The equations of motion of an elastico-viscous medium in which the material constants
vary with position are deduced. These can be put into the form of a wave equation only
when the gradients of the constants are small. By the method of Sommerfeld and Runge
these equations are compared with the equation of the characteristic function, whence the
condition for the validity of the ray method is obtained. It is similar to De Broglie's criterion
in wave mechanics. Expressed in terms of measurable quantities in seismology, the condition
is applied to the data recently obtained by Gutenberg for the upper layers of the earth's
crust. The equation of the characteristic function is used in deriving the forms of the ray
paths for several particular velocity functions, following a method previously used by
Epstein
