236 ANNUAL EEPOET SMITHSONIAN INSTITUTION, 1916. 



distances from the origin ; these times are collected in the transmis- 

 sion curves. At first sight this seems an insoluble problem; but, 

 thanks to a remarkable mathematical theorem of Abel, it is not. It 

 is clear that the time of arrival of an earthquake disturbance at a 

 distant station will depend on the path followed and the velocity in 

 different parts of the path, and if we make the reasonable assump- 

 tion, which is borne out by observation, that the velocity is every- 

 where the same at the same depth, then it is evident, if the velocity 

 increases continuously with the depth, that the transmission curves 

 will be continuous without breaks, and their curvatures will no- 

 where make a sudden change. The mathematical solution of the 

 problem has been obtained by Wiechert, Bateman, and others; and 

 concrete results have been obtained by Wiechert and his assistants, 

 so that we now know the paths of the waves and their velocities with 

 a fair degi*ee of accuracy, at least to a considerable distance below 

 the surface. But the questions arise. Do the velocities increase con- 

 tinuously with the depth; and if so, How? questions which could 

 be answered by the study of perfect transmission curves; but even 

 imperfect curves yield some information ; which, however, may be so 

 faulty that it must be received with great caution. Milne, who has 

 done such excellent pioneer work in seismology, was the first to pro- 

 pose and attempt to answer these questions.^ He thought the trans- 

 mission curve could be satisfied by supposing the earth to consist of 

 a solid core having a radius of nineteen twentieths of the earth's 

 radius, and surrounded by a thin shell. The core was of uniform 

 density and elasticity, so that the velocity of propagation in it was 

 uniform, and the paths of the rays would be straight lines. The 

 velocity in the shell was much less than in the core. These condi- 

 tions satisfied fairly well the very imperfect transmission curve of 

 1902, but they may be dismissed without further consideration, for 

 such an earth could not satisfy the astronomic requirements, which 

 exact, at the same time, the proper mean density and moment of 

 inertia. 



Benndorff in 1906 thought he found evidence of a central core 

 of about four-fifths the earth's radius, surrounded by two shells, 

 the outer one having the same thickness as Milne's.- In the same 

 year Oldham deduced from the transmission curves a central core 

 of not more than four-tenths the earth's radius in which the velocity 

 was distinctly less than in the surrounding shell.^ Neither of these 

 arrangements have been shown to conform to the astronomic require- 

 ments. Oldham's conclusions are based on what he considers a 



1 Rep. of the Com. on Seismol. Investigation, B. A. A. S., 1903, p. 7. 



" " Ueber die Art der Fortpflanzungsgeschwindigheit der Erdbebenwellen in Erdinnern," 

 Mitt. d. Erdheben Com. k. Akad. Wiss. in Wien, 1905, Nos. 29 and 31. 



s Constitution of the Interior of the Earth, Quart. Jour. Oeol. Soc, 1906, vol. 62, 

 p. 456. 



