310 Mr Hicks, On the motion of a mass of liquid [Jan. 29, 



If now the shell changes its axes so that 



(p- 



*fM^-*i)=°(*-*f) » 



the pressure along the inside of the shell is uniform ; in other 

 words a uniform pressure over the surface will make the axes 

 change according to the laws given by this equation. We may 

 therefore suppose the shell removed, and the fluid will move by itself 

 so that it always preserves the ellipsoidal form, and the magnitude 

 of the axes at any time will be determined by equations (1) 

 and (2). 



A first integral is easily obtained, for each member of (2) is 

 equal to 



, .... 77 - .... , /B£lda SCldb B£l dc\ 

 -2 {aa+hh + CC) - k {jaTt + Wdt + Jc-dt) 



a b 6 

 -+j + - 

 a o c 



and since the denominator vanishes by (1) and II contains t only as 

 entering through a, b, c it follows that 



whence d 2 + ¥+ c 2 = 4k (fl - fi ) (3), 



H being a constant. This is the equation of the conservation of 

 energy, and might have been thus deduced. Two particular cases 

 are at once completely solved by means of (1) and (3), viz. the 

 case of two axes always equal, or spheroids ; and the case where one 

 is infinite, in which case we get an elliptic cylinder. In the first 

 case we have immediately, if b = c, 



a 2 = lf = 2aT+b^ n ~~ n ^ 



and ab 2 = r 3 , if r is the radius of the sphere whose volume is the 

 same as that of the liquid. Putting in this value we have 



07. 3 



d ' = 2^P < Q -°J («> 



The relation between the greatest and least values of a is found 

 from the equation 



