159 
1918-19.] The Propagation of Earthquake Waves. 
defy 
ds 
dO p dv 
ds r dr 
giving the curvature * 
1 _ d(<f> + 6) 
p ds 
p dv 
r dr 
(7) 
The parameter p is constant along each brachistochronic path or ray 
but varies from ray to ray, having by definition the value dT/dO. Thus 
along any ray the curvature depends on the expression r^dv/dr. It is 
conceivable that, through a limited interval from r = r 1 to r=r 2 , this 
expression may be constant. This consideration, which is referred to by 
Rudzki in a note at the end of his paper, is made the foundation on which 
Wiechert and Zoppritz construct their numerical solution of the problem 
of finding the law connecting v and r. f 
Assuming constancy, they put 
1 ^ 
r dr 
~ +/ 5 
the sign being negative or positive according as v diminishes or increases 
with distance from the earth’s centre. Integration gives 
tr=/J/(R 2 + r 2 ), 
either assumed law of variation carrying with it the circular or constant 
curvature form of path within each of a succession of concentric spherica] 
layers of the earth. By taking three layers and properly piecing together 
the results for the successive layers, Wiechert and Zoppritz worked the 
problem out in full detail. It is to be noted, however, that this process 
of solution of equation (4) is a tentative one, being based on the assump- 
tion of an analytical form of v as a function of r. 
Passing back now to equations (6), let us continue the analytical 
discussion of the integral equation. In the first place, if we put 0 = ?, 
lj 
we find 
1 ==^— or z—pv . . . . , (8) 
Now for any given ray p = 3T/30, a relation which holds for the special 
case when T and 0 refer to the point on the surface at which the ray 
emerges. This value of T is determined by observation at stations where 
delicate seismographs are installed, and may be tabulated as a numerical 
* This result, among others, was first given by Rudzki (see Gerland’s Beitrdge zur 
GeophysiJc, vol. iii, p. 496 (1898)). 
+ See their paper, “ Ueber Erdbebenwellen,” Nachrichten von der Konig. Gesell. d. Wissen- 
schaften zu Gottingen , 1907. p. 491 et seq. 
