Page:BatemanConformal.djvu/9

the transformation as a change of axes in a space in which the coordinates are the spherical coordinates $a_{r}$.^[1] It is important to notice that the angle between two manifolds in this space is equal to the angle between the corresponding manifolds in the space to which the conformal transformations are applied. In the case of a space of four dimensions, we have, in fact,

$da_{1}^{2}+da_{2}^{2}+da_{3}^{2}+da_{4}^{2}+da_{5}^{2}+da_{6}^{2}\equiv dx^{2}+dy^{2}+dz^{2}+dw^{2},$

$da_{1}da'_{1}+da_{2}da'_{2}+da_{3}da'_{3}+da_{4}da'_{4}+da_{5}da'_{5}+da_{6}da'_{6}$
$\equiv dx\ dx'+dy\ dy'+dz\ dz'+dw\ dw'$,

from which the result easily follows.

A change in the sign of $a_{6}$ corresponds to an inversion, a change in the sign of $a_{6}$ coupled with a change in the sign of $a_{4}$ corresponds to the other transformation we have mentioned. It is evident that each of these transformations is of period 2. In general, a reflexion in a linear manifold in the a space corresponds to an inversion with regard to the corresponding circle, sphere, or hypersphere, in the space of four dimensions. A displacement of period n in the a space may be obtained by taking successive reflexions in two plane five-folds which cut at an angle π/n.^[2] It evidently corresponds to a periodic conformal transformation made up of inversions with regard to two hyperspheres which cut at an angle π/n.

3. The Relation between Riemann's General Hypergeometric Function and the Group of Conformal Transformations of a Space of Four Dimensions.

We shall now endeavour to satisfy the differential equation

${\frac {\partial ^{2}U}{\partial l\partial \lambda }}+{\frac {\partial ^{2}U}{\partial m\partial \mu }}+{\frac {\partial ^{2}U}{\partial n\partial \nu }}=0$

by means of a function of the form

$U=l^{-\alpha }\lambda ^{-\alpha '}m^{-\beta }\mu ^{-\beta '}n^{-\gamma }\nu ^{-\gamma '}P,$

(1)

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1. Reference should be made to Darboux, Théorie des Surfaces, Tome I., p. 213. Jessop's Treatise on the Line Complex, p. 251. Koenig's La Géometrie réglée, p. 125, and to an article by Borel on the "Transformations of Geometry in Niewenglowsky's Solid Geometry."
2. This is equivalent to a rotation through an angle 2π/n just as successive reflexions in two planes are equivalent to a rotation about their line of intersection.