Page:Philosophical magazine 23 series 4.djvu/112

92 varied motion is

{{MathForm2|(145)|$$\left.\begin{array}{l} Y=k_{2}\frac{dy}{dz}-\frac{1}{4\pi}\mu r\frac{d\alpha}{dt}.\\ \\X=k_{1}\frac{dx}{dz}+\frac{1}{4\pi}\mu r\frac{d\beta}{dt}.\end{array}\right\} $$}}

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,

{{MathForm2|(146)|$$\left.\begin{array}{l} \rho\frac{d^{2}x}{dt^{2}}=\frac{dX}{dz}=k_{1}\frac{d^{2}x}{dz^{2}}+\frac{d}{dz}\frac{1}{4\pi}\mu r\frac{d\beta}{dt},\\ \\\rho\frac{d^{2}y}{dt^{2}}=\frac{dY}{dz}=k_{2}\frac{d^{2}y}{dz^{2}}-\frac{d}{dz}\frac{1}{4\pi}\mu r\frac{d\alpha}{dt}.\end{array}\right\} $$}}

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. to determine these quantities in terms of the motion. We find there at equation (68),

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

{{MathForm2|(147)|$$\left.\begin{array}{lc} & \frac{d\alpha}{dt}=\gamma\frac{d^{2}x}{dz\ dt};\\ \mathrm{and\ in\ like\ manner}\\ & \frac{d\beta}{dt}=\gamma\frac{d^{2}y}{dz\ dt};\end{array}\right\} $$}}

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

{{MathForm2|(148)|$$\left.\begin{array}{l} \frac{d^{2}x}{dt^{2}}=a^{2}\frac{d^{2}x}{dz^{2}}+c^{2}\frac{d^{3}y}{dz^{2}dt},\\ \\\frac{d^{2}y}{dt^{2}}=b^{2}\frac{d^{2}y}{dz^{2}}-c^{2}\frac{d^{3}x}{dz^{2}dt}.\end{array}\right\} $$}}

These equations may be satisfied by the values provided

{{MathForm2|(149)|$$\left.\begin{array}{l} x=A\cos(nt-mz+\alpha),\\ y=B\sin(nt-mz+\alpha),\end{array}\right\} $$}}