286 Archdeacon Pratt on the Thickness of 



These, when substituted, make 



Annual Precession = 51" # 36. 



9, Were the material of the earth not highly rigid, the par- 

 ticles would yield to the strain which I have shown would be per- 

 petually coming upon them, going through its changes every 

 twelve hours, and would describe reentering curves on the sur- 

 faces of infinitesimal ellipsoids. The result would be that the 

 strain could not be transmitted through the mass, and preces- 

 sion and nutation would not occur. 



Observation makes the annual precession =50". And the 

 result of the above calculations, viz. 51 ;,, 36, is very near this. 

 This near coincidence shows how highly rigid the earth's mass 

 must be. 



10. But it may be thought that if the interior is fluid, and 

 the solid part only of a certain thickness t, the crust to be moved 

 being a less mass than the whole earth, the observed precession 

 might still coincide with what calculation would bring it out, as 

 the action of the sun and moon on the solid crust would be less, 

 and the pressure of the fluid would perhaps compensate for any 

 difference. This I propose now to examine. I will, however, 

 previously, in this paragraph, calculate the quantity / for a shell 

 or solid crust. 



Let a and a be the mean radii of the outer and inner surfaces 

 of the crust. In this case 



- f a d.a 5 e _ f a d.a 5 , 



This ratio, it will be easily seen, is a larger quantity than when 

 a = 0, or for the whole earth. For the larger a is the larger is e ; 

 and therefore the elementary terms in the numerator off for 

 only the crust are the largest of all the terms of/ for the whole 

 earth ; and they correspond term by term with the terms in the 

 denominator. Hence for the crust /is larger than for the whole 

 earth. 



Suppose, as usual, that p = Qsin qa 4- a; then by the Figure 

 of the Earth, 



(3 \ 3 



q l a z J qa qa 1 



e= ti \ ±-> — ± — = 1 —z, 



tan qa— qa tan qa 



where H is independent of a. 



The advocates for a thin crust consider 100 miles sufficient. 

 This is one fortieth of the radius, the square of which may be 

 neglected. Call, then, t the thickness, and expand; 



