OF THE EAETH'S AXIS OF FIGURE. 37 



the point D returning to its original position after an interval of 

 about 300 days. 



It is plain therefore that the question before us will be answered 

 if we determine the position of the earth's axis of figure after the 

 elevations and depressions have been effected ; i. e. we need do no 

 more than determine the angle C C. In doing this it appears that 

 approximate formulas and numbers will answer every useful purpose. 



4. If A, B, C, denote the moments of inertia of the earth about 

 principal axes through her centre of gravity, it is known that (ap- 

 proximately) A=B and 300 (C— A)=C. It will be assumed that 

 C — A equals 



5ir -1 



6 ™nS 



*pa 5 , C 1 ) 



where p denotes the average density of rocks at the surface*, a the 

 equatorial radius, and tc the number 3*14159. 



5. The steps in the requisite calculation may be briefly indicated 

 as follows: — 



(a) The moments and products of inertia are found with respect 

 to the earth's principal axes of a zone passing round the earth of 

 small uniform height (h) enclosed between two parallel planes inclined 

 at an, angle a to the equator, and at assigned perpendicular distances 

 from the centre ; for the purpose of this determination it appears that 

 the ellipticity of the earth may be neglected. 



(6) In applying these formulas to the case suggested in the extract 

 (art. 1), the width of the elevated zone is assumed to be 23° 6', for 

 convenience (the sine of half this angle being 0*2) ; it is also assumed 

 that the submerged parts of the elevation produce the same effect as 

 if they were above water, and the depressions as if they were empty ; 

 it is also assumed that the matter removed to form the depressions is 

 transferred to form the elevation : all these assumptions are favourable 

 to the effect contemplated in the extract. The result arrived at is, 

 that the moments of inertia A, B, C become respectively, 



A + Jg. ma 2 (2-3 cos 2 a), 

 B - a, ma\ 



C + JL-™« 2 (2-3sin 2 a), 



and that the axes with reference to which A and C are taken are no 

 longer principal axes, because there is a product of inertia (Zmzx) 

 equal to -f-^ma 2 sin a cos a, where m denotes the mass of the elevation. 

 {c) If now is taken to denote the displacement of the axis of 

 figure (COC), it is easily shown that 



. 3m« 2 sin2a 



tan 20 



25 (C- A) + 3m« 2 cos2a. 



* The expression for C-A is deduced from the suppositions usually made 

 in treatises on the figure of the earth (Airy's Treatise, p. 189). 



