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



11611(36 R2 © 2 = " 2=V2 -^ 



= V 2 -RV[X 2 -(1-M/R 2 )] 2 



= R 2 a) 2 [X 2 -^ 2 ][^ 2 -fC] ? . . (6*4) 



where C is a constant, which may be positive or negative, 

 and \ is the latitude in which the particle is moving to the 

 east. 



Thus J = ± W [(V-X 2 )(\ 2 + C)] V2 . • • (6-41) 



Three cases have to be considered, as C may take either 

 sign or vanish. 



7. Case I. — C positive. 



Let C = \oV 2 5 so th^ 



dX 



[(V-^ 2 )(^+/* 2 V)] 



<°dt— r/V2_>2W^2_ L '„2-\.2\ll/2' • • ( 7 '1) 



The northerly component of the velocity vanishes only in 

 latitude X North or South, so that the track must cross and 

 recross the equator. The negative sign is introduced as, if 

 the time is measured from the instant at which X=X , X will 

 be decreasing to begin with. 



To effect the integration, elliptic functions must be used. 



We write 



X=X cn<\/r (7*21) 



(\ 2 -X 2 ) 1 /2 = X sn^ ..... (7-22) 



( /A V + X 2 ) 1 ^=(/* s + l) 1 ^o(int, • • (7-23) 

 so that the modulus k is given by 



k 2 =(fji 2 + l)-\ ..... (7-24) 

 and the differential equation is satisfied if 



^V/^ + l) 1 ^. .... (7-25) 

 Comparison with equations (6*4), (6 23) shows that 



Y = PU0u 2 + 1)V (7-26) 



and w=R«[\ 2 + i(/* 2 -l)V] .... (7*27) 



= Ro)X 2 (^ 2 + l)[dn 2 t-i]. . . . (7*28) 

 The longitude <f> can be found from the equation 



b 2=« w 



