80 Mr Basset, On the Stability of [Nov. 28, 



The last two integrals are known to be equal, because they 

 represent the mutual potential energy of the original figure and the 

 stratum. To evaluate this integral, let V be the gravitation 

 potential of the original figure at an external point ; then if A- be 

 a length measured along the normal 



where g' is the normal component of the attraction at the surface. 

 But if g be the total force in this direction due to gravitation and 

 centrifugal force, 



9 = 9 - (^\i ^g'- h'rj/i;' ; 

 whence 



jj'^~^=jv:df.-jg\df.-^,frAdf^ (9). 



Now F is constant at the surface, also since the mass of the 

 stratum is zero, 



jdfji = 0, 



accordingly from (7) we get 



[ h^ C 



jVo'dfi = -^,jr^'d^, 



and (9) becomes 



jjdn^ = - ^, j(n + IXf df. - jgXdf. + ^, fl\^d^ . . .(10). 



Now d/j, = pdXdS, whence the last integral in (10) is of the 

 third order of small quantities and may be neglected ; also if a be 

 the thickness of the stratum, and / the moment of inertia of the 

 disturbed figure about the axis of rotation, the right-hand side of 

 (10) becomes 



K\I-h) 1 



2/o^ 



-\p^\gaHS. 



Whence if W be the potential energy of the disturbed figure 

 due to gravitation and W^ be that of the original figure 



and 



Tf.= F.-|//*^' (11), 



lf=F.-i//'^ + |p//K<iS + ^^<|^> (12), 



The total energy E of the disturbed figure, which represents 

 its capacity to do work upon itself, is obtained by adding the term 



