326 
MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROID. 
(dv . dv n . dv \ (du dv duo \ 
— p cos p + H ( 0 - cos a + cos p + 07 cos yj -fill,- cos a + ^ cos p -j- xp cos yj 
= — cos (3, 
, /dw die „ , die \ . fdu dv n . dw \ 
— p cos y + R [ g- cos a -f- -p cos p + gT cos y j + IT /pp cos a + "01 cos p + ^7 cos y ) 
= — cos y. 
These equations express the fact that the surface-tractions at the mean surface are 
equivalent to a normal stress equal to the weight of the harmonic inequality, 
and they might have been written down at once from this consideration. It has, 
however, been thought preferable to verify them at length ; it will be seen that the 
shorter procedure is only justifiable in the case where the material is initially in 
a state of hydrostatic equilibrium. If in the zero configuration this condition is not 
satisfied the form of the boundary equations will be much more complicated. 
Let us now replace p by the function xp of the previous section. By the definition 
of xp we have 
p/p = V + l ad (x 3 + if) — i /t 
= non-periodic terms -)- v — xp, 
where v' denotes the potential due to the harmonic inequalities. 
The non-periodic terms vanish at the surface in virtue of the conditions for steady 
motion, and therefore the boundary equations may be written 
| du du du du dv p dw 
xp cos « + n cos a -f- cos p -+- ^ cos y + - - cos « + ,r cos p -J- ^ cos y 
—• ( v ' — 9l) cos a 
1 Pi i Sr dv dv du dv dm , 
xp cos/3 + n jppeosa + g— cos p -f- pp cos y ~j- ^ cos a -f cos p + 0y cosy| ^ 
= i v ' ~ fft) cos £ 
, jdw dw dw 
xp e°s y + n COS a + ^ COS/3 + jfCOSy + 02 
= (v — gi) cos y 
t, — u cos a -\- V COS /3 -f- IV cos r.(10)- 
dw „ . dw . du dv n . dw 
cos a + ^ - cos p + x; cos y 
wher< 
The rigorous method of procedure would be first to solve the equation (8) for xp. 
Having found xp we could introduce its value into the right-hand members of (7) and 
proceed to find solutions of these equations consistent with the boundary conditions 
(9). Unfortunately the form of equation (8) is such as to make it appear hopeless 
to carry out this process, and we must have recourse to some method of approxima¬ 
tion before we can advance further. 
