Page:BatemanConformal.djvu/20

1908.]
89
The conformal transformations of a space of four dimensions.

The linear magnification is given by

 ${\displaystyle m'_{1}={\frac {\frac {B_{1}Q''_{1}}{\mu ''}}{\frac {B_{1}Q'_{1}}{\mu '}}}={\frac {\mu '\xi ''\left(\zeta '+c-2\xi '\right)}{\mu ''\xi '\left(\zeta '+c-2\xi ''\right)}}.}$

hence

 ${\displaystyle m'_{1}=m'{\frac {\zeta '+c-2\xi '}{\zeta '+c-2\xi ''}}=m'{\frac {z-2x'}{z-2x''}},}$

which is of the same form as before.

If M and ${\displaystyle M_{1}}$ be the total linear magnifications for the two instruments, we have

 ${\displaystyle M_{1}=M{\frac {z-2x}{z-2x^{(n)}}}.}$

The relation between two corresponding points is evidently a homographic one; hence we have the following theorem:—

If the points on the axis of a symmetrical optical instrument be transformed by means of a homographic transformation, any pair of conjugate points for the instrument are transformed into a pair of points which are conjugate with regard to a second instrument. The centres of curvature of the interfaces and the points in which the interfaces meet the axis correspond in the two instruments, and the refractive indices of corresponding media are the same.