124-125] STABILITY. 189 



with three planes of symmetry, as in Art. 123, 3, slightly dis 

 turbed from a state of steady motion parallel to x, we find, 

 writing u = u + u , and assuming u t v, w, p, q, r to be all small, 



...(1). 



D d 2 v A (A-B) A 



Hence B ~j- + A ^ - u&amp;lt;?v = 0, 



with a similar equation for r, and 



with a similar equation for q. The motion is therefore stable only 

 when A is the greatest of the three quantities A, B, C. 



It is evident from ordinary Dynamics that the stability of a 

 body moving parallel to an axis of symmetry will be increased, or 

 its instability (as the case may be) will be diminished, by 

 communicating to it a rotation about this axis. This question 

 has been examined by Greenhill*. 



Thus in the case of a solid of revolution slightly disturbed from a state of 

 motion in which u and p are constant, while the remaining velocities are 

 zero, if we neglect squares and products of small quantities, the first and 

 fourth of equations (1) of Art. 121 give 



du/dt=0, dp/dt = 0, 

 whence u- = u , p=p Q ........................... (i), 



say, where U Q , p are constants. The remaining equations then take, on 

 substitution from Art. 123 (3), the forms 



&quot;Fluid Motion between Confocal Elliptic Cylinders, &c.,&quot; Quart. Journ. 

 Math., t. xvi. (1879). 



