Page:Philosophical magazine 21 series 4.djvu/314

$\left.{\begin{array}{lll}&&{\frac {dQ}{dz}}-{\frac {dR}{dy}}=\mu {\frac {d\alpha }{dt}},\\\mathrm {Similarly,} \\&&{\frac {dR}{dx}}-{\frac {dP}{dz}}=\mu {\frac {d\beta }{dt}},\\\mathrm {and} \\&&{\frac {dP}{dy}}-{\frac {dQ}{dx}}=\mu {\frac {d\gamma }{dt}}.\end{array}}\right\}$

(54)

From these equations we may determine the relation between the alterations of motion ${\tfrac {d\alpha }{dt}}$, &c. and the forces exerted on the layers of particles between the vortices, or, in the language of our hypothesis, the relation between changes in the state of the magnetic field and the electromotive forces thereby brought into play.

In a memoir "On the Dynamical Theory of Diffraction" (Cambridge Philosophical Transactions, vol. ix. part 1, section 6), Professor Stokes has given a method by which we may solve equations (54), and find P, Q, and R in terms of the quantities on the right-hand of those equations. I have pointed out^[1] the application of this method to questions in electricity and magnetism.

Let us then find three quantities F, G, H from the equations

$\left.{\begin{array}{l}{\frac {dG}{dz}}-{\frac {dH}{dy}}=\mu \alpha ,\\\\{\frac {dH}{dx}}-{\frac {dF}{dz}}=\mu \beta ,\\\\{\frac {dF}{dy}}-{\frac {dG}{dx}}=\mu \gamma ,\end{array}}\right\}$

(55)

with the conditions

${\frac {1}{4\pi }}\left({\frac {d}{dx}}\mu \alpha +{\frac {d}{dy}}\mu \beta +{\frac {d}{dz}}\mu \gamma \right)=m=0,$

(56)

and

${\frac {dF}{dx}}+{\frac {dG}{dy}}+{\frac {dH}{dz}}=0.$

(57)

Differentiating (55) with respect to $t$, and comparing with (54), we find

$P={\frac {dF}{dt}},\ Q={\frac {dG}{dt}},\ R={\frac {dH}{dt}}.$

(58)

1. Cambridge Philosophical Transactions, vol. x. part 1. art. 3, "On Faraday's Lines of Force."