32 
ME. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
vertical co-ordinates, so that for instance all systems, whether in equilibrium or not, 
which have the same angular momentum as the critical Jacobian ellipsoid, must be 
represented in the horizontal plane through O (this plane having as many 
“ dimensions ” as are necessary). In this diagram the pear-shaped series of figures 
will he below this plane in the neighbourhood of 0. 
There are two a priori alternatives, represented on the right-hand and left-hand sides 
of fig. 1. In the first place, it is possible that after passing a certain distance OR' 
along the pear-shaped series with decreasing angular momentum, we come to a region 
ITS' in which the angular momentum again increases. Any horizontal line in the 
diagram ought, on the principle of stable and unstable configurations of equilibrium 
occurring alternately, to meet stable and unstable branches of the linear series 
alternately. Thus the branch R'S' ought to be stable, so that B/ would be a 
point of bifurcation on this series, and the other series through R', say R'T, would 
be unstable. 
The alternative possibility is that the pear-shaped series of figures proceeds 
continually downwards in the diagram, so that its angular momentum continually 
decreases. 
26. Either of these two possibilities removes a difficulty to which Sir G. Darwin 
has drawn attention. # In what is commonly referred to as Roche’s problem an 
attempt is made to discover the other end of the pear-shaped series of figures, on the 
supposition that this other end represents two detached masses revolving about one 
another. Two such series of figures appear to existf ; in one the satellite is spheroidal 
except for tidal distortion, while in the other it is ellipsoidal. The former series has 
been shown to be stable, the latter unstable. 
As the angular momentum decreases on passing along these series, the distance 
between the two masses also decreases until a point is reached at which the two 
series coalesce, the configuration of bifurcation being one in which the radius vector 
from the centre of the primary to that of the satellite is equal to 2'457 radii of the 
primary.| If the distance between the masses is decreased still further, the remaining 
configurations form an unstable series. Sir G. Darwin found a difficulty in the 
instability of this series, since he believed it to be the far end of the pear-shaped 
series which he thought stable. We now see, however, that this series may, without 
change of stability, join up with either the series TIT on the right-hand of our 
diagram, or the series RO on the left. 
* ‘Coll. Works,’ III., pp. 515-524. 
t Roche’s problem has only been solved strictly by imposing sphericity on the primary and assuming 
the satellite to be infinitesimal. Sir G. Darwin’s work (‘ Coll. Works,’ III., p. 436), leaves little room 
for doubt that Roche’s result may be extended in the way I have stated. 
\ If the satellite is not infinitesimal, the radius vector depends on the ratio of the masses, but always 
lies between the narrow limits 2 - 457 and 2 - 514 times the radius of the primary (see Darwin, Joe. cit., 
p. 507). 
