322 MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROID. 
to the time, on the supposition that as 0 , y 0 , z 0 , remain constant, and d/dt be used when 
x, y, z, are the other independent variables, we have 
d d du 0 dv 0 dw 0 
dt dt dt dx dt o/j dt. dz’ 
Hence, if we neglect squares and products of the small quantities u, v, iv, we 
obtain 
u = du/clt = du/dt, u = cV’ujdV 2 = d 2 ujdt~, &c. 
Thus, the only modification necessary on the right will be to replace x 0 -f- u by x, 
and y Q + v by y, and the equations of motion become 
orx) 
"V) > .(!)• 
l + | + 5 + ^ X 
ST , SS SR 
+ Ty + S + 
p (u — 2(ov 
p (v -f- 2 mu 
pw 
We have retained the fluxional notation, it being understood that the dots now 
denote differentiation with regard to the time on the supposition that x, y, z remain 
constant. 
Since the stress-strain relations do not involve differentiation with regard to the 
time, they may be written down in the same manner as when the axes are fixed. To 
the degree of approximation to which we are going, we may replace du/dx 0 , &c., by 
du/'dx, &c., and therefore when the material is isotropic and incompressible, the 
components of stress are given by 
where p denotes the hydrostatic pressure at x, y, z, and It the rigidity. 
