OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
195 
ther the earth’s rotation, provided we introduce the force (5.), in addition to the ac- 
celerating- forces, whatever they may be, which are really in action upon the point (m); 
it being understood that the common centrifugal force is allowed for in g. 
(92.) In (5.) ^ is the apparent velocity of the point u (apparent, that is, to an ob- 
server unconscious of the earth’s motion) ; and if n denote the earth’s angular velo- 
city, and y' the direction of the polar axis, u=.ny' . Thus (5.) becomes 
^ -r^ , du 
( 6 .) 
du 
Fig. 36. 
Now this represents a force at right angles to y' and i. e. to the polar plane in 
which the apparent velocity is taking place at the instant t. Also the magnitude of 
this force is 2w times the apparent velocity multiplied by the sine of the angle 
it makes with the polar axis (/). 
(93.) This force may be expressed with reference to horizontal and 
vertical coordinates at any place, as follows : 
Let O be the place, y the vertical at O, and then che plane (aj3) 
will be horizontal : also let a be chosen so as to lie in the meridian 
plane ; and let I denote the latitude of O (/. e. the angle y' makes 
with a). Then 
y'=acos/d-y sin / (7.) 
Also if we take u to denote the distance of the moving point from O at any time t, 
and therefore put 
u=xa-\-y^-^zy, (8.) 
^ will be the same as the in (6.), for all that we have to express by ^ is the ap- 
parent velocity of the point u. 
Hence, differentiating (8.), and performing the operations indicated by 2wDy'., (6.) 
becomes 
dx . dy ^ . dz 
or 
-2«D(a cos Z+y sin Z). (^t«+ J|3+^y), 
2n{(| sin ;)«+ (-§ sin l+~ cos i)/3- (f cos /)y}. 
(9.) 
Hence the ordinary equations of motion will be 
g=X+ 24 sin/ 
rr ^ dy , 
Z— 2r(^COS I 
( 10 .) 
* Originally u was measured from the earth’s centre. 
2 c 2 
