184 Prof. Ohallis on the Hydrodynamical Theory 



17. We have next to take account of the effect of the velo- 

 city W. Resolving it into parts W sin and W cos respec- 

 tively along and perpendicular to the surface of the sphere, 

 the latter may, for reasons already given, be left out of account, 

 so that we have only to calculate the accelerative action on the 

 sphere resulting from the velocity W sin 0. This velocity 

 takes place along the intersection of the spherical surface by 

 any plane passing through the axis from which is measured, 

 inasmuch as the motion is evidently symmetrical with respect 

 to this axis. Thus we have here a case of constrained motion 

 uninfluenced by the action upon the fluid of any extraneous 

 accelerative force. Hence, if V be the velocity at any point 

 at any time t, and a f be the corresponding condensation, we 

 shall have by a known formula, 



K 2 a 2 d<r f (<fT\~ 



(l+a')bdd + \dt) ' 

 or 



da' b dN JQ YdV 



K?a? 



- 2 ~ 2 dt 



But since Y = W sin 0, we get by means of the value of W in 

 art. 16, 



_— = nrkno 1 sin V cos" 1 u I -=- + 7 , 9 bcosV 1 . 

 dt \dt icadr / 



Hence, by substituting in the last equation and integrating, 

 Nap.log(l + </) = F(0 



nkntf /dT cos 3 , b d 2 T cos 4 

 K 2 a 



1 \dt 3 + Ka dt 2 ' 4 J 2/eV 



Since by the conditions of the question the velocity V and con- 

 densation g' pertain only to the first hemispherical surface, we 



IT 



must have </ = and V = where the angle is ^-. Conse- 

 quently ¥{t) — 0. Hence, if we omit periodic terms and those 



b 2 

 which as containing the factor r^ are ^ 00 small to be signifi- 



cant, the foregoing equation gives, to the second order of ap- 

 proximation relative to the constant m, 



aV = - ^ = - Z 2 T 2 sin 2 cos 4 0. 

 2/r 2k 2 



But T 2 = m 2 sin q(c — Ka£)—~^ (\ — cos 2q(c — /cat)*), terms 

 containing higher powers of m than the second being omitted. 



