50 
MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
(A being constant), and take q' a = 150°, we obtain a value of s 1 (the ellipticity of 
the surface) which coincides very nearly with its observed value, as shown in the 
common treatises on the figure of the earth. This expression for f also gives us the 
ratio of the mean to the superficial density equal to 2‘4225 *, which agrees very nearly 
with the value determined by Cavendtsh. It therefore appears extremely probable 
that this formula represents very approximately the actual law of the earth’s density. 
The above expression for § gives us 
£ 
tan q' a x — q ' a x 
tan q' a — q' a 
/ 3 \ 3 
( 1 55—5 ) tan q' a + -j- 
\ q n crj q a 
( 1 jt—o ) tan q' a, + - - 
V q n a{) J q a x 
If we here substitute the above value of q' a x , and put a = — a x , we obtain a value 
of s which nearly satisfies equation (3.) (Art. 1.). 
If we take q' a x = 160° we obtain the mean density more than three times the su- 
perficial density, and a value of z x not so nearly coinciding with the observed value as 
in the former case. In this case the formula probably gives us a density increasing 
too rapidly with the depth, and therefore also a too rapid decrease of ellipticity in the 
surfaces of equal density. To obtain a value of s, which will satisfy equation (3.), we 
must put a equal to about y a x . 
§. Thickness of the Earth's Crust. 
7. If the surfaces of equal solidity were coincident with those of equal density, and 
we adopted the first value of q' a x mentioned in the preceding article, we should 
obtain the effective thickness of the crust (= a x — a) — — 1000 miles ; or if we 
adopt the other value of q 1 a x as less favourable to a great thickness of the crust, we 
shall have that thickness = 800 miles. But the surface of equal solidity through 
any point must be intermediate between those of equal density or pressure, and of 
equal temperature through the same point ; and we have seen (Art. 5.) that the 
ellipticity of the latter increases with the distance from the external surface. Conse- 
quently the ellipticity of every surface of equal solidity must be greater than that of 
the corresponding surface of equal density, and, therefore, the effective thickness of 
the crust must be greater than that above determined, in order that it may be con- 
sistent with the observed amount of precession. 
The thickness of the actually solid portion of the earth’s crust will necessarily be 
less than what I have termed the effective thickness, but there cannot, I conceive, be 
any reasonable doubt that the difference between these quantities is small compared 
with either ; for if r x be the highest temperature at which any substance retains the 
property of solidity, and r 2 the lowest at which it acquires that of fluidity, r 2 — t x is 
* Airy’s Tracts, p. 178. 
