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.