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



2. For many purposes the departure of the earth from a 

 spherical form can be neglected, and it is therefore in- 

 teresting to approach our problem by considering, in the 

 first instance, the motion of a particle on the surface of a 

 smooth rotating sphere. The reaction between the particle 

 and the sphere is in this case radial and so is the gravita- 

 tional force. Accordingly, the motion of the particle in 

 space is the same as it would be if the globe did not 

 revolve. The path relative to the centre of the sphere is 

 accordingly a great circle. To fix our ideas we take a 

 particle starting at relative rest in latitude X. Its velocity 

 in space is H&) cos X, where B is the radius and co the 

 angular velocity of the sphere. The time of revolution of 

 the particle in its orbit is 2ttB,/(R&) cos X) or T sec X, where 

 T is the period of the rotation of the sphere. Let us sup- 

 pose that the particle starts from a point in the northern 

 hemisphere. The track on the sphere leads southward to 

 begin with. To find the angle at which it crosses the 

 equator we notice that the relative velocity at the crossing 

 is compounded of R&>cosX, making the angle X with the 

 equator and of Rw along the equator. The relative velocity 

 is therefore R<w sin X, in a direction making an angle X with 

 the meridian. The particle approaches the equator from the 

 north-east. 



At the most northerly and southerly points of the track 

 there are cusps. The difference of longitude between con- 

 secutive cusps on either side of the equator is (o>T secX — 2it) 

 or (secX — l)2ir. If the range of latitude is small, this angle 

 is approximately X 2 7r. 



The track can be specified in polar coordinates. Let X, $' 

 be coordinates relative to axes fixed in space, X being the 

 latitude and $>' the longitude reckoned towards the east 

 from a meridian which does not turn with the globe and 

 which passes through the initial position of the particle. 

 The coordinates of this position are X , 0. 



The angular distance moved by the particle is cot cos X , 

 and we have the equations 



sin X = sin X cos (cot cos > )> 

 cos (p ' — cot X tan X. 



Let (f> be the longitude reckoned towards the west from 

 the meridian turning with the globe and passing through 

 the initial position of the particle, then 



cp + fiz^wt. 



