186 Proceedings of the Royal Society of Edinburgh. [Sess. 
that as the nucleus is approached the material of the earth is becoming 
less of an elastic solid and more of an elastic, highly compressed liquid. 
The change probably comes on gradually within a comparatively thin 
shell whose outer and inner radii are, say, 0-5 and 04 of the earth’s radius. 
Within this nucleus of radius 04 the material of the earth has lost its 
elastic solid character and can transmit only compressional waves with 
speed \/k/p, where k is the incompressibility and p the density of the 
material. 
It is not possible to make any definite calculations as to the manner in 
which elastic waves of both types are transformed during transmission 
across this gradually changing layer. It may be spoken of as a semi- 
permeable layer, for no distortional wave can pass through it. 
We may, however, gain some idea of what occurs by considering the 
limiting case of an elastic solid passing abruptly into a non-rigid elastic 
substance of equal compressibility and density. This is one of a general 
type of problem which I discussed as early as 1888 in a paper read before 
the Seismological Society of Japan and published in their Transactions. 
The paper was reprinted with additions in the Philosophical Magazine 
(July 1899) under the title “ Reflexion and Refraction of Elastic Waves 
with Seismological Applications.” 
The details of the present calculation need not be given. It is a simple 
enough matter to construct the special forms of the equations determining 
the energies of the refracted and reflected waves in terms of the energy 
of the incident wave as they apply to the case now imagined. The data 
are the velocities of the compressional and distortional waves in the elastic 
shell ; and the assumption is that the non-rigid nucleus has the same 
compressibility as the enclosing shell. If V and U represent the speeds 
of propagation of the two types of wave in the shell, then 
y2 = fe + 4«/3 and U2 = ^ 
P P 
Hence the square of the speed of propagation of the compressional wave 
in the nucleus is 
V' 2 = ^ =V 2 -4U 2 /3. 
P 
Now V and U are given from the seismic calculations, hence the value 
of V' is found, and from these the angles of reflection and refraction. 
Thus taking from the Tables IV and V the values 12-89 and 6 88 for 
V and U, we find for V 7 the value 10*15. The ratios of these, taken in 
pairs, give either the refractive index in the usual sense, or what might 
