14-2
200 , 201 ]
Pear-shaped Configurations
219
Adding corresponding sides of the three equations (200*6), we again obtain
the value for 6 already obtained in equation (190*3), namely,
and on inserting this value for 6 , equation (201‘1) becomes an equation in
a, b, c alone. It is this equation which determines the first point of bifurcation
on the Jacobian series of ellipsoids.
The numerical solution of this equation is an arduous piece of work, for
the integrals c lf c 2> c 3 cannot be evaluated in finite terms, and the solution of
the equation can only proceed by trial and error. The necessary computations
were carried through by Darwin*, who obtained the solution
Darwin’s calculations were based on the equations he obtained by harmonic
analysis, not on equations (201*1) and (201*2), but I have verified*}- that his
solution satisfies these equations.
The shape of the critical ellipsoid defined by equations (201*3) is shewn
* Coll. Works, hi. p. 288; or Phil. Trans. 198 A (1901), p. 301. + Phil. Trans. 215 A (1915), p. 53.
6 =
( 201 * 2 ),
— = 1*885827, -=0*814975, - = 0*650659 (201*3)
r 0 r 0 r 0
where r 0 3 = abc, so that r 0 is the radius of the sphere of equivalent volume.
The corresponding value of « is found to be given by
A
t
B
c
-© 1 — ; !_
O N c
A
C
M
0
M
C
Fig. 26.