Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/184

152 given their relative position is as completely determined as if they formed part of the same rigid body.

512.] If we use rectangular coordinates and make $$x$$, $$y$$, $$z$$ the coordinates of $$P$$, and $$x^\prime$$, $$y^\prime$$, $$z^\prime$$ those of $$P^\prime$$, and if we denote by $$l$$, $$m$$, $$n$$ and by $$l^\prime$$, $$m^\prime$$, $$n^\prime$$ the direction-cosines of $$PQ$$, and of $$P^\prime Q^\prime$$ respectively, then

where $$\epsilon$$ is the angle between the directions of the elements themselves, and Rh

Again

and differentiating $$r\frac{dr}{ds}$$ with respect to $$s^\prime$$,

We can therefore express the three angles $$\theta$$, $$\theta^\prime$$, and $$\eta$$, and the auxiliary angle $$\epsilon$$ in terms of the differential coefficients of $$r$$ with respect to $$s$$ and $$s^\prime$$ as follows,