ON THE PHYSICAL STRUCTURE OF THE EARTH. 217 



solid spheroitl would be 57". This was the result of Mr. Hopkius's cal- 

 culations; and tbe difference, amouutiug" to between six and seven sec- 

 onds between it and the observed value, formed the basis of all his 

 conclusions relative to tbe Earth's internal condition. Hitherto I liavo 

 not seen any reason for doubting the above numerical result; but on 

 looking more closely into the question it appears probable that we 

 must reduce the precession for the hypothetical solid spheroid to about 

 55". If the Earth were a spheroid perfectly rigid, the amount of pre- 

 cession can be calculated from formula'- given in Airy's Tracts^ Pratt's 

 Mechanical Philosophij, Pontecoulant's Theoric Analytiqiie du SSystcmedu 

 Monde, or Eesal's Traite do Mecanique Celeste. In the two latter works 

 Poisson's memoir on the rotation of the Earth about its center of grav- 

 ity is very closely followed, and the formuU^ are those which I have 

 generally employed. From these writings we have 



where J is the inclination of the equator to the ecliptic, y the ratio of 

 the Moon's action on the Earth compared to that of the Sun, m the 



Earth's mean motion around the Sun,— the ratio of this mean motion 



n 



to the Earth's rotation, and A, B, G the three principal movements of 



the inertia of the Earth. When the Earth is supposed to be a spheroid 



of revolution, A=B, and the above becomes 



(1) 1"=^^' (i + r)cos 7. 

 Pratt gives the formula 



(2) P=^f(^A) j 1+^^; IHL^nM ) 



^ 2/iV ^ y ' n^ 1+r ' 



where i is the inclination of the Moon's orbit to the ecliptic, y the ratio 

 of the Earth's mass to that of the Moon. 



In all these formula?, or in any others by which the precession can 

 be calculated, the Moon's mass enters directly or indirectly. When 

 Mr. Hopkins made his calculation more than forty years ago, he appears 

 to have taken the value of the Moon's umss and all his other numerical 

 data from the early editions of Air;y's Tracts. He uses 36G.2G for 

 the Earth's period, 27.32 for the Moon's. He makes 7=23^ 28', i=5o 

 8' 5(»", and the Moon's mass yV of the Earth's mass. All of these values 

 require revision, and it may be remarked that Sir George Airy has 

 more recently expressed the opinion that J-^j may be taken as the value 

 of the Moon's mass.* On this question I may be permitted to remark 



^Monthly Notices of the Royal Astronomical Society, December, 1878, p. 140. 



