Original and Actual Fluidity of the Earth and Planets. 273 



which might have been deduced directly from (31), by making £ = e, = e on the left-hand side, 

 6 = (e) on the right, and A = 2p„'. Solving this equation for a, we find, 



_3^ _ 1 10(e) -3e-5mi 0^ + 3e 



5a ~ 5e(0'+ 1) • ^'^' 



Eliminating a from equations (2) and (3), we obtain finally, 



Sm- 2e ^ 5e 



^ "^ 10(e) - 3e - 5m 10(f) - 3e - 5m ^' ^ ' 



]n this equation (ji is the reciprocal of Qi, and the other letters are the same as the corresponding 

 letters used in equation (1). Equation (4) diifers widely from the equation (1) obtained by Mr. 

 Hennessey; the hypotheses used in obtaining it are the four hypotheses used by him; and yet I 

 am unable to perceive any error in the process by which (4) is found. 

 Substituting for m, e, (c) ; their values ^gg i ogg t and ^-^ , we find, 



0»+ 1-58425 = 2-48290 0^ (5) 



Applying Sturm's theorem to this equation, I find that it has three real roots, one negative and 

 two positive; the latter lying between = 1 and = 2. These roots are 



= 10436; 0=1-3626. 



Rejecting the negative root, as being not applicable to the question in hand, it would appear at 

 first sight as if there were two solutions, corresponding to the two real positive roots just found ; 

 but it is evident, by referring to equation (28), that the second value of 0, being greater than 

 1-2407, is to be rejected as well as the negative root; in fact, the second value of would give a 

 thickness to the crust of the Earth greater than the depth to which the density of the rocks at the 

 surface can extend; and such a thickness, as has been already shown, is inconsistent with the 

 supposition of a fluid nucleus. Calculating the thickness of crust corresponding to the least 

 positive root of equation (5), we find, 



a - a, = 166 miles. (6) 



This result difiers materially from that obtained from the same data by Mr. Hennessey, but as the 

 hypothesis on which it is founded is untenable, the result itself is of little value, except so far as it 

 illustrates the use of the equations already given. As I have before stated, the thickness of the 

 crust would be zero, if we were to admit the first three statements, and combine with them au 

 assertion that the surface of the Earth is perpendicular to gravity. This I believe to be the true 

 minor limit of the thickness of the Earth's crust; and the major limit appears to me to require for 

 its numerical calculation a knowledge of facts, respecting which we must be content to remain in 

 ignorance. 



