Page:BatemanElectrodynamical.djvu/25

 These formulae are well known in the differential geometry of a surface.

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

then by multiplication we may obtain an integral form of the second order

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,

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