FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
31 I 
of potential in passing from any one j^oint to any other lying on the same orthogonal 
curve in the neighbourhood of a finite double layer. 
Again, the system D, consisting of — li and -j- C, may be built up by an infinite 
number of finite double layers. Hence by a second integration we may find the 
difference between the potential of D at any point inside R and the point lying on J 
where the orthogonal curve through the first point cuts the surface of J. 
Finally, it may be proved that the lost energy ^DD is equal to half the difference 
of potentials just determined multiplied by the density and integrated throughout 
the region R. The required expression of this portion of the energy is found to 
consist of two parts, of which one depends on magnetic moment and the other on the 
force (§ 9). The reduction of this part of the energy to calculable forms is not very 
simple ; it is carried out in §§11, 12. 
The calculation of the moment of inertia of the pear is comparatively easy, since it 
only involves those harmonic inequalities of J which are expressible by harmonics of 
the second degree (§ 13). On multiplying the moment of inertia by ^-Sco^, we obtain 
the last contribution to the expression for E. 
The energy function cannot involve e^, since the vanishing of the coefficient of that 
term is the condition whence the critical Jacobian was determined. If f denotes the 
coefficient of any harmonic inequality other than the third zonal one, the part of E 
independent of Sco^ is found to contain terms in e^, elf and (/)'^. The coefficient of 
Sco® consists of a constant term, a term in and terms in /I and where these /’s 
denote the coefficients of the second zonal and sectorial harmonics. This last part 
does not contain the coefficient of any hai’inonic of odd degree, and in the first ]3art 
the coefficient of the term in eyfor all such harmonics is found to vanish. 
The condition for the figure of equilibrium is that the variations of E for variations 
of and of each f shall vanish. On differentiating E with respect to the f of any 
harmonic of odd degree and equating the result to zero, we see that that f must 
vanish. Hence it follows that the pear cannot involve any odd harmonic excej^ting 
the third zonal one. Again, the symmetry of the figure negatives the existence of 
any even functions involving sine-functions of the quasi-longitude measured from 
the prime meridian (as I may call it) of symmetry through the axis of rotation. 
The same consideration negatives the existence of even functions involving cosine 
functions of odd rank. Accordingly the only functions to be considered are the even 
ones of even rank, involving the cosine functions of the longitude. 
The equation to zero of the variations of E for all the /’s, excepting ff', gives 
at once all those /’s in terms of The equations to zero of the variations for /g, 
// give three equations for the determination of as multiples of e". We 
thus have the means of finding the angular velocity and all the /’s in terms of the 
parameter e, which measures the amount of departure of the j)ear from the critical 
Jacobian ellipsoid (§ 14). 
It seems unnecessary to give here any explanation of the methods adopted for 
