460 Mr. F. J. W. Whipple on the Motion of a 



It can be shown that p, the geodesic radius of curvature, is 

 given * by the equation 



V 2 / / o = 2fi>Vsin\, (3*1) 



The curvature is to the right in the northern hemisphere, 

 to the left in the southern. The equation indicates that if* 

 the velocity is small, so that variation in latitude is negli- 

 gible, the track is a circle which is described in the anti- 

 cyclonic sense, i. e. clockwise in the northern hemisphere, 

 anti-clockwise in the southern. 



The time of one revolution is 2ir/2co sin X or -J cosec X days, 

 the length of one (sidereal) day being 27r/co. 



At the poles the period is \ day, in latitude 60° it is 

 •58 day, and in latitude 30° it is one day. 



4. The case of motion in the neighbourhood of the pole is 

 of special interest. 



If the pole is the centre of the circle, then the velocity of 

 the particle relative to the earth being 2cop the velocity in 

 space is cop, i. e. the particle moves with the same speed 

 as the ground over which it is travelling but in the opposite 

 direction. The horizontal component of gravity, being suffi- 

 cient to maintain in relative equilibrium a particle resting 

 on the surface, is also sufficient to maintain in circular 

 motion a particle moving with the same speed in the oppo- 

 site direction. Evidently two such particles pass one another 

 twice in the course of a day, so that the statement that the 

 period of the relative motion is half a day is verified. 



If the track of the moving particle crosses the pole, the 

 motion in space is simple harmonic. The acceleration due 

 to the horizontal component of gravity is oo 2 r at a distance r 

 from the pole, and the period of this motion is one day. 

 The same point of the surface of the globe is, however, 

 under the particle at each end of its swing, so that the 

 period of the relative motion is again seen to be half a day. 



5. In low latitudes the assumption that the variation in 

 X may be neglected is not legitimate. In such latitudes a 

 good approximation to the path may be found by writing X 

 for sinX, so that p = Y/2co\. If we think of a belt near the 

 equator as developed into a plane, B.X is the ordinate of a 



* Various proofs of this formula have been given. Three of them will 

 "be found in the ' Computer's Handbook ' of the Meteorological Office. 



