LAMBERT: CONSTITUTION OF THE EARTH 1 27 



The value of 6 used in computing the above table is 2.5066 

 radians or 143.618°, which gives a flattening of 1/296.5. The 

 column showing the modulus of rigidity will be explained later. 



Another effect of the arrangement of density within the earth 

 is the precession of the equinoxes. Theory shows that the annual 

 precession, which is known accurately from the long series of 

 available observations, is proportional to (C — A)/C, C and A being 

 principal moments of inertia of the earth. Evidently {C—'A)/C 

 depends on the distribution of density within the earth. With the 

 law we have assumed, its value comes out i /304.3 . The observed 

 precession requires more nearly {C — A)/C = 1/305.3. Agree- 

 ment between computed and observed values can be obtained by 

 increasing B a little, thus changing very slightly the quantities 

 in the above table and making the flattening equal to 1/297.2, 

 which is in excellent agreement with the flattening derived from 

 pendulum observations and from triangulation. 



It might be supposed that this agreement is at least some evi- 

 dence that the type of formula assumed for the law of density 

 is nearly correct. It is a curious fact however, that almost any 

 law of density will do exactly as well, so far as any of our means 

 of observation go. That is, assume any type of law that y^ou 

 please that gives a density decreasing from center to surface, 

 for example: 



p = a — bx' (4) 



a, b and c being constants to be determined, assume further that 

 the hydrostatic equilibrium prevails and determine the constants 

 a, b and c of your assumed law so that (C—y4)/C shall be equal to 

 its observed value 1/305.3, then your flattening comes out almost 

 exactly 1/297.2. This fact was first observed to be true when 

 various hypothetical laws were tried, and mathematical demon- 

 strations have been given by Poincare and others." These dem- 

 onstrations set limits within which the flattening must He for any 

 permissible law of density, provided (C— A) /C has its observed 



^ Poincare. Figures d'equilibre d'une masse fluide (Paris, 1902). Chap. IV. 

 Veronnet. Journal des Mathematiques. pures et appliquees 77: 331. 1912. 

 TissERAND. Mecanique Celeste, 2: 221. 



