228
Mr. H. Bateman
[March 11,
A term such as
may now be understood to mean[1]
and a term such as
to mean
The equation (II) may now be written
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Transforming the left-hand side by means of Green's theorem, we get
![{\displaystyle {\begin{array}{ll}\int \int \int \left[{\frac {d}{dx}}\left(E_{x}-H_{y}{\frac {\partial t}{\partial z}}+H_{z}{\frac {\partial t}{\partial y}}\right)\right.&+{\frac {d}{dy}}\left(E_{y}-H_{z}{\frac {\partial t}{\partial x}}+H_{x}{\frac {\partial t}{\partial z}}\right)\\\\&\left.+{\frac {d}{dz}}\left(E_{z}-H_{x}{\frac {\partial t}{\partial y}}+H_{y}{\frac {\partial t}{\partial x}}\right)\right]dx\ dy\ dz.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0764364d2bb88da0af2f664c6838fd54d9304822)
Now
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hence the above integral may be written
![{\displaystyle {\begin{array}{l}\int \int \int \left[\left({\frac {\partial E_{x}}{\partial t}}-{\frac {\partial H_{z}}{\partial y}}+{\frac {\partial H_{y}}{\partial z}}\right){\frac {\partial t}{\partial x}}+\left({\frac {\partial E_{y}}{\partial t}}-{\frac {\partial H_{x}}{\partial z}}+{\frac {\partial H_{z}}{\partial x}}\right){\frac {\partial t}{\partial y}}\right.\\\qquad \left.+\left({\frac {\partial E_{z}}{\partial t}}-{\frac {\partial H_{y}}{\partial x}}+{\frac {\partial H_{x}}{\partial y}}\right){\frac {\partial t}{\partial z}}+\left({\frac {\partial E_{x}}{\partial x}}+{\frac {\partial E_{y}}{\partial y}}+{\frac {\partial E_{z}}{\partial z}}\right)dx\ dy\ dz,\right]\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2424761dae831440f3ed045a7db982abf42c81cd)
- ↑ The relation
is a particular ease of a general relation given in Scott's Determinants, p. 134.