MOVING FRAME OF REFERENCE 199 



Let co be the angular velocity of the earth, i.e. let it turn through 

 ft> radians per unit time. Then the time of P describing a com- 

 plete circle is the same as the time required for the earth to perform 



, , , . , 2 7T , . . , 27rttCOSX 



a complete revolution, namely This time is also 



TT 1_ ^ 



Hence we have 



i) = aco cos X. 



The acceleration of the frame of reference is now seen to be 



= co 2 a cos X 

 a cosX 



along PN. The motion of any particle referred to a frame moving 

 with P may accordingly be calculated as though with reference to 

 a fixed frame, provided the component of force in the direction 

 PN is diminished by mcaPa cos X. 



Thus the total force acting may be supposed to consist of the 

 forces which actually do act, combined with a force mca?a cos X 

 along NP. Compounding this last force with the earth's attrac- 

 tion, we obtain a force which may be called the apparent force of 

 gravity at P. Thus the motion of the frame of reference may be 

 allowed for by using the apparent force of gravity in place of the 

 true attraction of the earth. It is this apparent gravity which is 

 determined experimentally, and which is always meant in speaking 

 of the weight of a particle at any point. 



To find the apparent weight of a 

 body at the point P, we have to com- 

 pound its true weight, say mG acting 

 along PC, with a force mco 2 a cos X along 



NP. Let the latter force be resolved 



C 

 into its components FIG. 112 



mco^a cos 2 X, m&> 2 a cos X sin X 

 along PC, PT respectively, PT being the tangent at P. 



Compounding with the force mG along PC, we find for the com- 

 ponents X, Y of the apparent weight along PC, PT respectively, 



X = m (G - a> 2 a cos 2 X), (53) 



Y = rao> 2 a cos X sin X. (54) 



