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LXXII. Note on Mr. Batemans Paper on Earthquake* 

 Waves. By Robekt E. Baynes *. 



TOWARDS the end of his interesting paper on Earth- 

 qoake-waves in the April number of the Phil. M ig. 

 (p. 585), Mr. Bateman connects the times of transit T of the 

 iirsf-phase waves to stations at angular distance 6 from the 

 source, as given by Prof. Milne, by a formula of the type 

 T = C-f- A0 — B# 2 , and, by assumption of the relation 

 dT/d6 = (R/\J) cose, where R is the earth's radius, U the 

 speed at the surface, and e the angle of emergence, shows 

 that U = R/A if is measured in radians, and, by further 

 application of Abel's transformation to the equation of the 

 path, that the speed v at distance R.r from the centre is 

 given by f 



(l-UV/V^-sech- 1 (Ua/v) = (2ttB/A) log x. . (1) 



The same procedure is of course equally applicable to tho 

 equation for the time along the path which is given on p. 583 

 and which takes the form 





cf)(t)dt 



on putting frl-UV/^ (ll 2 x/v 2 )dx^^(t)dt, sEEsm 2 e; 

 the relation assumed above then gives 



T=OR/2U)(E*+2F), 



where 



E-R/2ttBU and F-CU/7rR + (A 2 U 2 -R 2 )/47rBRU, 



and the transformation gives 



whence 



log l r = E^"(E + F)tanh- 1 ^ 



= E(l-U 2 <r 2 A' 2 )^-(E + F)sech- 1 (LVr). . (2) 



Comparison of (2) with (1) requires E= A/277 B and F = 0, 

 i, e. IT = R/A as before and = 0. The latter result is doubly 

 obvious; for T must vanish with 0, and the assumed relation 

 is not true except with this condition. 



* Communicated by the Author. 



t In three plnces the factor R has dropped out by a slip and in the 

 Table jj//3 should be substituted for q. 



