272 The Rev. Samuel Hadghton on the 



which we wish to compare them, and then to assume that the diiFerence between the mean so found 

 and that quantity is a real diiFerence. 



Adopting the four hypotheses above mentioned, Mr. Hennessey has deduced from his formulse 

 the following value for the ratio of the radius of the nucleus to the radius of the exterior surface, 

 p. 54.5 ; 



«!=:; + 



336 



4 ' 



(1) 



In this equation a, denotes the ratio of the radius of the nucleus to the radius of the external sur- 

 face, which is assumed equal to unity ; m = ^ is the ratio of centrifugal force to gravity at equa- 

 tor ; e = r^ is the ellipticity of the actual surface of the Earth ; and (c) = — (the mean of the frac- 

 tions — and — ,obtainedfromthependulumandlunarobservations),istheellipticityofthesurface, 

 288 300 



if perpendicular to gravity. Substituting these values in equation (1), Mr. Hennessey obtains 

 0|i= 0-97714, and a^ = 0-99539, 1 - a, = 0-00461, from which he infers, that "consistently with 

 observation, the least thickness of the Earth's crust cannot be less than 1 8 miles." It is very easy 

 to prove, that if the shell be bounded by similar surfaces, both of which are perpendicular to gra- 

 vity, that its thickness is zero ; this I believe to be the true minor limit of the thickness of the 

 crust. 



But even admitting Mr. Hennessey's assumption, that the outer surface of the Earth is not 

 perpendicular to gravity, I am unable to agree with him as to the formula from which its thick- 

 ness should be calculated. In equation (1), which is deduced from the previous equations, a^ is 

 the reciprocal of the quantity I have called 0. This equation contains only the lifth power of a, or 

 0, whereas, the equation deducible from the investigation which I have given contains both the 

 fifth and third powers of 0, and gives a numerical result which differs materially from Jlr. Hen- 

 nessey's. The investigation is as follows. Assuming e, = c = e in equation (30), which asserts that 

 gravity is perpendicular to the inner surface of the crust and is deduced from (14), and solving 



for <r, we find, making A = 1p^ , 



_3_ ^ {(lf+\)e -m 



5a (0^ + 1) e ■ ^ ' 



In equation (15), the external surface is supposed perpendicular to gravity, and, therefore, the 

 ellipticity e of its right-hand member must be replaced by (c); the integral at the left-hand 

 side of this equation is proportional to the difference of the moments of inertia of the Earth with 

 respect to its polar and equatorial axes (16), and does not require the surface to be perpendicular to 

 gravity; in fact, the left-hand side of this equation may be supposed to belong to any body having 

 the same difference of moments of inertia as that belonging to the Earth. Separating the integral 

 into two parts, belonging respectively to the shell and nucleus of the Earth, the external surface 

 being supposed similar to the inner, and not perpendicular to gravity, we find, 



3(0'- l)« + 3(0' + l)- = 5 12(e)- m| 0^; 



