# Page:BatemanConformal.djvu/18

1908.]
87
The conformal transformations of a space of four dimensions.

From a study of the nodes in the ether produced by a vibrating atom or molecule, it appears that various systems of nodes may be obtained from one another by inversions which correspond to the same frequencies of vibration. The investigation will be reserved, however, for another paper.

5. Application of the Preceding Results to a Symmetrical Optical Instrument.

Let QR (Fig.) be the incident ray, Q'R the refracted ray, OR the normal to the spherical interface, and let C be the centre of inversion.

We shall suppose that the incident ray makes a small angle with the axis.

Let

 ${\displaystyle AC=z,\ AO=a,\ AQ=x,\ AQ'=x',}$

and let the velocity of light for the first medium be represented by 1/μ, the times corresponding to the points Q, Q', O, A respectively may then be taken to be -μx, -μx', -μa, 0, respectively, and the corresponding quantities ct are simply the reduced distances (-x, -x', -a, 0).

Let ${\displaystyle Q_{1},\ Q'_{1},\ O_{1},\ A_{1}}$ be the points corresponding to Q, Q', O, A in the transformation, ${\displaystyle x_{1},\ x'_{1},\ z_{1}-a_{1},\ z_{1}}$ their distances from C.

Now the sphere centre Q and radius QA inverts into the sphere centre ${\displaystyle Q_{1}}$ and radius ${\displaystyle QA_{1}}$; hence, if the radius of inversion be equal to k³, we have for the points in which these spheres meet the axis

 ${\displaystyle z_{1}={\frac {k^{2}}{z}},\ z_{1}-2x_{1}={\frac {k^{2}}{z-2x}}.}$

We also have the relations

 ${\displaystyle A_{1}Q_{1}=-x_{1}={\frac {k^{2}x}{z(z-2x)}},}$ ${\displaystyle A_{1}Q_{1}'=-x'_{1}={\frac {k^{2}x'}{z(z-2x')}},}$ ${\displaystyle A_{1}O_{1}=-a_{1}={\frac {k^{2}a}{z(z-2a)}}.}$