Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/442

 410 MAGNETIC ACTION ON LIGHT. [825.

The next step in our hypothesis is the assumption that the kinetic energy of the medium contains a term of the form

2C (aa&amp;gt; 1 + j8a&amp;gt; 2 + yfi&amp;gt; 3 ). (3)

This is equivalent to supposing that the angular velocity acquired by the element of the medium during the propagation^ of light is a quantity which may enter into combination with that motion by which magnetic phenomena are explained.

In order to form the equations of motion of the medium, we must express its kinetic energy in terms of the velocity of its parts, the components of which are ,77, f We therefore integrate by parts, and find

2 C 1 1 I (aco 1 + /3a) 2 -I ya&amp;gt; 3 ) das dy dz

-av) dxdy

��The double integrals refer to the bounding surface, which may be supposed at an infinite distance. We may, therefore, while in vestigating what takes place in the interior of the medium, confine our attention to the triple integral.

825.] The part of the kinetic energy in unit of volume, expressed by this triple integral, may be written

iv Cfa+W + faf ( 5 )

where u, v, w are the components of the electric current as given in equations (E), Art. 607.

It appears from this that our hypothesis is equivalent to the assumption that the velocity of a particle of the medium whose components are f, r/, (f, is a quantity which may enter into com bination with the electric current whose components are u, v, w.

826.] Returning to the expression under the sign of triple inte gration in (4), substituting for the values of a, /3, y, those of a, /3&quot;, y , as given by equations (1), and writing

d d d d

ji for a -=- + 0-7- H-y-v- ; (6)

dh dx dy dz

the expression under the sign of integration becomes

L( d A_ d ^\, d ( d _ d c\+i d (**-

d& \d,y dz&amp;gt; + ^ dh (dz dx) + C dh (dx In the case of waves in planes normal to the axis of z the displace-

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