Velocity of Propagation of an Earthquake- Wave. 577 



an interesting case in which it is assumed that the rays are 

 circles. 



For theoretical purposes it is desirable to have a general 

 method of solving the inverse problem, and so an attempt 

 has been made in this paper to study the properties of the 

 integral equation to which the problem is reduced when 

 it is assumed that the earth may be divided into a series 

 of concentric spherical shells within each of which the 

 material is uniform. In this ide;il case the elastic con- 

 stants, and consequently the velocities of the seismic waves, 

 are functions only of the distance from the centre of the 

 earth. 



The integral equation may be easily identified with one 

 which was partially solved by Abel* in 1823, and later by 

 Liouville t in 1832. 



By a few transformations we may obtain a formula con- 

 necting the speed of propagation with the times to different 

 points on the Earth's surface. A simple case has been 

 worked out for purposes of illustration. The problem has 

 been solved on the assumption that the seat of the disturb- 

 ance can be regarded as a point on the surface of the Earth. 

 This condition is only approximately satisfied in actual cases, 

 for the hypocentre may be several miles below the surface, 

 and the disturbances may radiate from a crack instead of a 

 point. This may be allowed for roughly by assigning the 

 time T = to the points of a small circle surrounding the 

 epicentre. 



1 wish to express my gratitude to Dr. Schuster for sug- 

 gesting the investigation and for the interest which he has 

 taken in it. 



2. In order to simplify the mathematical analysis it is 

 convenient to treat the Earth as a sphere of radius R, and to 

 suppose that the elastic constants of the substance of which 

 it is composed are the same at all points of a concentric 

 sphere. The velocity of propagation of a disturbance at any 

 point is then a function of the distance of the point from the 

 centre of the Earth. 



Let v be the speed at distance r from the centre,and T the 

 time the disturbance has taken to travel from the seat of the 

 earthquake to the point in question, Then by Hamilton's 



* Collected Works (Svlow and Lie edition), vol. i. p 11. 

 t Liouville's Journal,\o\. iv. (1839) p. 233. Journal de VEcole Poly- 

 technique, cahier21 (1832) p. 1. 



Phil. Mag. S. 6. Vol. 19. No. 112. April 1010. 2 P 



