238 



Dr. C. Chree. 



dxx dxy dxz 

 dx + dy + dz 



dx dy dz 



Qpy, 



(3); 



dxz dyz dzz 

 dx + dy + dz 



Vxpz 



where 



P/> EE g* P ' ( P - P ') (1 - Vl2 - ^ 13 )/E! , 1 

 Qp = g 2 p ( P - p') (1 - m - 7/ 23 )/E 2 , }> 



(9). 



= 9 2 P(P-P)0- ~V31 -^/32)/E 3 J 



The surface equations to be satisfied by the supplementary solution 



Xxx + p>xy + va-v = Xxy + /x«/?/ + vyz = Xxz + + vzz = . . . (10). 



The problem thus resolves itself into that of an ellipsoid acted on 

 solely by bodily forces derivable from the potential 



This problem was solved by me in 1894 for isotropic* materials, and 

 in 1899 I extended the solution to seolotropicf ellipsoids. We can 

 thus derive a satisfactory supplementary solution from the sources 

 specified. Finally adding the stresses of the supplementary solution 

 to the stresses (3), and the displacements to the displacements (7), we 

 have a consistent and complete solution of the problem presented by 

 a heavy ellipsoid in a homogeneous liquid, when the action of the 

 liquid is supposed that given by elementary hydrostatics. 



§ 3. The supplementary solution, though simple in type, contains 

 terms which are of great length when the ellipsoid has three unequal 

 axes, and is of a complex kind of seolotropy. It will thus perhaps 

 suffice to select for illustration the simple case of an isotropic sphere of 

 radius a. 



Denoting Young's modulus by E, Poisson's ratio by t], and writing 

 r 2 for x 2 + y 2 + z 2 , we have in full 



are 



i(?x 2 + Qy 2 + Hz 2 ). 



* 'Roy. Soc. Proc.,' vol. 58, p. 39; ' Quarterly Journal of Pure aud Applied 

 Mathematics,' vol. 27, p. 338. 



f ' Camb. Phil. Soc: Trans.,' vol. 17, p. 201. 



