220 The Configurations of Rotating Liquid Masses [ch. viii 
in fig. 26. The dotted lines represent the surface distorted by the third zonal 
harmonic displacement through which instability first sets in. 
202. Proceeding along the series of Jacobian ellipsoids from its junction 
with the Maclaurin spheroids, the configurations remain stable up to the 
critical ellipsoid defined by equations (201'3). Through this point two linear 
series pass, namely, the series of Jacobian ellipsoids and the series of pear-shaped 
configurations which are specified, in the immediate neighbourhood of the 
Jacobian series, by giving small values to e in equation (199’5). This speci 
fication of course breaks down as soon as the values of e 2 become appreciable. 
It would be a feasible, although extremely laborious task, to use the 
formulae already given to calculate the whole series of pear-shaped configura 
tions. I have calculated* these as far as terms in e 3 , but find no new features 
are introduced beyond those shewn in fig. 30, until terms in e 4 and higher 
powers of e become appreciable and the calculation fails. 
The corresponding problem in two dimensions—the determination of the 
analogous figures for infinitely long cylinders rotating under no forces beyond 
their own gravitational attractions—is incomparably simpler, because the 
formula for the gravitational potential of a two-dimensional cylinder is far 
simpler than that for a three-dimensional body. I have calculated f the two- 
* Phil. Trans. 217 A (1916), p. 1. 
f Problems of Cosmogony and Stellar Dynamics, p. 102, and Phil, Trans. 200 A (1902), p. 67. 
The original calculation in the Phil. Trans, contained a number of numerical errors, mainly quite 
unimportant, which were corrected in the later presentation in Problems of Cosmogony.