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



The elliptic functions degenerate into hyperbolic functions, 

 and the solution is given by 



\ = \osechi/r ....... (8'21) 



and <£ = \ [tanh<f-^], .... (8*22) 



where i|r=\ &)£ (8*23) 



For large values of t, X tends to zero, so that the track 

 approaches the equator asymptotically. 



9. Case III. — C negative. 

 Let C= — Ws so ^at 



w ^ = ~[(V-V)(A 2 -\ y)] 1/2 " ' ' (9 * 1} 



The northerly component of the velocity vanishes for \ = A 

 and also for X = X oy ct, so that the path consists of loops on one 

 side of the equator. 



The integration is effected by taking 



X=Xodn^ (9-21) 



(X 2_ X 2 ) i/ 2= (1-^)1/2^^ . . . (9-22) 



(\ 2 -X V 2 ) 1/2 =(l-^ 2 ) 1/2 ^ocnf. . . (9-23) 

 The modulus is given by 



P=l_^ (9.94) 



and the argument by ^ = \ cot (9*25) 



Comparison with (6*4) and (6*23) shows that 



V=iRo)(l- A 6 2 )X 2 (9-26) 



w = R w A 2 [dn 2 ^-i(H-^ 2 )].. . . (9-27) 

 The longitude (/> is found from the equation 



</»=Xo[zn^+{|-i(l + / a 2 )}f]. . . (9-3) 



The period for the velocity is given by -^ = 2K, and the 

 change in longitude during the period is therefore 



\ [(l + yu, 2 )K-2E] towards the west. . (9'31) 



10. Numerical Examples. 



In the following examples the velocity of the moving 

 particle is taken as 10 metres per second. The dimensions 



