MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
205 
differentiating the previous equation subject to these conditions. Hence 
dt 2 a 9 g h W \ dt) » 
= fl ? (*i ~ one da y) ; 
or if s l = the space which a body would describe in one day by the action of a con- 
stant force = g, 
40 7 r 2 
- f F" (p) a ; 
d 2 r 
Jtf- 
9 
or, writing r instead of its mean value a, 
d 2 r 
— 9 
d?r 40 w 2 a, 
= ~^ p W r - 
40 ra t 3 fx. 
9 s, 2 r 3 
sin 2 A cos 0 sin 6 cos <p . r 
together with another term involving cos 2 <p, which will, therefore, disappear after the 
final integrations in determining the effect of this force on the motion of the shell, as 
in article 6. 
f • (ffi T 
The above term in the value of is precisely similar to the only effective term in 
that part of R which arises from the disturbing force of the sun (Art. 6.), with which 
term it may therefore be combined. The numerical value, however, of the factor 
40 7r 2 a, 1 . . . . d 2 r . , 1 * _ , . „ 
• — • 7 T is so small that the term in -r-& is about Trw.th of the term in R. 
9 Si 2 'f 
dt 2 
250 
An exactly similar investigation is manifestly applicable to the case of the moon’s 
T 
action, from which it follows that the term in -jj 2 depending on this cause will be 
about QT^tli of the corresponding term in R. 
The correction to be applied to the value of R obtained in article 6 is additive 
( d*r \ 
f^since ^ is negative), and, instead of the results given in that article, we shall have 
(A3) = h A 3 (l + 
(A 4 ) = h A 4 (l + nearly, 
with similar modifications of the quantities B 3 B 4 , & c. 
8. If the interior fluid were homogeneous, it is manifest that the attraction of its 
different particles on those of the shell could have no tendency to turn the shell 
about the axis of y, whatever might be the position of the spheroidal axis with respect 
to the axis of rotation of the fluid. This will not be accurately true when the fluid 
is heterogeneous ; but it may easily be shown that this effect may be neglected in our 
results. For, in the first place, the attraction between the whole shell and any fluid 
particle would vanish if the surfaces of equal density were all spheroidal with the 
same ellipticity, and can therefore only be of an order of quantities depending on the 
