50 



SCIENCE 



[N. S. Vol. XL No. 263. 



and the arguments by which he reached it 

 throw no light on the distribution of the 

 earth's mass, a slight extension of his 

 methods gives a formula which shows that 

 any considerable difference in the equator- 

 ial moments of inertia of the earth would 

 produce a variation in the acceleration of 

 gravity dependent on the longitude of the 

 place of observation.* Thus it is possible 

 by means of pendulum observations alone 

 to reach the conclusion that the mass of the 

 earth is very nearly symmetrically dis- 

 tributed with respect to its equator and 

 with respect to its axis of revolution. 



A question of great interest with which 

 the acceleration of gravity at the sea sur- 

 face is closely connected is that of the 

 earth's mass as a whole. About two years 

 ago I published a short paper which gives 

 the product of the mean density of the 

 earth and the gravitation constant in terms 

 of the coefBcients of Laplace's formula and 

 the dimensions of the earth. f It was shown 



*See Helmert, Geodiisie, Band II., p. 74. The ex- 

 pression for the acceleration is 



g^a -\- ji sin^ (* + }' cos' <p cos 2/1, 

 where a, /3, y are constants, and f, A are latitude and 

 longitude respectively ; and the constant j; involves 

 the difference of the equatorial moments of inertia as 

 a factor. 



t See The Astronomical Journal, No. 424. This prod- 

 uct is expressed thus : 



27r 3K+^) . 



wherein k is the gravitation constant, /j is the mean 

 density of the earth, T is the number of mean solor 

 seconds in a sidereal day, a and /3 are the first two 

 constants in the formula p = a -(- ;3 sin' ^ + } cos'0cos2/i 

 for the acceleration of gravity at the sea surface in 

 latitude p and longitude A ; a is the half major axis 

 and e is the eccentricity of the earth's spheroid ; and 



1 + : 



2e 



log : 



1— e^f 2e , , 1 — e 



The resulting numerical value is 



/^/3 = 36797 X 10 -"/(second) 2. 



that this product can be easily computed 

 from existing data to five significant figures 

 with an uncertainty of only one or two 

 units in the last figure ; thus making it 

 po.ssible to obtain the mass of the earth to 

 a like degree of precision if the constant of 

 gravitation can be equally well determined. 

 In a subsequent communication to this So- 

 ciety it was explained that the product in 

 question is equal to 3^ divided by the square 

 of the periodic time of an infinitesimal 

 satellite which vi^ould pass around the earth 

 just grazing the equator if there were no at- 

 mosphere to impede its progress. The peri- 

 odic time of such a satellite would be 1 

 hour, 24 minutes, 20.9 seconds. Attention 

 is called to this subject with the hope that 

 some mathematician may point out another 

 possible relation between the gravitation 

 constant and the mean density of the earth 

 which can be accurately observed, or that 

 some physicist may show how the gravita- 

 tion constant can be measured directly with 

 a precision extending to five significant 

 figures. 



The lithosphere is the special province of 

 the geologist, and we may hence pass on to 

 the nucleus, or chief part of the mass of the 

 earth. Much time and attention have been 

 devoted to the study of the important but 

 intricate problems which the geometers of 

 the early part of the century left to their 

 successors. But while the obscurities and 

 vagaries of our predecessors have been 

 cleared away, it must be confessed that our 

 improved mathematical apparatus has not 

 brought us very far ahead of the positions 

 of Laplace and Fourier as regards the con- 

 stitution and properties of the nucleus. 

 With respect to the law of the distribution 

 of density in the nucleus it may be said that 

 although Laplace's law* is probably not ex- 



* The Laplacian distribution of pressure, densityi 

 and potential in the earth are defined essentially (neg- 

 lecting the effect of rotation) by the three following 

 equations : 



