MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
197 
force on the fluid, and the disturbing forces of the sun and moon. If p denote the 
fluid pressure at any point x, y , z, and the density there, we have 
d p 
S 
*?)<**+( Y-g) dy+(z~ 
x d^" if d? z 
The accurate determination of the values of jp and ^55 wou ld require that of 
the motion of the fluid ; but since these quantities will be very small, it will be suffi- 
cient to determine their approximate values. The angular motion denoted by a! (First 
Series, Art. 15.) may be neglected in determining the value of p, as shown in Art. 23. 
(First Series). The motion to be taken account of is the internal tidal oscillation. 
The direction of each particle’s tidal motion will be approximately in a line through 
the earth’s centre ; and therefore, if we suppose a portion of the fluid to be contained 
in a rectilinear and rigid canal of small diameter, passing through the centre of the 
earth, the motion of the fluid and the fluid pressure at any point within the canal will 
not be affected. Let r be the distance of an element of the fluid contained in this 
canal from the centre of the earth ; R the sum of the impressed forces acting on this 
element resolved in the direction of the canal ; then, instead of the general equation 
above, we shall have 
and 
p= n +/ r f( R -S)^> 
II being the pressure at the centre, r the value of r at the interior surface of the shell, 
and p the pressure there ; and we shall have the moment of p round the axis of y 
(First Series, Art. 21.) 
= 2 g 2 *&Scos£(n+^ r £ (R - jp)dr). 
The value of R is given by the equation 
R = X cos d" + Y cos 6' + Z cos 6, 
6, 0" being the angles which the direction of the canal makes with the axes of z, y, 
and x respectively. We may consider separately the effects of the different forces 
above mentioned which combined produce the force R. 
4. Let us first take that part of R which depends on the mutual attractions of the 
particles constituting both the fluid and solid portions of the mass. 
If we examine the part of the expression under the sign 2 for the moment about 
the axis of y, given in article 23 of my first memoir, we observe that it consists partly 
of terms involving (3 as a factor, and partly of terms independent of (3, but that all the 
latter disappear when the integration is performed between the proper limits, leaving 
a result containing (3 as a factor. It was easy to foresee that this must be the case, 
