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XII. On Mr. Hopkins's Method of determining the Thickness of 

 the Earth's Crust. By Archdeacon Pratt, M.A., F.R.S. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



IN my letter in your Number for July of last year I gave a 

 reply to M. Delaunay on the above subject in a popular 

 form, — and in that way explained that, even if the fluid interior 

 of the earth's mass at any moment revolved with the crust as if 

 the two were one solid mass, this state could not continue ; the 

 crust under the action of the precessional force would slip over 

 the fluid, not being solidly connected with it. 



M. Delaunay (as reported in 'Nature/ March 16, 1871, 

 column 1) has again said that " calculations prove" that the 

 thickness of the crust has " no influence on the revolution of the 

 earth." I therefore now send you a calculation to show that 

 the crust, with an interior fluid nucleus, both following the law 

 of density adopted by Laplace, cannot move as it would if the 

 crust and nucleus were one solid mass. The method I pursue is 

 this. At the epoch from which t (the time) is measured I assume 

 that things are exactly as M. Delaunay supposes, viz. that in- 

 ternal friction and viscosity have reduced the fluid to entire 

 obedience at that moment to the movements of the crust. I 

 then show by the equations that this state of things cannot con- 

 tinue. This mode of taking the problem enables me easily to 

 calculate the effect of the fluid pressure on the crust at the epoch ; 

 and any minute motion, gradually generated in the fluid after 

 this for a short time, would enter into the equations as a quan- 

 tity of the second and higher orders and may be neglected. The 

 slowness of the disturbed motion and the viscosity would have 

 the effect of prolonging the period through which my equations 

 would apply. 



2. The forces acting on the crust are the attraction of the sun 

 and moon from without, and the pressure of the fluid against its 

 interior surface; and the pressure of the fluid is produced by 

 the centrifugal force and the attraction of the sun and moon on 

 the fluid. As the crust is supposed to be made up of spheroidal 

 shells, it will have no effect on the fluid. 



Let (o v o) 2 , o> 3 be the angular velocities of the solid crust round 

 any line at right angles to the earth's axis, another axis at right 

 angles to it and lying in the plane of the equator, and the axis 

 of rotation ; i. e. the axes of x, y, z fixed in the body ; A, A, C 

 the moments of inertia about those three axes ; L, M the mo- 

 ments of the forces acting on the crust about the axes of % 

 and y. There will be no moment of forces about z, because 

 the resultant effect of each set of forces passes through the axis 



