552 On Forced Precession Sfc. of a Rotating Ellipsoidal Shell. 



ration of the values of (j, >/r given in (18), (19) shows that 

 with the actual values of I//M 7 , a/n and the supposed value 

 of e, for all thicknesses of the shell the semiaxes of the ellipse 

 described by its polar axis have values between those given 

 above from the two extreme hypotheses, i. e. the semiaxis 

 major lies between 8"" 91 and ' 9"'22, the semiaxis minor 

 between 6"*48 and 6"'86. 



8. In the case of the half-yearly nutation the couples about 

 the axes OA', OB' are of the forms — K sin st, K cos a cos st, 

 s denoting twice the earth's mean angular velocity round the 

 sun*. Here L / /M / = 1/cosa. If we replace this ratio by unity we 

 see from Art. 6 that the nutation of the massless shell (e=^-J ) 

 would be approximately! times what it would be if the whole 

 were rigid, and would not be reversed in direction. This 

 differs from Lord Kelvin's result. More accurately, taking 

 the semiaxes of the ellipse described by the axis of a rigid earth 

 to be 0""55 perpendicular to and 0"*51 parallel to the plane 

 of the ecliptic, those for the shell would be (0"*55 x 2*49 = ) 

 1""37 perpendicular to, and (0"'51 x 2*77=) 1"'41 parallel to 

 the plane of the ecliptic. 



If the mass of the shell be taken into account, as now s is 

 greater than en, the sign of the denominator of the second of 

 the two fractions in (18), (19) depends on the thickness of 

 the shell. Accordingly, if the thickness be gradually increased 

 from zero, for a certain value the nutation perpendicular to 

 the ecliptic vanishes and changes sign; for a certain greater 

 value the nutations both parallel and perpendicular to the 

 ecliptic become infinite and change sign; for another greater 

 value the nutation parallel to the ecliptic vanishes and 

 changes sign ; and for all greater values the nutations in both 

 directions remain of the same sign as if the whole were 

 rigid. 



9. In the case of the fortnightly nutations the couples are 

 of the same type as in that of the half-yearly; but s is twice 

 the moon's mean speed in longitude. Taking the period to 

 be 13§ days, it appears that the nutation of the massless shell 

 ( e= 3oo) * s approximately 21 times what it would be if the 

 whole were rigid, and is not reversed in direction. This differs 

 from Lord Kelvin's result. 



It may be pointed out that for the fortnightly and other 

 short-period nutations of a rigid earth s cannot be neglected 

 in comparison with n in equations (22), (23), without intro- 

 ducing a sensible proportionate error ; the calculations of 



* See e. g. Routh, l Rigid Dynamics/ 4th edit. vol. ii. p. 275, where 

 the couples causing the half-yearly nutation are given along with that 

 causing the solar precession. 



