With the values above of u, v, w, u´, v´, w´, the equations become of
the form
1 dp
----- -- + 4[pi][rho]Ax + [alpha]x + hy + gz = 0, (11)
[rho] dx
1 dp
----- -- + 4[pi][rho]By + hx + [beta]y + fz = 0, (12)
[rho] dy
1 dp
----- -- + 4[pi][rho]Cz + gx + fy + [gamma]z = 0, (13)
[rho] dz
and integrating
p[rho]^(-1) + 2[pi][rho](Ax² + By² + Cz²)
+ ½([alpha]x² + [beta]y² + [gamma]z² + 2fyz + 2gzx + 2hxy) = const., (14)
so that the surfaces of equal pressure are similar quadric surfaces,
which, symmetry and dynamical considerations show, must be coaxial
surfaces; and f, g, h vanish, as follows also by algebraical
reduction; and
4c²(c² - a²) / c² - a² \²
[alpha] = ------------[Omega]2² - ( -------[Omega]2 - [eta] )
(c² + a²)² \ c² + a² /
4b²(a² - b²) / a² - b² \²
- ------------[Omega]3² - ( -------[Omega]3 - [zeta] ), (15)
(a² + b²)² \ a² + b² /
with similar equations for [beta] and [gamma].