220 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
pagation of the surface trace of the disturbance is obviously greater than 
the true speed of the wave in the ratio of unity to the cosine of the angle 
of emergence. Hence a is the reciprocal of the Active speed of propagation 
of the surface trace. This relation, which can be established almost 
intuitively by consideration of the manner in which an impinging wave 
runs along a boundary, must be as old as Fresnel, and is made use of 
explicitly by Green and many later writers on the laws of reflexion and 
refraction of light and elastic waves. 1 It gives, indeed, the most lucid 
explanation of the phenomenon of total reflexion. Nevertheless, both 
V. Kovesligethy and Benndorf think it necessary to prove the relation 
analytically, and to enunciate it as something altogether new in the theory 
of wave motion. 
It is clear that a knowledge of the angle of emergence can give no 
information regarding the speeds of propagation in the deeper parts of the 
earth. Benndorf has, however, by certain plausible assumptions as to the 
limiting values of the quantities involved, obtained a solution which 
satisfies the values of the angles of emergence determined experimentally 
by Schliiter. The experimental determination of the angle of emergence 
depends on the comparison of the vertical displacement of the ground with 
the simultaneous maximum horizontal displacement, and involves a 
mathematical reduction. The displacements were measured on appropriate 
forms of seismometer. They were very small and subject to large errors. 
Taken as a whole, Schliiter’s values for the angles of [emergence at one 
locality due to earthquake tremors coming from sources at different 
distances are in accordance with what would be expected from the nature 
of the problem ; but the data are too meagre to establish any peculiarity 
of detail such as seems to be indicated. 
I purpose to work out a definite case, assuming for the speed v the 
expression 
v 2 WV 
To determine V and jul we have the two conditions, (1) the value of v at the 
surface, (2) the time of transit across a diameter. 
After some trials I chose the following expression as satisfying these 
two conditions fairly well, namely, 
v = fxJd*^==l3‘Q Jl-2-x 2 . 
Putting 0 = jua, we get equations (2) and (3) in the somewhat simpler 
forms 
1 See, for example, my paper of 1888 on Earthquakes and Earthquake Sounds, etc., 
Trans. Seism. Soc. of Japan, vol. xii., p. 123 ; also Phil. Mag., July 1899, p. 71. 
