1892.] Maclaurins Liquid SpJteruid. 27 



On reduction it will be found that (6) becomes 



d\ 4 i"^ dX - - 



1-f^ -rz — Ti>0, 



which is the condition that the spheroid should coincide with the 

 limiting Jacobian ellipsoid (Hydrodynamics, vol. ii. p. 113). The 

 excentricity is determined by the equation 



e (1 - e'f (8 + We') - (3 + 8e' - 8e') sin-^e = 0, 



which gives e = '8127. 



When the liquid is frictionless, the condition of stability is 



fdQ dQ\ 



giving e = "9529, which shows the difference between the two con- 

 ditions. 



4. We can also show that when the liquid is viscous a Jacobi's 

 ellipsoid, which differs very slightly from the limiting form, is 

 stable for an ellipsoidal displacement. Since Va, Vj, are zero in 

 steady motion, the value of S F is 



SF= i (VaaBa^ + 2Vab8a8b + V,,Sb^). 



Let cio be the equatorial radius of the spheroid for which 



e = -8127, 



and let the values of a and b in the Jacobi's ellipsoid be 



a = ao + a, b = ao + ^, 



where a and yS are very small quantities ; then the value of SFmay 

 be written 



8V = i{Vaa+ Va,\ {Sci + 8bf + i ( Vaa - F„,)o (ScL - 8by 



+ terms of the third and higher orders, 



where the suffix denotes that a and ^ are to be put equal to 

 zero. But we have already shown that in this case Vaa — Fa& = 0, 

 whence 8F is positive unless Sa = — Sb. It, therefore, follows that 

 the Jacobi's ellipsoid is stable for all ellipsoidal disturbances for which 

 Ba + 8b is not zero ; but when 8a + 8b is zero, the stability depends 

 on terms of the third order, which must be examined. It is, how- 

 ever, probable that the ellipsoid is stable in this case also. 



Since the limiting spheroid is a surface of bifu7xatioti, we have 

 an example of figure in which there is an exchange of stabilities, 

 the figures belonging to one series becoming unstable at this 

 point, whilst the figures belonging to the other series remain 

 stable. 



