1918-19.] The Propagation of Earthquake Waves. 
157 
XIV. — The Propagation of Earthquake Waves through the 
Earth, and connected Problems. By Professor 0. G-. Knott, 
D.Sc., LL.D. 
(MS. received July 10, 1919. Read November 4, 1918, and January 20, 1919.) 
This paper is a continuation of two papers on Seismic Radiations published 
in the Proceedings of the Royal Society of Edinburgh, vol. xxviii, pp. 217- 
230 (1907-8) and vol. xxx, pp. 23-37 (1909). The object of the present 
communication is to place on record a new determination of the laws of 
propagation of seismic waves based upon a method of calculation in which 
no assumptions are made as to the functional relation between velocity of 
propagation and distance from the earth’s centre. References to the work 
of others will be given incidentally as occasion arises.* 
To make the present discussion intelligible in itself, it is necessary to 
reproduce from my earlier paper the mathematical investigation on which 
the calculations are based, along with the fruitful transformation given 
by Dr Bateman in a paper published in the Philosophical Magazine for 
April 1910. 
The earth is assumed to be a sphere, the elastic properties at any point 
being a function only of the distance from the earth’s centre. 
The disturbance, assumed to originate near the surface, will be propa- 
gated in a succession of waves, each trajectory or ray having the property 
of a brachistochrone meeting the surface at a point depending on the 
initial direction of the ray, and lying wholly in the plane containing the 
centre, the source, and the point of emergence. The position of any point 
may therefore be determined by the polar co-ordinates r and 0, referred 
to the earth’s centre and to any convenient line in the plane of the ray 
passing through the point. 
If v is the velocity of propagation at any point, then Hamilton’s general 
method applied to brachistochronic problems gives 
T being the time. 
The discussion is simplified when the earth’s radius is chosen as the 
unit length, so that r is a fraction and v is expressed in the unit earth - 
radius per second. 
* See, however, the closing paragraph of the second paper referred to above for an 
account of the important work of Wiechert and Zoppritz in this connection. 
