44 
MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
the whole of this imperfectly fluid mass as entirely solid, we should manifestly take 
the thickness of the shell too large to represent the actual phenomena depending on 
that thickness ; and if, on the contrary, we should consider the whole of the imper- 
fectly fluid portion as perfectly fluid, we should take the thickness of the shell too 
small. Hence there must be some surface of equal fluidity (or, if we please so to 
term it, of equal solidity ) intermediate to the above surfaces, such that if the whole 
mass superior to it were entirely solid, and that inferior to it entirely fluid, the phe- 
nomena of precession and nutation would be the same as in the actual case of a 
gradual transition from the solidity of the superior to the fluidity of the inferior por- 
tions of the mass. When, therefore, we speak of the interior surface of the solid 
shell with reference to our previous investigations as applicable to the case of the 
earth, it is this intermediate, or effective surface , which is always to be understood ; 
and the thickness of the earth’s crust, as defined by this surface, I shall term its ef 
fective thickness. 
3. In order that the value of P' may agree with that determined by observation, 
we must have approximately 
and, therefore, referring to the equation of Article 1, we must have 
An approximate value of ^ will be obtained by making constant in the expres- 
sion given for it in Article 5, Mem. II. We then have 
which gives 
It has been stated (Art. 2. Mem. II.), that if y be ever negative, it can only be so 
when a — a x very nearly. But in this case q 5 — 1 is extremely small, and therefore 
the value of the second factor on the left-hand side of the above equation will be very 
nearly = unity. In all other cases it will be less than unity. Let it = 1 — j3 ; then 
, _ _L_! i 
e i 8 *1-/3* 
and 
If, as an approximation, we omit (3 (which will necessarily be considerably smaller 
