312 Mr Hicks, On the motion of a mass of liquid, etc. [Jan. 20, 

 From (4) we have 



*"2V(2A;)JV K 



2a 3 + r 3 

 (a 3 (n-Xl )_J 



where it must be remembered that Q has different forms for the 

 cases of a > r and a < r. We consider these separately and deter- 

 mine the time from the spherical state to the greatest elongation 

 or depression. If g denote the acceleration due to gravitation on 

 the surface of the liquid when it has the spherical form it can 

 easily be shewn that ik = Sgr 2 . 



Putting in this value of k, it can be shewn, after some numerical 

 calculations, that if t v t 2 denote the times from greatest elongation 

 to the spherical form, and from the spherical form to the greatest 

 compression respectively, then to the second order of u, v 



*-i /Efc-uJl , [ 101S , 304 N J 

 1 " ¥ V g [2 + 126 U ° + V42336 "" + S9Q9J "° j 



/_i /?rfe_l? , / 1013 589 \ J 



2 2 V g (2 126 %+ 1,42336 "" 2 x 3969/ V ° } ' 



whence if T denote the time of semi-oscillation, we have, from (7), 



2Y ^ I 21168 



the first term of this agrees with the result obtained by Sir W. 

 Thomson* for the time of oscillation of a liquid sphere, slightly 

 deformed according to a zonal harmonic of order 2. 



Similarly in the case of the elliptic cylinder, it follows that 

 a + r v 0/ 



where 2k = 



gr, 



i _ r °° a\ 



ioV{(a 2 +X)(6 2 +X)} ; 

 fl is here infinite but O — O is finite, viz. 



2 log 



«o+j>o 



a + b 



Hence *= #* log «£±£ « . 



a 4 + ?' & « 2 + ?' 2 ff 



* " Oscillations of a liquid sphere." Phil. Trans. Boy. Soc. (18G3), p. G08. 



