1894.] Forces derivable from a Potential of the Second Degree. 27 



whose density p is uniform, the statical resultant reduces to a couple 

 whose components about the axes of x, y, z are respectively 



4irabc (6 2 c 2 ) /S/15, 47rabc (c 2 a 2 ) //T/15, and 47rabc (a 2 fc 2 ) />TJ/15. 



These components vanish in the case of a sphere, but in an 

 ordinary ellipsoid equilibrium will not exist unless S, T, U all 

 vanish. 



The problem solved in the present memoir, viz., that of an isotropic 

 elastic solid ellipsoid under the action of bodily forces derived from a 

 potential 



is thus, for an ordinary ellipsoid, the most general case of equilibrium 

 under forces derived from a potential of the second degree. The above 

 potential covers forces arising from mutual gravitation or from rota- 

 tion about a principal axis in an ellipsoid of any shape. 



The method of solution reverses the usual order of procedure, the 

 stresses being first determined and then the strains and displace- 

 ments. The solution obtained satisfies without limitation or assump- 

 tion of any kind all the elastic solid equations. Unless the ratios 

 a:b:c are assigned definite numerical values, the constant coeffi- 

 pients in the expressions for the stresses and strains are of course 

 somewhat cumbrous ; but for any specified case, whether of gravita- 

 tion or rotation, or both combined, the solution becomes easily 

 manageable. It enables the variation in the effects of gravitation and 

 rotation with the change of shape of the ellipsoid to be completely 

 traced. 



The comprehensiveness of the problem solved forbids more than a 

 brief consideration of the general solution with illustrations of its 

 application to a few of the more interesting special forms of ellipsoid. 

 The results obtained for the very oblate and very oblong forms seem 

 to show that in many cases of bodily forces the assumptions usually 

 made in the treatment of thin plates and long rods would not be 

 justified. 



By comparison with the author's previous researches, a close 

 similarity is shown to exist between the phenomena in rotating flat 

 ellipsoids and thin elliptic discs on the one hand, and rotating 

 elongated ellipsoids and long elliptic cylinders on the other. 



Various results confirmatory of the accuracy of the present solution 

 are obtained by the application of the general formulae for the mean 

 strains in elastic solids. It is also shown that some of the results 

 may be arrived at by the use of approximate but simple methods. 



The Society adjourned over the Whitsuntide Recess to Thursday, 

 May 24. 



