Page:BatemanConformal.djvu/3

expressed in terms of Riemann's general hypergeometric function, and new light is thrown upon the theory of the transformations of the hypergeometric equation into itself.

In § 4 the applications to geometrical optics are considered. When applied to a symmetrical optical instrument, the transformation reduces to a homographic transformation of the points on the axis.

2. The Conformal Transformations of a Space of Four Dimensions.

The study of the conformal transformations of a space of four dimensions is simplified by the introduction of the six homogeneous coordinates^[1]:

$\left.{\begin{array}{lll}l=x-iy,&m=z+iw,&n=x^{2}+y^{2}+z^{2}+w^{2}\\\\\lambda =x+iy,&\mu =z-iw,&\nu =-1\end{array}}\right\}$

(1)

connected by the identical relation

$l\lambda +m\mu +n\nu =0.$

(2)

A function F(x, y, z, w) can be expressed by means of them as a homogeneous function of arbitrary degree. For instance, we may write

$V=F(x,y,z,w)=F\left(-{\frac {l+\lambda }{2\nu }},-{\frac {l-\lambda }{2i\nu }},-{\frac {m+\mu }{2\nu }},-{\frac {m-\mu }{2i\nu }}\right),$ $U=f(x,y,z,w)=-{\frac {1}{\nu }}f\left(-{\frac {l+\lambda }{2\nu }},-{\frac {l-\lambda }{2i\nu }},-{\frac {m+\mu }{2\nu }},-{\frac {m-\mu }{2i\nu }}\right).$

In the first representation V is a homogeneous function of degree zero, and in the second U is a homogeneous function of degree -1. The coordinate n may be introduced into the representations by means of the identical relation (2), the homogeneity of the expression being thereby unaltered.

Conversely, any homogeneous function of the six variables (l, m, n, λ, μ, ν) can be expressed as a function of (x, y, z, w). We shall now consider under what circumstances such a function can satisfy one of the differential equations—

${\begin{array}{r}\left({\frac {\partial V}{\partial x}}\right)^{2}+\left({\frac {\partial V}{\partial y}}\right)^{2}+\left({\frac {\partial V}{\partial z}}\right)^{2}+\left({\frac {\partial V}{\partial w}}\right)^{2}=0,\\\\{\frac {\partial ^{2}U}{\partial x^{2}}}+{\frac {\partial ^{2}U}{\partial y^{2}}}+{\frac {\partial ^{2}U}{\partial z^{2}}}+{\frac {\partial ^{2}U}{\partial w^{2}}}=0.\end{array}}$

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1. These bear the same relation to the hexaspherical coordinates of a point as the ordinary line coordinates of a line bear to the system of coordinates introduced by Klein.