342 
MR. S. S. HOUGH ON THE ROTATION OF AN ELASTIC SPHEROID. 
force, if distortion were resisted by gravitation as well as by elasticity. The quantity 
e is the ellipticity which would be induced if elasticity were the only resisting factor. 
The reason Professor Newcomb’s hypothesis is at fault is that the ellipticity e, 
called by him the “ natural ” ellipticity of the spheroid, is itself partly maintained by 
centrifugal force, and that if the rotation were annulled, the spheroid would be 
distorted so that its ellipticity would no longer be e, but E, say. 
The ellipticity induced by rotation about a displaced axis has to be superposed on 
the ellipticity E about the original axis of rotation, and not on the ellipticity e. 
Thus, in ( b ), e should be replaced by E. Now it is obvious that e = E + E', and by 
a formula given by Thomson and Tait # 
Thus 
and therefore 
1 
E 7 
E'^ and E = —- > 
e + e e + e 
E' : E = e' : e. 
With the above correction, then, the laws («) and (b) become identical. 
The spheroid will be distorted not only by centrifugal force about a displaced 
axis but by the relaxation of centrifugal force about the original axis. The second 
disturbing factor has been neglected by Professor New 7 comb. 
The disturbing potential will be the difference of the rotation-potentials due to 
rotation with angular velocity w about 0 z and about 02], that is to 
icw ( X 3 + if) — -\co : (xf + y x ~) 
= W + f/ 2 ) — W {(x — z9 2 )~ + (y + 
= (6„xz — 9 x yz). 
It is obvious that the equations (19), (20) are the equations for the distortion of 
an elastic sphere when distorted by forces throughout its mass derivable from this 
potential function. 
Finally the angular velocity of It about P is X, and therefore the angular velocity 
of R as viewed from P r is 
e + e' 
-= we, 
which is the third hypothesis made by Newcomb. 
> 
* ‘ Natural Philosophy,’ Part II., § 840. 
