470 ME. R. D. OLDHAM ON THE CONSTITUTION [Aug. I906, 



important questions to be answered, namely, the size of the core 

 and the rate of transmission of the waves in it. 



As regards the size of the core, we have seen that it is not 

 penetrated by the wave-paths which emerge at 120° ; and the great 

 decrease at 150° shows that the wave-paths emerging at this 

 distance have penetrated deeply into it. Now, the chord of 120° 

 reaches a maximum depth from the surface of half the radius, and 

 we have seen that the wave-paths up to this distance are convex 

 towards the centre of the earth, so it may be taken that the central 

 core does not extend beyond about *4 of the radius from the 

 centre. 



As regards the rate of transmission of the waves, the data hardly 

 deserve elaborate mathematical treatment until more have been 

 collected, but it is easy to arrive at an approximate estimate of the 

 rate of transmission and the nature of the wave-paths. The chord 

 of 150° has a length of 12,297 kilometres, of which 8413 km. lies 

 in the outer *6 of the radius and, at a mean rate of 6 km. sec, 

 requires 23'4 minutes, leaving 21*6 minutes for the remaining 

 3884 km., or a mean rate of 3 kilometres per second. 



This reduction in speed has an important and unexpected result, 

 for it means a refractive index of 2*0 and a great deviation of the 

 wave-paths as they enter the central core. As a first approximation 

 to the actual course of the wave-paths, I give, in fig. 2 (p. 468), a 

 representation of what they would be on the supposition of a 

 central core, occupying '4 of the radius, in which the rate of pro- 

 pagation is one half of that in the outer shell ; in this it will be seen 

 that the wave-paths emerging at 150° reach their emergence after 

 passing on the opposite side of the centre of the ea.rth, and exhibit 

 that concavity towards the centre which Dr. Rudzki's investigation 

 requires where increase in depth of wave-path is accompanied by a 

 decrease in the rate of propagation. 1 The actual wave-paths, 

 however, are not, as has been shown, composed of straight lines, 

 and the real wave-paths must be more like what is indicated in fig. 3 

 (p. 469), which may be taken as correct in kind, though perhaps 

 wrong in detail, as to the actual position of the wave-paths. 



The high index of refraction prevents the formation of a complete 

 shadow-band, for the most extreme of the rays which enter the 

 central core suffer so great a deviation that their point of emergence 

 at the surface overlaps that of the last rays which miss the central 

 core ; but an inspection of figs. 2 & 3 will show that there should 



1 The wave-paths shown in fig. 2 give, for an emergence at 150° and at a 

 time-interval of 45 minutes, a rate of transmission of 3*5 km. sec. in the central 

 core, and 7*0 km. sec. in the outer shell. These values are higher than can be 

 admitted ; the explanation probably lies in the shortening of these long-distance 

 wave-paths which results from their curvature, as shown in fig. 3, and possibly 

 also in a lesser ratio than that of 2 : 1 of the rates of transmission, or a lesoer 

 size of the central core. It may also be noticed that rates of 7 and 3'5 km. 

 sec. respectively give an interval for the diameter of about 42"5 minutes ; and 

 although this value cannot be accepted, it indicates a possibility that the 

 emergence of the second-phase waves at the antipodes of the origin may actually 

 be earlier than at a distance of 150°. 



