442 Professor Cargill G. Knott [Feb. 14, 



Unfortunately the expressions on which the value of the time 

 and the form of the ray depend contain an unintegrable definite 

 integral, involving the speed v as an unknown function of the 

 distance from the centre of the earth. By assuming manageable 

 forms for this function, various investigators have obtained empirical 

 solutions sufficiently satisfactory to lead to conclusions not far from 

 the truth. Such indirect and empirical methods are, however, not 

 entirely satisfying. It is claimed that in the present discussion the 

 problem has been worked out for the first time by an absolutely 

 rigorous method from the data of observation, no assumptions being 

 made as to the functional relationship connecting the speed of 

 propagation and the distance from the earth's centre. The method 

 is a numerical solution of an integral equation, giving each speed 

 and the corresponding distance in figures. It will be found explained 

 in detail in a paper read before the Royal Society of Edinburgh in 

 November 1918 (see Proceedings R.S.E., vol. xxxix, pp. 157-208). 

 It will suffice to indicate a few of the results obtained. 



With the exception of the seismic ray which starts downwards 

 towards the earth's centre and presumably emerges at the antipodes 

 of the epicentre, all the seismic rays begin by curving so as to have 

 the concavity towards the surface. This means that the speed of 

 propagation of each type of wave increases as the depth below 

 the surface increases. The outward concavity characterises the 

 whole length of all rays which emerge at distances less than about 

 G0° from the epicentre. But for greater arcual distances the ray 

 straightens out at its middle and deeper part, and even becomes dis- 

 tinctly convex towards the surface. This change in the sign of 

 curvature takes place at a depth equal to about three-tenths of the 

 earth's radius. At this depth accordingly the speed of the wave 

 which for shallower depths increases with the depth begins to 

 diminish with increase of depth. 



The forms of several of these seismic rays are shown in the 

 accompanying figure, which represents fully a quadrant of a diametral 

 section of the earth through the position of the epicentre. The 

 sinuosity begins to appear in Ray VI. and is most marked in Ray 

 VII, which emerges at an arcual distance of 73° from the epi- 

 centre. Ray VIII becomes practically straight throughout a con- 

 siderable length, indicating that the speed is not changing with the 

 depth. 



The rays shown in the figure belong to the Primary wave. If 

 the rays of the Secondary wave were placed on the same figure they 

 would deviate very slightly from those of the Primary. In the 

 figure the wave-fronts of the Secondary waves are entered in 

 broken lines to distinguish them from the wave-fronts of the Primary 

 waves which are drawn in full. The numbers attached are the 

 times in minutes for the wave-fronts to pass from the earthquake 

 origin to the marked position. The broad similarity between the 



