Forces derivable from a Potential of the Second Degree. 4 1 



2 ' ' f 



H-L(6 2 +c 2 )}] ... (8). 



2. Let t x . . . . denote the component stresses across the tangent 

 plane at#, y, z to a quadric concentric with, and similar and similarly 

 situated to the bounding surface (1), for whose equation we take 



a-V + &-y + c-V= X (9), 



then we easily find 



= i-X .......... (10), 



where p^ is the perpendicular from the centre on the tangent plane. 

 Thus the resultant stresses across parallel tangent planes to the 

 system (9) at the points of contact are all parallel, and their intensity 

 varies as 1 \. 



3. If the ellipsoid be rotating with uniform angular velocity to 

 about the axis 2a, we have 



while if it be gravitating, the force between unit masses at unit 

 distance being taken as unity, 



with symmetric il expressions for Q and R. If the ellipsoid be 

 gi-avitating, and at the same time rotating about a principal axis, we 

 have only to add the respective values of P, Q. R. Substituting the 

 values of P, Q. R in the expressions for L, M, N, and inserting the 

 consequent values of L, M, N in the formulae (4), (6), (7), \ve have 

 the complete values of the stresses, strains, and displacements. 



Gravitating nearly Spherical Ellipsoid. 



4. Denoting by ft, the force between two unit masses at unit dis- 

 tance, we may take 



P= -- 

 3 



with symmetrical expressions for Q and R. 

 We thence find 



* Thomson and Tait's ' Natural Philosophy ' vol. 1, Part II, p. 47. 



E 2 



