Page:BatemanElectrodynamical.djvu/25

These formulae are well known in the differential geometry of a surface.^[1]

We shall now show that the above theory may be extended to transformations in any number of variables. In the first place we must show that the law of multiplication still holds. Let there be n variables $x_{1},\dots ,x_{n}$ and suppose that

$\sum a_{r}dx_{r}=\sum a'_{r}dx'_{r}$ $\sum b_{r}dx_{r}=\sum b'_{r}dx'_{r};$

then by multiplication we may obtain an integral form of the second order^[2]

$\sum \left(a_{r}b_{s}-a_{s}b_{r}\right)dx_{r}dx_{s}=\sum \left(a'_{r}b'_{s}-a'_{s}b'_{r}\right)dx'_{r}dx'_{s}$

the multiplication being performed by Grassmann's rule. To verify this we have only to replace $dx_{r}dx_{s}$ by ${\frac {\partial \left(x_{r},x_{s}\right)}{\partial (\alpha ,\beta )}}d\alpha \ d\beta$, and notice that the last equation may be written

$\left|{\begin{array}{ccc}\sum a_{r}{\frac {\partial x_{r}}{\partial \alpha }},&&\sum b_{r}{\frac {\partial x_{r}}{\partial \alpha }}\\\\\sum a_{r}{\frac {\partial x_{r}}{\partial \beta }},&&\sum b{\frac {\partial x_{r}}{\partial \beta }}\end{array}}\right|{\begin{array}{c}d\alpha \ d\beta =\end{array}}\left|{\begin{array}{ccc}\sum a'_{r}{\frac {\partial x'_{r}}{\partial \alpha }},&&\sum b'_{r}{\frac {\partial x'_{r}}{\partial \alpha }}\\\\\sum a'_{r}{\frac {\partial x'_{r}}{\partial \beta }},&&\sum b'_{r}{\frac {\partial x'_{r}}{\partial \beta }}\end{array}}\right|{\begin{array}{c}d\alpha \ d\beta .\end{array}}$

Similarly, if we take three integral forms,

$\sum a_{r}dx_{r}=\sum a'_{r}dx'_{r},$

(α)

$\sum b_{r}dx_{r}=\sum b'_{r}dx'_{r},$

(β)

$\sum c_{r}dx_{r}=\sum c'_{r}dx'_{r},$

(γ)

and multiply them together by Grassmann's rule, we obtain the integral

1. Darboux, Théorie générale des Surfaces, t. 3, p. 193. In particular, if

   $dx^{2}+dy^{2}=\lambda \left[dx'^{2}+dy'^{2}\right],$

   we may deduce from the identity

   ${\frac {\partial V}{\partial x}}dx+{\frac {\partial V}{\partial y}}dy={\frac {\partial V}{\partial x'}}dx'+{\frac {\partial V}{\partial y'}}dy',$

   that

   ${\frac {\partial V}{\partial y}}dx-{\frac {\partial V}{\partial x}}dy={\frac {\partial V}{\partial y'}}dx'-{\frac {\partial V}{\partial x'}}dy',$

   and on multiplying by (9), we get the well known relation

   $\left({\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}\right)dx\ dy=\left({\frac {\partial ^{2}V}{\partial x'^{2}}}+{\frac {\partial ^{2}V}{\partial y'^{2}}}\right)dx'\ dy'.$

2. An integral form of the second order must be carefully distinguished from a quadratic differential form.