Page:BatemanConformal.djvu/16

1908.]
85
The conformal transformations of a space of four dimensions.

Consider a ray of light travelling from ${\displaystyle \left(x_{0},\ y_{0},\ z_{0}\right)}$ in a direction (l, m, n) with velocity c. At time t the wave has reached a point (x, y, z) on the ray, where

 ${\displaystyle x=x_{0}+l.ct,\ y=y_{0}+m.ct,\ z=z_{0}+n.ct.}$

The corresponding point (X, Y, Z) derived from this by the transformation also travels along a straight line, for

 ${\displaystyle {\begin{array}{cl}X&={\frac {x_{0}+lct}{\left(x_{0}+lct\right)^{2}+\left(y_{0}+mct\right)^{2}+\left(z_{0}+nct\right)^{2}-c^{2}t^{2}}}\\\\&={\frac {x_{0}}{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}+\left[{\frac {ct}{\left(x_{0}+lct\right)^{2}+\left(y_{0}+mct\right)^{2}+\left(z_{0}+nct\right)^{2}-c^{2}t^{2}}}\right]\times \left[l-{\frac {2x_{0}}{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}\left(lx_{0}+my_{0}+nz_{0}\right)\right]\\\\&=X_{0}+LcT,\end{array}}}$,

where

 ${\displaystyle T={\frac {t}{r^{2}-c^{2}t^{2}}}.}$

The corresponding ray thus passes through the inverse point ${\displaystyle \left(X_{0}\ Y_{0}\ Z_{0}\right)}$ on the inverse surface at which it may be supposed to be incident. Its direction cosines (L, M, N) are connected with those of the former ray by means of the equations

 ${\displaystyle {\begin{array}{cl}L&=l-{\frac {2x_{0}}{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}\left(lx_{0}+my_{0}+nz_{0}\right)\\\\M&=m-{\frac {2y_{0}}{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}\left(lx_{0}+my_{0}+nz_{0}\right)\\\\N&=n-{\frac {2z_{0}}{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}\left(lx_{0}+my_{0}+nz_{0}\right).\end{array}}}$

These relations establish a correspondence between the sheafs of rays through the points ${\displaystyle \left(x_{0},\ y_{0},\ z_{0}\right)}$, ${\displaystyle \left(X_{0},\ Y_{0},\ Z_{0}\right)}$ respectively. This correspondence is such that the angle between two rays (l, m, n), (l', m', n') is equal to the angle between the two corresponding rays (L, M, N), (L', M', N'), for we have identically

 ${\displaystyle LL'+MM'+NN'\equiv ll'+mm'+nn'.}$

Since the transformation enables us to derive the surfaces which are parallel to one surface from the surfaces which are parallel to the inverse surface, it is natural to expect that the above relation between the direction cosines will make the normals to the two surfaces correspond.