LAMBERT: CONSTITUTION OF THE EARTH 137 



In the actual case of elastic yielding the variation of latitude 

 gives us the quantity k by itself. Tidal observations give us 

 h — k, since the observed motion is to the theoretical motion for 

 arigidearthin theratio I — (/z— fe) : i. The value of k from the 

 variation of latitude is about 0.275. There are several diffi- 

 culties in connection v/ith the values of h — k deduced from 

 the tides, which I have not mentioned; probabl)^ the best 

 value is Ji — k = 0.29, from Michelson's pipe. 



What we should like to be able to do is to find the theoretical 

 values of h and k corresponding to any system of values, varying 

 from point to point, of the earth's density and elastic constants, 

 so as to find by trial some plausible law of distribution that would 

 fit the observations. What we are able to do is much less. 

 The compressibility, in particular, introduces mathematical diffi- 

 culties and the usual assumption is to make the earth incompres- 

 sible. The errors due to this assumption are not so serious as 

 might be supposed at first sight. If we further assume that the 

 earth is of uniform density and has the same modulus of rigidity 

 throughout the whole mass, its modulus of rigidity that will 

 represent the lengthening of the Eulerian period comes out 

 16.3 X 10^^ C. G. S. units. We can get rid of the assumption of 

 uniform density by using Wiechert's hypothesis of a metal nu- 

 cleus and an outer shell of rock, assuming, which is not very 

 satisfactory, that both nucleus and shell have the same modulus. 

 The latter must be 11.7 X 10^^ to represent the latitude varia- 

 tion.-' The hypothesis of a continuous change of density ac- 

 cording to Roche's law-^ one of the many laws of density I 

 mentioned earlier and one of the simplest, gives about the same, 

 still supposing the rigidity constant. Roche's law is the one law 

 of continuously varying density for which the theory has been 

 worked out, and a rather formidable theory it is — a differential 

 equation of the sixth order, and twelve new transcendental 

 functions defined by infinite series.^" 



^ For this result and the preceding one see LovE, Proc. Royal Soc, A, 82: 73. 

 1909. 



-" Roche's law assumes that the density falls from center to surface proportionally 

 to the square of the distance from the center, or c = 2 in equation (4). 



-' HeRGL,OTZ. Zeitschr. Math. u. Physik 52: 275. 1905- 



