Page:Philosophical magazine 21 series 4.djvu/365

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applied to Electric Currents. 341

along three axes properly chosen, x', y', z', the nine direction-cosines of these axes with their six connecting equations, which are equivalent to three independent quantities, and the three rotations about the axes of x, y, z.

Let the direction-cosines of with respect to x, y, z be , those of and those of ; then we find

(64)

with similar equations for quantities involving and .

Let be the values of referred to the axes of ; then

(65)


We shall then have

(66)
(67)

By substituting the values of , and comparing with equations (64), we find

(68)

as the variation of due to the change of form and position of the element. The variations of and have similar expressions.

Prop. XI. — To find the electromotive forces in a moving body.

The variation of the velocity of the vortices in a moving element is due to two causes — the action of the electromotive forces, and the change of form and position of the element. The whole variation of is therefore

(69)

But since a is a function of x, y, z and t, the variation of may be also written

(70)