Method of determining the Thickness of the Earth's Crust. 99 

 of z } owing to the symmetry of figure. Hence 



A-^ + iC-J^w^L, 



A^-iC-A^co^M, 



C d -^ =0*. 



at 



The third equation gives <w 3 =#, constant =n; and the others 

 become 



A-^+{C— A)np 9 =*L, 



A^-(Q-A)m k =M. 

 at 



3. The values of L and M for the sun and moon are known 

 from the ordinary problem of precession and nutation ; they are 

 easily shown to be 



go o e 



(C— A) -g- sin 6 cos sin <j>, — (C—A) -^ sin 6 cos 6 cos <j> 



for the sun, where S is the sun's mass, c his mean distance, 6 

 and (f> his colatitude and right ascension f. Similar expressions 

 are true for the moon's action. Let them be 



(C—A) — g-sin t cos t sin y , — (C—A) — g- sin t cos ^cos^. 



4. I will now find L and M for the pressure of the fluid. 

 Let r, &, <j>' be the coordinates to any point in the inner surface 

 of the crust, andjo the fluid pressure. Then pr*sm6' d<f>'dd r is 

 the pressure on an element of the surface, and acts in the nor- 

 mal. Let I be the angle the normal makes with the earth's 

 axis. Then by conies 



Z=<9'«-2ecos0'sin<9', 



e being the ellipticity of the inner surface of the crust. Hence 

 the pressures parallel to the axes are 



pr 2 sin &dtfd& . sin I cos <£', pr 2 sin O'dtfJdO 1 . sin / sin <£', 

 and 



pr 2 sin O'dfidO'. cos I; 

 also 



x = r sin ! cos (/>', y — r sin & sin <£/, z = r cos 6 l . 



* These equations will be found in the Mecanique Celeste, or any work 

 on the motion of a rigid body. I have taken them from my ' Mechanical 

 Philosophy,' second edition, p. 425, making B=A. 



t Mechanical Philosophy, p. 426. 



H2 



