198 
ME. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
because the moment considered in the article referred to was that of the centrifugal 
force, and would manifestly vanish with (3. In like manner it is manifest that the 
moment about the axis of y, produced by that part of R which depends on the mutual 
attractions of the particles of the whole mass, must vanish with (3, i. e. when the in- 
stantaneous axis of rotation of the fluid coincides with the spheroidal axis of the shell. 
Consequently all the terms which need be retained in R, in the above expression, must 
involve (3 as a factor, and, therefore, all terms in R involving s may be neglected ; for 
since the terms retained must involve (3, those involving s would be of the order s (3 
under the integral sign ; and since the whole integral is multiplied by s, the corre- 
sponding terms in the result would be of the order s 2 (3, terms which have always been 
rejected. With this restriction the attraction on a point at the distance r from the 
centre in the direction of r, or the value of R, will 
4 7r C q' r 12 dr 1 
«/ o 
4 7r <f> (r) # 
and the corresponding moment about the axis of y 
= 2 s 2 x S S cos £ (n + 4 g d r) . 
Again, in the expression for g we may reject all terms involving s, for the same 
reason as they have been rejected in the expression for R, which will reduce the ex- 
pression for g to the same form as if the surface of equal pressure were a sphere, i. e. 
g will become a function of r. Hence, since the quantity under the sign J ” will now 
be a function of r, and all terms in the definite integral which do not involve the 
factor (3 must disappear, the above expression for the moment about the axis of y 
mav be written 
2 s 2,r&Scos £ (EH- ^3¥(r)). 
Also ( a being, as heretofore, the axis of the interior surface of the shell) 
r = a-f terms involving s ; 
and therefore for ¥ (r) we may substitute 'F (a) ; and the moment becomes 
= 2s(n + (3F»)2.zSScos£ 
= 0 
when integrated between the proper limits. 
5. Let us now consider that part of R which depends on the centrifugal force on 
the fluid. Taking those terms in the expression for the resolved parts of this force 1 
which are independent of the factor s (3 (First Series, Art. 23.), we have for these 
parts co 2 x ' and co 2 y ', parallel to the axes of x' and y' respectively, the axis of rotation 
