ME. S. S. HOUGH ON THE ROTATION OF AN ELASTIC SPHEROID. 
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§ 3. Change of Variables. 
An elastic body, such as that with which we are dealing, will, of course, be capable 
of an infinite number of independent normal types of vibration. The equations of 
motion we have found in § 1 are applicable to any one of these types, while the 
substitution of the solutions in the boundary equations should lead to an equation 
for the determination of the frequencies. In general, the values of u, v, w will 
become very small when n is large in such a manner that nu, nv, nw approach finite 
limits, while the admissible values of X become large of the order n\ That type of 
oscillation with which we are concerned is, however, unique in character in that it 
continues to exist even when the rigidity is perfect. If, then, in the expressions for 
u, v, iv in terms of n we suppose n to be made infinite, u, v, w should approach finite 
limits which we will denote by u 0 , v 0 , iv 0 . 
The quantities u 0 , v 0 , iv Q denote the displacements of a body which is supposed 
perfectly rigid. Now the most general small displacement of such a body consists of 
a translation whose components we may denote by g, £, and a rotation whose 
components we denote by 9 X , 9. 2 , # 3 . Thus, the most general values of u Q , v Q , w 0 
geometrically possible are 
£ — yO 3 + zO,, y — 4- £ — xd. 2 + y9 x . 
When, however, we are dealing with the rotation of a rigid body not subject to 
external disturbing force the quantities f, g, £, 9 3 will not appear, and we may take 
u 0 = z9. 2 , v 0 = - z9 x , w 0 = — x9 2 + y9 1 .( 11 ). 
Let us now suppose that 
u = u 0 + v = Vq + w — Wq + 
where u Q , v Q , iv 0 have the values (llj. This is equivalent to supposing that the body 
is first displaced by rotation as a whole through small angles 9 X , 9. 2 about Ox, 0 y and 
is then subjected to elastic distortion, the displacements due to the distortions 
being u x , v v w x . 
Further, let £ = £ 0 + ( x , v' — v\ + v\, where 
£ 0 = cos a + v 0 cos /3 + iv 0 cos y 
£ : u x cos a + v x cos /3 + iv x cos y 
and where v' 0 , v\ denote the parts of v due to the harmonic inequalities £ 0 , £ L 
respectively. 
If V 0 denote the non-periodic part of Y, V 0 -f v' Q will be the potential at the point 
