Particle on the Surface of a Smooth Rotating Globe. 463 

 If cf> be reckoned from the meridian crossed at £ = 



cl) = \ (^ + l)V 2 (\&n 2 f-i)d^. . . (7-31) 



The longitude <p A of the point A, where the path crosses the 

 equator for the first time, is given by writing ^ = K, where 

 K is the quarter period of the elliptic function, and, 

 therefore, 



^=X (^ + i)i/2(E_iK), . . . (7-32) 



where, as is usual, E= p dn ^'^ 



Tables of the function zn ijr defined by 



znf= P(dn 2 ^-E/K)d^ 

 Jo 

 have been published. 

 In this notation 



0=\„(M 2 +l)^[W + (|-|)f]- • (7-33) 



The angle a at which the track crosses the equator is 

 given by 



C0Sa= V = /?+T' ' * ' * (r4) 



so that , a .a ._ ... 



cot ~ = fju and sm ^ = i . . . {I'^l) 



If the track is regarded as determined by V and a, then \ 

 may be found from equation (7'26), which takes the form 



"° = VRS sm 2 (7 ' 5) 



X 



From (7*25), (7*26) it follows that the periodic time is 



2Y da 7 s ( 7 ' 6 ) 



2K /; 



7V: 



The different subcases which occur as a is given different 

 values will be considered when we come to numerical 

 examples. 



8. Case II.— C = 0. 

 In this case 



Biil= '%'-xf • ' • • (8 ' 1} 



