82
[Nov. 12,
Mr. H. Bateman
where
$\theta +\phi +\psi =0,$


we see that
$(za)^{\theta }(zb)^{\phi }(zc)^{\psi }P\ \left\{{\begin{array}{ccccccc}a&&b&&c\\\\\alpha &&\beta &&\gamma &&z\\\\\alpha '&&\beta '&&\gamma '\end{array}}\right\}$


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
$(b'c')(z'a')={\frac {(ADBC)^{2}}{(Ca+D)(Cb+D)(Cc+D)(Cd+D)}}(bc)(za),$


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