Forces derivable from a Potential of the Second Degree. 47 



are so far in harmony with the assumptions usually made in theories 

 of thin plates. It should be noticed, however, that dzxjdz is of the 



same order of magnitude as dxxjdx, and so of equal importance in 

 the first body-stress equation. 



When R alone exists, or the forces are parallel to the small dimen- 

 sion, zz at points in the interior is of the same order of magni- 

 tude as xx, yy, and xy. 



If there be bodily forces both parallel and perpendicular to the 

 small dimension, and the two sets be of the same order of magnitude, 

 then to a first approximation the stresses due to the former set of 

 forces may be neglected. 



Gravitating very Oblate Spheroid. 



7. Taking /i as before for the gravitation constant, we have as a 

 first approximation * 



T> ^_ f\ ^^ 2 nlft "R A /"-IQ^ 





As there is symmetry round the axis of z, we employ cylindrical 



^ N ""N 



coordinates r, 0, z. The notation rr, 00, . for the stresses 



explains itself. The displacements are u along r, and 7 parallel 

 to z. The strains are 



s r = dujdr, s<t> = u/r, s z = d^/dz, 



r$ = 0, ff <t>z = 0, ff er = du/dz + dy[dr 



For the values of the stresses we find 





zz = 27T/4/> 2 c 2 (l r-ja z z 2 / 

 c 



(35). 



r0 = 0z = 



The strains which differ from zero are 



* Thomson and Tait's ' Natural Philosophy,' yol. 1, Part II, Art. 527. 



