Page:Electromagnetic phenomena.djvu/9

$k^{2}{\frac {w}{c^{2}}}x+{\frac {k}{l}}{\frac {r_{0}^{'}-r^{'}}{c}}=k^{2}{\frac {w}{c^{2}}}x+{\frac {k}{l}}{\frac {1}{c}}\left(x{\frac {\partial r'}{\partial x}}+y{\frac {\partial r'}{\partial y}}+z{\frac {\partial r'}{\partial z}}\right)$

units of time. In this last expression we may put for the differential coefficients their values at the point A.

In (17) we have now to replace $\left[\varrho \right]$ by

$\left[\varrho \right]+k^{2}{\frac {w}{c^{2}}}x\left[{\frac {\partial \varrho }{\partial t}}\right]+{\frac {k}{l}}{\frac {1}{c}}\left(x{\frac {\partial r'}{\partial x}}+y{\frac {\partial r'}{\partial y}}+z{\frac {\partial r'}{\partial z}}\right)\left[{\frac {\partial \varrho }{\partial t}}\right]$,

(25)

where $\left[{\frac {\partial \varrho }{\partial t}}\right]$ relates again to the time t₀. Now, the value of t' for which the calculations are to be performed having been chosen, this time t₀ will be a function of the coordinates x, y, z of the exterior point P. The value of $\left[\varrho \right]$ will therefore depend on these coordinates in such a way that

$\left[{\frac {\partial [\varrho ]}{\partial x}}\right]=-{\frac {k}{l}}{\frac {1}{c}}{\frac {\partial r'}{\partial x}}\left[{\frac {\partial \varrho }{\partial t}}\right]$, etc.,

by which (25) becomes

$\left[\varrho \right]+k^{2}{\frac {w}{c^{2}}}x\left[{\frac {\partial \varrho }{\partial t}}\right]-\left(x{\frac {\partial [\varrho ]}{\partial x}}+y{\frac {\partial [\varrho ]}{\partial y}}+z{\frac {\partial [\varrho ]}{\partial z}}\right)$.

Again, if henceforth we understand by r' what has above been called $r_{0}^{'}$, the factor ${\frac {1}{r'}}$ must be replaced by

${\frac {1}{r'}}-x{\frac {\partial }{\partial x}}\left({\frac {1}{r'}}\right)-y{\frac {\partial }{\partial y}}\left({\frac {1}{r'}}\right)-z{\frac {\partial }{\partial z}}\left({\frac {1}{r'}}\right)$,

so that after all, in the integral (17), the element dS is multiplied by

${\frac {\left[\varrho \right]}{r}}+k^{2}{\frac {w}{c^{2}}}{\frac {x}{r}}\left[{\frac {\partial \varrho }{\partial t}}\right]-{\frac {\partial }{\partial x}}{\frac {x[\varrho ]}{r'}}-{\frac {\partial }{\partial y}}{\frac {y[\varrho ]}{r'}}-{\frac {\partial }{\partial z}}{\frac {z[\varrho ]}{r'}}$.

This is simpler than the primitive form, because neither r' , nor the time for which the quantities enclosed in brackets are to be taken, depend on x, y, z. Using (23) and remembering that $\int \varrho \ dS=0$, we get

$\varphi '=k^{2}{\frac {w}{4\pi c^{2}r'}}\left[{\frac {\partial {\mathfrak {p}}_{x}}{\partial t}}\right]-{\frac {1}{4\pi }}\left\{{\frac {\partial }{\partial x}}{\frac {\left[{\mathfrak {p}}_{x}\right]}{r'}}+{\frac {\partial }{\partial y}}{\frac {\left[{\mathfrak {p}}_{y}\right]}{r'}}+{\frac {\partial }{\partial z}}{\frac {\left[{\mathfrak {p}}_{z}\right]}{r'}}\right\}$,

a formula in which all the enclosed quantities are to be taken for the instant at which the local time of the centre of the particle is $t'-{\frac {r'}{c}}$.

We shall conclude these calculations by introducing a new vector ${\mathfrak {p}}'$, whose components are