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82 where

we see that

is a multiple of

Again, if we write

we have

so that

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

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

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