Particle on the Surface of a Smooth Rotating Globe. 461 



point on the track and the curvature of the track is propor- 

 tional to the ordinate. The track is therefore an elastica, 

 the curve made by a thin rod strained by forces applied at 

 the two ends. The various forms which the elastica can take 

 were classified by Euler. An account of these is to be found 

 in Love's ' Theory of Elasticity/ vol. ii. The most detailed 

 study of the subject is that of Hess (Math. Ann. 1885). 



As we wish to obtain numerical results, it is desirable to 

 work out our problem without assuming a knowledge of the 

 elastica. 



6. Let u, v be the components of velocity towards the 

 east and towards the north respectively. As the velocity 

 relative to the earth is a constant V, 



u 2 + v 2 = V 2 (6-1) 



The forces acting on the particle are in the meridian and 

 therefore they have no moment about the axis of the globe. 

 Accordingly, by the Conservation of Angular Momentum, we 

 can write down the equation 



B/ cos \[w + B/a> cos X]= constant, . . (6*2) 



where B/ is the length of the normal terminated by the 

 polar axis. 



If o>M be written for the constant angular momentum 



CO 



L B'cosX J' • • * (621) 



whence rR' 2 sin 2 X-(R' 2 -M)-] ,. .„, 



u= 4 — w±x — J- • • f6 ' 22 ) 



This equation holds good in the general case, whatever the 

 latitude may be. When A is small throughout the motion 

 1 — M/R/ 2 is also small, and it follows that we may use the 

 approximation 



u=Rg>[\ 2 -(1-M/R 2 )], . . . (6-23) 



where R is the equatorial radius. 

 The velocity in the meridian is given by 



. »= B t <* 8 > 



