Page:Philosophical magazine 23 series 4.djvu/112

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92

Prof. Maxwell on the Theory of Molecular Vortices

varied motion is

\left.{\begin{array}{l}Y=k_{2}{\frac {dy}{dz}}-{\frac {1}{4\pi }}\mu r{\frac {d\alpha }{dt}}.\\\\X=k_{1}{\frac {dx}{dz}}+{\frac {1}{4\pi }}\mu r{\frac {d\beta }{dt}}.\end{array}}\right\}

(145)

The whole force acting upon a stratum whose thickness is $dx$ and area unity, is ${\tfrac {dX}{dz}}dz$ in the direction of $x$ , and ${\tfrac {dY}{dz}}dz$ in direction of $y$ . The mass of the stratum is $\rho dz$ , so that we have as the equations of motion,

\left.{\begin{array}{l}\rho {\frac {d^{2}x}{dt^{2}}}={\frac {dX}{dz}}=k_{1}{\frac {d^{2}x}{dz^{2}}}+{\frac {d}{dz}}{\frac {1}{4\pi }}\mu r{\frac {d\beta }{dt}},\\\\\rho {\frac {d^{2}y}{dt^{2}}}={\frac {dY}{dz}}=k_{2}{\frac {d^{2}y}{dz^{2}}}-{\frac {d}{dz}}{\frac {1}{4\pi }}\mu r{\frac {d\alpha }{dt}}.\end{array}}\right\}

(146)

Now the changes of velocity ${\tfrac {d\alpha }{dt}}$ and ${\tfrac {d\beta }{dt}}$ are produced by the motion of the medium containing the vortices, which distorts and twists every element of its mass; so that we must refer to Prop. X.^[1] to determine these quantities in terms of the motion. We find there at equation (68),

\delta \alpha =\alpha {\frac {d}{dx}}\delta x+\beta {\frac {d}{dy}}\delta x+\gamma {\frac {d}{dz}}\delta x.

(68)

Since $\delta x$ and $\delta y$ are functions of $z$ and $t$ only, we may write this equation

\left.{\begin{array}{lc}&{\frac {d\alpha }{dt}}=\gamma {\frac {d^{2}x}{dz\ dt}};\\\mathrm {and\ in\ like\ manner} \\&{\frac {d\beta }{dt}}=\gamma {\frac {d^{2}y}{dz\ dt}};\end{array}}\right\}

(147)

so that if we now put $k_{1}=a^{2}\rho ,\ k_{2}=b^{2}\rho$ , and ${\tfrac {1}{4\pi }}{\tfrac {\mu r}{\rho }}\gamma =c^{2}$ , we may write the equations of motion