Page:BatemanConformal.djvu/13

where

$\theta +\phi +\psi =0,$

we see that

$(z-a)^{\theta }(z-b)^{\phi }(z-c)^{\psi }P\ \left\{{\begin{array}{ccccccc}a&&b&&c\\\\\alpha &&\beta &&\gamma &&z\\\\\alpha '&&\beta '&&\gamma '\end{array}}\right\}$

is a multiple of

$P\ \left\{{\begin{array}{ccccccc}a&&b&&c\\\\\alpha +\theta &&\beta +\phi &&\gamma +\psi &&z\\\\\alpha '+\theta &&\beta '+\phi \ &&\gamma '+\psi \end{array}}\right\}.$

Again, if we write

${\begin{array}{cl}a'={\frac {Aa+B}{Ca+D}},&b'={\frac {Ab+B}{Cb+D}},\\\\c'={\frac {Ac+B}{Cc+D}},&z'={\frac {Az+B}{Cz+D}},\end{array}}$

we have

$(b'-c')(z'-a')={\frac {(AD-BC)^{2}}{(Ca+D)(Cb+D)(Cc+D)(Cd+D)}}(b-c)(z-a),$

so that

${\frac {l\lambda }{(b'-c')(z'-a')}}={\frac {m\mu }{(c'-a')(z'-b')}}={\frac {n\nu }{(a'-b')(z'-c')}},$

This shows that P is the same function of the quantities a', b', c', z' as it is of a, b, c, z; that is

$P\ \left\{{\begin{array}{ccccccc}a'&&b'&&c'\\\\\alpha &&\beta &&\gamma &&z'\\\\\alpha '&&\beta '&&\gamma '\end{array}}\right\}=P\ \left\{{\begin{array}{ccccccc}a&&b&&c\\\\\alpha &&\beta &&\gamma &&z\\\\\alpha '&&\beta '&&\gamma '\end{array}}\right\}.$

Hence the general hypergeometric function is unaltered if the quantities a, b, c, z are replaced by quantities a', b', c', z' which are derived from them by the same homographic substitution.

4. Applications to Geometrical Optics.

Let us consider a series of waves of light traversing a homogeneous or heterogeneous medium, and let

$V=\int \mu ds$

be the reduced path from a standard orthotomic surface or wave front to the point (x, y, z). Let us suppose, moreover, that V is expressed only in