224 PROFESSOR A. E. H. LOVE ON THE 



surface harmonics which occur are those of the first degree and the following 

 harmonics of the third degree : 



cos 3 6 f cos 0, (cos 2 ) sin 9 cos <j>, (cos 2 ^) sin sin <f>. 



To each of these terms there corresponds a term of the same form in ', and 

 therefore also in P, or p +\; and it follows that the displacement of the bounding 

 surface from its initial form (which is a slightly elliptic oblate spheroid appropriate to 

 the rotation) is expressed by a radial displacement, which consists of a part propor- 

 tional to a spherical surface harmonic of the first degree, together with parts propor- 

 tional to the above surface harmonics of the third degree. In like manner all the 

 terms of the additional potential W are the products of functions of r and surface 

 harmonics, which are either of the first degree or are the above harmonics of the 

 third degree ; but the coefficients of the various harmonics in W are different from 

 their coefficients in The equation of the equipotentials under gravity, modified by 

 the rotation, is 



V e + W + <o 2 (x? + y 3 ) = const. , or p /p a + W = const. ; 



and thus the situation of the bounding surface relative to the equipotentials is 

 expressed by a difference of radii at corresponding points, this difference being a sum 

 of terms of the form &S, where b denotes a constant and S denotes a surface 

 harmonic ; and the surface harmonics which can occur are those of the first degree 

 and the three of the third degree written above. 



51. It appears from this investigation that, if a gravitating body, which is rotating 

 about an axis, has so small a modulus of compression that, if the body were at rest, a 

 spherically symmetrical distribution of density would be unstable, it would tend to take 

 up a state in which the distribution of density would not be exactly hemispherical, 

 but the excess density would also contain terms expressed by spherical harmonics of 

 the third degree. The figure of the body would differ from the oblate spheroidal 

 figure appropriate to the rotation by a radial displacement at each point ; and this 

 displacement would be expressed partly by spherical surface harmonics of the first 

 degree, indicating that the centre of gravity does not coincide with the centre of 

 figure, and partly by spherical harmonics of the third degree. If the body were 

 entirely devoid of rigidity, the oblate figure appropriate to the rotation would be the 

 same as that of an equipotential surface under gravity, modified by the rotation ; and 

 the figure of the body, as determined by difference of level above or below a certain 

 equipotential surface, would be an harmonic spheroid of the third degree, and the 

 situation of the body would be that of such a spheroid when displaced towards one 

 side. If the body possessed some rigidity, the oblate figure appropriate to the 

 rotation would differ a little from that of a nearly coincident equipotential surface, 

 and the shape of it, determined as before, would be that derived from a certain oblate 

 spheroid of small ellipticity by a displacement proportional to a surface harmonic of 

 the third degree. The surface harmonic would be of a somewhat specialised type. 



