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
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(64)
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with similar equations for quantities involving and .
Let be the values of referred to the axes of ; then
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(65)
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We shall then have
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(66)
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(67)
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By substituting the values of , and comparing with equations (64), we find
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(68)
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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
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(69)
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But since a is a function of x, y, z and t, the variation of may be also written
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(70)
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