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LXT. The Solution of the Integral Equation connecting the 

 Velocity of Propagation of an Earthquake- Wave in the 

 Interior of the Earth with the Times which the Disturbance 

 takes to travel to the different Stations on the Earth's 

 Surface. By H. Bateman *. 



1. rriHE records of instruments at different places on the 

 JL Earth's surface indicate that the seismic disturb- 

 ances produced by an earthquake are of several distinct 

 kinds t- The disturbances that are first recorded are called 

 by Milne the " preliminary tremors." They are supposed to 

 be propagated through the body of the Earth with a speed 

 depending on the elastic properties of the material through 

 which they pass. In reality there are two speeds, for the 

 seismic waves in which the vibrations take place in a longi- 

 tudinal direction generally travel faster than those in which 

 the vibrations are transverse to the direction of motion. 

 The speed of propagation in either case is given by the 

 formula 



where D is the density of the material and E the appropriate 

 elastic modulus. 



It has been suggested that this formula may provide us 

 with a means of estimating the elastic constants of the 

 materials of which the Earth is composed. The problem to be 

 solved is the inverse one of calculating the speed of propa- 

 gation at different points inside the Earth from the observed 

 times of transit to different places on the Earth's surface. 



RudzkiJ and V. Kovesligethy § treated this problem 

 mathematically by regarding the rays along which the 

 energy is propagated as brachistochronic paths ; their ma- 

 thematical assumptions, however, were made so as to get a 

 soluble case, and had no dynamical harmony with known 

 seismological facts. 



Recently Prof. 0. Gr. Knott || has discovered a law of 

 velocity which gives results agreeing closely with the obser- 

 vations, while Wiechert and Zoeppritz If have worked out 



* Communicated by the Author. 



t A good account of the subject is given in Prof. Knott's book, 'The 

 Physics of Earthquake Phenomena,' Clarendon Press (1908). 



\ Beitrdge zur Geophysik, iii. (1908). 



§ Mathem. u. Naturw. Berichte aus Ungarn, xiii. (1897) ; xxiii. (1905). 



j| Proceedings of the Royal Society of Edinburgh, vol. xxviii. Part 3 

 (1907). 



•f] Gottinger Xachrichlen (190/), Heft 4. 



