Page:On Governors.pdf/12

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

(61)	${\begin{aligned}\Phi +P\Xi +QH&=L'{\frac {d2\theta }{dt^{2}}}+M'{\frac {d2\phi }{dt^{2}}}\\\Phi +R\Xi +SH&=M'{\frac {d2\theta }{dt^{2}}}+N'{\frac {d2\phi }{dt^{2}}}\\\end{aligned}}$

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

(62)	$LPQ+MRS=0$

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.