufficient to keep the governor in a proper state of efficiency, is applied to the axis $$\scriptstyle\ \eta$$, and the motion of this axis is made to work a valve or a break on the main shaft of the machine. $$\scriptstyle\ \Xi$$ in this case is merely the friction about the axis of $$\scriptstyle\ \zeta$$. If the moments of inertia of the different parts of the system are so arranged that $$\scriptstyle\ M'=0$$, then the disturbance produced by a blow or a jerk on the machine will act instantaneously on the valve, but will not communicate any impulse to the governor.