MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
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where 
x' = x sin A + « cos A, 
y =y> 
z' = z sin A — x cos A ; 
and therefore, by substitution, 
X' = ^3 { (-£• “ -f cos 2 a) x + sin 2 A . z j , 
Z ("^ +4 cos2 A ) 2 + 4 sin 2 A.* j. 
In the formula 
R = X cos 0" + Y cos 0' + Z cos 0, 
putting for X, Y, Z the above values of X', Y', Z', we find 
R = ^4 sin 2 A cos 0 sin 0 cos <p r + terms which involve even powers of cos <p, and 
r \ 
which, therefore, when multiplied by x h S cos £ (= a 3 sin 2 0 cos 0 cos <p d <p cL 0), will 
contain odd powers of cos <p, and will therefore disappear after the final integration. 
Hence then the moment about our present axis of y 
— 2s2.z&Scos£ (n + 3^ sin 2 A sin 0 cos 0 cos <p grdr), 
— 6 ^ s sin 2 A a?f (a) ^y^sin 3 0 cos 2 0 cos 2 d 0 d<p, 
2 f(a) 4i a . . , 
= iy 6Si n2 A. a 
(Art, 5.). 
If § be constant and = unity, 2 f (a) = a 2 , and the expression becomes the same 
as that for the homogeneous spheroid. 
Dividing the above quantity by the moment of inertia of the shell, we have the 
angular accelerating force 
= M = JM^4.i jiSin2A) 
v > cr (flj) — <r (a) 2 ? i } 
3 j x 
2 rj 3 q 5 — 1 
sin 2 A 
(Art. 5.), 
and therefore 
Hence we have 
and similarly 
(First Series, Art. 21.) ; 
y) = _A_ 
« q b — 
(A 3 ) = A A 3 ; 
(-^3) = ^ ®3 j 
(D3) = A D 3 . 
MDCCCXL. 
