Page:On Governors.pdf/8

entering into the calculation of the general equations of motion of these pieces, we may confine ourselves to the case of small disturbances, and write the equations

(29)	${\begin{aligned}A{\frac {d^{2}\theta }{dt^{2}}}+X{\frac {d\theta }{dt}}+K{\frac {d\phi }{dt}}+T\phi +J\psi =P-R,&\\B{\frac {d^{2}\phi }{dt^{2}}}+Y{\frac {d\phi }{dt}}-K{\frac {d\theta }{dt}}=0,&\\C{\frac {d^{2}\psi }{dt^{2}}}+Z{\frac {d\psi }{dt}}-T\phi =0&\\\end{aligned}}$

where $\scriptstyle \ \theta$, $\scriptstyle \ \phi$, $\scriptstyle \ \chi$ are the angles of disturbance of the main shaft, the centrifugal arm, and the moveable wheel respectively, $\scriptstyle \ A$, $\scriptstyle \ B$, $\scriptstyle \ C$ their moments of inertia, $\scriptstyle \ X$, $\scriptstyle \ Y$, $\scriptstyle \ Z$ the viscosity of their connexions, $\scriptstyle \ K$ is what was formerly denoted by $dA/d\phi$, and $\scriptstyle \ T$ and $\scriptstyle \ J$ are the powers of Thomson's and Jenkin's breaks respectively.

The resulting equation in $\scriptstyle \ n$ is of the form

(30)	${\begin{array}{|c c c|}An^{2}+Xn&Kn+T&J\\-K&Bn+T&0\\0&-T&Cn^{2}+Zn\\\end{array}}=0$

or

(31)	$n^{5}+n^{4}\left({\frac {X}{A}}+{\frac {Y}{B}}+{\frac {Z}{C}}\right)+n^{3}\left[{\frac {XYZ}{ABC}}\left({\frac {X}{A}}+{\frac {Y}{B}}+{\frac {Z}{C}}\right)+{\frac {K^{2}}{AB}}\right]$

$+n^{2}\left({\frac {XYZ+KTC+K^{2}Z}{ABC}}\right)+n{\frac {KTZ}{ABC}}+{\frac {KTZJ}{ABC}}=0$

I have not succeeded in determining completely the conditions of stability of the motion from this equation; but I have found two necessary conditions, which are in fact the conditions of stability of the two governors taken separately. If we write the equation

(32)	$n^{5}+pn^{4}+qn^{3}+rn^{2}+sn+t=0,$

then, in order that the possible parts of all the roots shall be negative, it is necessary that

(33)	$pq>y$ and $ps>t.$

I am not able to show that these conditions are sufficient. This compound governor has been constructed and used.

On the Motion of a Liquid in a Tube revolving about a Vertical Axis.

Mr C. W. Siemens’s Liquid Governor. Let $\scriptstyle \ \rho$ be the density of the fluid, $\scriptstyle \ k$ the section of the tube at a point whose distance from the origin measured along the tube is $\scriptstyle \ s$, $\scriptstyle \ r$, $\scriptstyle \ \theta$, $\scriptstyle \ z$ the coordinates of this point referred to axes fixed with respect to the tube, $\scriptstyle \ Q$ the volume of liquid which passes through any section in unit of time. Also let the following integrals, taken over the whole tube, be

(33)	$\int \rho kr^{2}ds=A,\int \rho r^{2}d\theta =B,\int \rho {\tfrac {1}{\alpha }}ds=C,$

the lower end of the tube being in the axis of motion.