134 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



the additive constant being zero if the axis of X be taken 

 coincident with, and not merely parallel to, the axis of the 

 impulse /. 



The exact solution of (43) involves the use of elliptic functions. 

 The nature of the motion, in the various cases that may arise, is 

 however readily seen from the theory of the simple pendulum. 

 For a full discussion of it we refer to Thomson and Tait, Arts. 333, 

 et seq. 



It appears from (43) that the motion of the solid parallel to its 

 axis is stable or unstable according as A &amp;gt; B. Since A denotes 

 twice the kinetic energy of the solid moving with unit velocity 

 parallel to its axis, and similarly for B, it is tolerably obvious that 

 if the solid resemble a prolate ellipsoid of revolution A &amp;lt; B, whilst 

 the reverse is the case if it resemble an oblate ellipsoid. Compare 

 Art. 111. 



The above analysis applies equally well to the somewhat more 

 general case (6) of a body with two mutually perpendicular planes 

 of symmetry, when the motion is altogether parallel to one of 

 these planes. If this plane be that of xy we must suppose the 



origin transferred to the point f -~- , 0, Oj ; if it be that of xz, 



f N \ 



to the point [ 77 1 0, OJ . 



118. The question of the stability of the motion of a body 

 moving parallel to an axis of symmetry is more simply treated 

 by approximate methods. Thus, in the case (d) of a body with 

 three planes of symmetry, and slightly disturbed from a state of 

 steady motion parallel to x, we have, writing u = c + u, and 

 assuming u , v, w, p, q, r to be all small, 



du dv 



Hence 



