MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
199 
of the fluid being the axis of Making the spheroidal axis of the shell the axis of z, 
and the plane of x z the same as that of x' z', we have 
x' = x cos (3 + z sin (3 
= x -}- z sin (3, 
Therefore 
Also 
and therefore 
y'=y- 
X = c^x' cos |8 = a 2 {^cos 2 |3 + z sin (3 cos (3), 
Y = u 2 y' — co 2 y, 
Z = or x' sin (3 = oj 2 {a? cos (3 sin (3 -f- zsin 2 ^3}. 
z — r cos 0, 
y — r sin 0 sin <p, 
x — r sin $ cos <p ; 
X 
cos sin 6 cos <p, 
cos O' 
y_ _ 
= sin 0 sin <p. 
Hence 
R = X cos 0" + Y cos O' -j- Z cos 0, 
= ai 2 1 (x cos 2 /3 + -|-sin2(3^ sin 0 cos<p -\-co 2 y sin 0 sin?) + sin 2 (3 + % sin 2 (3 ) cos 0 j> 
In this expression all the terms may be omitted except those which involve (3 as a 
factor, for the reasons assigned above (Art. 4.), and thus (omitting also terms of the 
order (3 2 ) the above expression becomes 
(XT 
— sin 2 {3 (z sin 0 cos <p + x cos 0), 
And 
= co 2 sin 2 (3 sin 0 cos 0 cos <p r. 
§ R d r = co 2 sin 2 /3 sin 0 cos 0 cos gr dr. 
Or putting r — a, we have the moment about the axis of y 
= 2 g 2 x S S cos £ (n -f* 2 sin 2 (3 sin 0 cos 0 cos rftrtr). 
But 
S . cos £ = a? sin 2 0 cos 0 cos (phcp'hO. (First Series, Art. 23.) 
and therefore, putting 
Jl\rdr-f{a), 
and omitting the first term in the above expression (since it vanishes between the 
