Page:BatemanElectrodynamical.djvu/26

form of the third order

${\begin{array}{c}\sum \limits _{r,s,t}\end{array}}\left|{\begin{array}{ccc}a_{r}&a_{s}&a_{t}\\b_{r}&b_{s}&b_{t}\\c_{r}&c_{s}&c_{t}\end{array}}\right|{\begin{array}{c}dx_{r}dx_{s}dx_{t}=\sum \limits _{r,s,t}\end{array}}\left|{\begin{array}{c}{\begin{array}{ccc}a'_{r}&a'_{s}&a'_{t}\\b'_{r}&b'_{s}&b'_{t}\\c'_{r}&c'_{s}&c'_{t}\end{array}}\end{array}}\right|{\begin{array}{c}dx'_{r}dx'_{s}dx'_{t}.\end{array}}$

It will be seen also that this equation may he obtained by multiplying the integral form of the second order derived from (α) and (β), by (γ). The question then arises whether integral forms of any order may be multiplied together by Grassmann's rule. To show that this is the case we shall consider the two integral forms

${\begin{array}{rcl}\sum \limits _{s,t,\dots }A_{s,t,\dots }dx_{s}dx_{t}\dots &=&\sum \limits _{s,t,\dots }A'_{s,t,\dots }dx'_{s}dx'_{t}\dots ,\\\\\sum \limits _{p,q,r,\dots }B_{p,q,r,\dots }dx_{p}dx_{q}dx_{r}&=&\sum \limits _{p,q,r,\dots }B'_{p,q,r,\dots }dx'_{p}dx'_{q}dx'_{r}\dots ,\end{array}}$

of orders k and m respectively. Choosing m + k variables $\alpha ,\beta ,\dots ,$, we have ${\frac {(m+k)!}{m!k!}}$ relations of the type

$\sum \limits _{s,t,\dots }A_{s,t,\dots }{\frac {\partial \left(x_{s},x_{t},\dots \right)}{\partial \left(\alpha ,\beta \dots \right)}}=\sum \limits _{s,t,\dots }A'_{s,t,\dots }{\frac {\partial \left(x'_{s},x'_{t},\dots \right)}{\partial \left(\alpha ,\beta \dots \right)}},$

and the same number of relations of the type

$\sum \limits _{p,q,r,\dots }B_{p,q,r,\dots }{\frac {\partial \left(x_{p},x_{q},x_{r}\dots \right)}{\partial \left(\gamma ,\delta ,\epsilon ,\dots \right)}}=\sum \limits _{p,q,r,\dots }B'_{p,q,r,\dots }{\frac {\partial \left(x'_{p},x'_{q},x'_{r}\dots \right)}{\partial \left(\gamma ,\delta ,\epsilon ,\dots \right)}}.$

These may be arranged in conjugate pairs in such a way that the whole set of variables $\alpha ,\beta ,\gamma ,\delta ,\dots$ occur in each pair. Multiplying conjugate pairs together, attributing proper signs to each product, and adding, we find from the properties of the minors of a determinant that the coefficient of a term such as $A_{s,t,\dots }B_{p,q,r,\dots }$, is zero, unless the quantities s,t,p,q,r,... are all different. When these quantities are different the coefficient is simply the determinant ${\frac {\partial \left(x_{s},x_{t},\dots ,x_{p},x_{q},x_{r},\dots \right)}{\partial \left(\alpha ,\beta ,\gamma ,\dots \right)}}$ with a proper sign. When multiplied by $d\alpha \ d\beta \ d\gamma \dots$ this yields $dx_{s},dx_{t},\dots ,dx_{p},dx_{q},dx_{r},\dots$ and it is clear from this that the multiplication of the integral forms by Grassmann's rule is justified.

Taking the case of a transformation from four variables (x, y, z, t) to (x', y', z', t'). If

${\begin{array}{l}H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\\qquad =H'_{x}dy'dz'+H'_{y}dz'dx'+H'_{z}dx'dy'+E'_{x}dx'dt'+E'_{y}dy'dt'+E'_{z}dz'dt',\end{array}}$