[ 3,27 ] 



LI. On the Thickness of the Crust of the Earth. By the Vene- 

 rable John Henry Pratt, M.A., Archdeacon of Calcutta. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 

 ^r^IIK question whetlier the earth's crust is of comparatively 

 -■- small or great thickness is one of considerable interest to 

 geologists, more especially because of its bearing upon the theory 

 of volcanoes. The only mathematicians who have taken up the 

 subject as a branch of physics are, as far as I am aware, Messrs. 

 Hopkins, Hennessy, and Haughton*. The result of Mr. Hop- 

 kins's investigations is, that the thickness is very considerable, 

 amounting to as much as 800 or 1000 miles. Mr. Hennessy 

 comes to the conclusion that the least possible thickness is 18 

 miles, and the greatest 600 miles. Mr. Haughton makes the 

 thickness less than 768 miles, but adds that, in fact, " the subject 

 would appear to be excluded from the domain of positive science, 

 and to possess an interest for the mathematician only." 



3. My present object is to point out what appears to me to 

 be a fallacy in the last gentleman's reasoning, and also an inad- 

 missible assumption in that of the second. Before doing this I 

 will trace the evidence upon which Mr. Hopkins's conclusion 

 rests. (The notation is changed, as it is not the same in the 

 papers I quote in this communication.) 



Mr. Hopkins has deduced the following formula. 



,d.a'\,<-s) 



da' 



F-P^^ E }/ da' 



^da' 



2e«3r'pV<^ff' + er 



where P is the precession of a homogeneous spheroid of ellip- 

 ticity e; P' of a heterogeneous shell, composed of nearly sphe- 

 rical strata, the outer and inner ellipticities being e and e, a and a 

 the mean radii of the bounding surfaces ; «' e' p' the mean radius, 

 ellipticity, and density of any stratum of the general mass, solid 

 or fluid. 



If we assume that the strata decrease in ellipticity in passing 

 downwards (which is the result of the usual theory of the earth's 

 figure), e'~g is never negative, and the fraction on the right 

 hand is never negative, and is never so large as unity. Let 

 it = /3. Calculation makes P = 57" nearly; observation makes 



• y\r. Iloijkins in Pliil. Trans, for 18.39, 1840, 1842; Mr. Hennessy in 

 I'liil. Trans, for IS.'Jl ; Mr. Ilanglitoa in tlie Transactions of the Koval 

 Irish Aradetiiy for 1852. It is only about a week aj!;o that 1 first saw the 

 lust of these jiupers. 



