GEOLOGICAL CHANGES ON THE EAETH’S AXIS OF EOTATION. 
273 
“ Let x , y, z be the coordinates of any particle of mass m at the time t , referred to 
axes fixed in space. Then we have the equation of motion 
£ 
II 
% I'* 
l 
^ ^ ^ 
w 
(1) 
and two similar equations. 
“ Let 
t / dy dx\ 
h 3 %m[x dt y dt J 
(2) 
with similar expressions for 7q, , 
K 
“ Then the equation (1) becomes 
^_3_ N 
dt 
. .... (3) 
“ Let the motion be referred to three rectangular axes Ox', O y\ Os' moving in any 
manner about the origin O. Let a, (3, y be the angles these three axes make with the 
fixed axis of z. Now h 3 is the sum of the products of the mass of each particle into 
twice the projection on the plane of xy of the area of the surface traced out by the radius 
vector of that particle drawn from the origin. Let h ! „ h\, h' 3 be the corresponding 
‘ areas ’ described on the planes y'z', z'x', x'y' respectively. Then by a known theorem 
proved in Geometry of Three Dimensions, the sum of the projections of hi, h' 2 , h' 3 on xy 
is equal to h 3 ; 
h 3 =h\ cos ot,-\-h! 2 cos cos y (4) 
“ Since the fixed axes are quite arbitrary, let them be taken so that the moving axes 
are passing through them at the time t. Then 
h\=h x , h' 2 =h 2 , h' 3 =h 3 ; 
and by the same reasoning, as in Arts. 114 and 115, we can deduce from equation 
(4) that 
dh 9 dh'o . 
~dt~~dt ~^ 1 ^ 2 + Mi (5) 
where 6 X , d 2 , d 3 are the angular velocities of the axes with reference to themselves. 
Hence the equations of motion of the system become 
$-*A+*A= L, ' 
^-/<A+M= N, 
“ These equations may be put under another form which is more convenient. 
y-. 
z' be the coordinates of the particle m referred to the moving axes, and let 
XT ( \dy' i d%'\ 
H,=Sm^ ar -y a? j. 
2 q 2 
( 6 ) 
Let 
