Page:Electromagnetic phenomena.djvu/6



The symbol $$\triangle'$$ is an abbreviation for $$\frac{\partial^{2}}{\partial x'^{2}}+\frac{\partial^{2}}{\partial y'^{2}}+\frac{\partial^{2}}{\partial z'^{2}}$$ and $$grad'\ \varphi'$$ denotes for a vector whose components are $$\frac{\partial\varphi'}{\partial x'},\ \frac{\partial\varphi'}{\partial y'},\ \frac{\partial\varphi'}{\partial z'}$$. The expresson $$grad'\ \mathfrak{a}_{x}^{'}$$ has a similar meaning.

In order to obtain the solution of (11) and (12) in a simple form, we may take x', y', z'  as the coordinates of a point P'  in a space S', and ascribe to this point, for each value of t' , the values of $$\varrho',\ \mathfrak{u}',\ \varphi',\ \mathfrak{a}'$$, belonging to the corresponding point P (x, y, z) of the electromagnetic system. For a definite value t'  of the fourth independent variable, the potentials φ' and $$\mathfrak{a}'$$ in the point of the system or in the corresponding point P'  of the space S' , are given by

Here dS'  is an element of the space S', r'  its distance form P'  and the brackets serve to denote the quantity $$\varrho'$$  and the vector $$\varphi',\ \mathfrak{u}'$$, such as they are in the element dS' , for the value $$t'-\frac{r'}{c}$$ of the fourth independent variable.

Instead of (15) and (16) we may also write, taking into account (4) and (7),

the integrations now extending over the electromagnetic system itself. It should be kept in mind that in these formulae r'  does not denote the distance between the element dS and the point (x, y, z) for which the calculation is to be performed. If the element lies at the point (x1,y1},z1), we must take

It is also to be remembered that, if we wish to determine φ and