373-376] 
The Tidal Theory 
401 
where r 0 is the radius of S in its undisturbed state. With a closer approach 
of S', the mass S becomes unstable and begins to break up. 
(3751) is rather less than that obtained by Roche’s investigation for the 
case in which the masses rotate around one another. Clearly this must be so, 
since in the latter case rotational forces join hands with tidal forces in trying 
to break up the mass. 
I have found it possible to follow out the dynamical motion after in 
stability has set in. No matter whether the mass S' comes closer or not, the 
mass S rapidly elongates until it has assumed the shape of a spheroid of 
eccentricity 0’9477. At this stage a new type of instability sets in. Just as 
when a Jacobian ellipsoid first becomes affected with instability in the 
rotational problem, so here also a third harmonic displacement supervenes, 
causing the formation of a waist on the spheroid which would gradually 
deepen until the mass broke into two. But before this happens other dis 
placements represented by fourth, fifth and higher harmonics become unstable 
in turn. These form furrows around the spheroidal mass at various points 
and as these steadily deepen, the mass breaks into a number of detached 
pieces. The sequence of configurations is shewn in fig. 60 (p. 402), the last 
configuration being almost entirely conjectural. 
376 . A somewhat different situation arises when the mass S' moves too 
rapidly for an equilibrium theory of the tides to give accurate results. In 
this case we have to add up the impulsive forces communicated to S by the 
tidal forces at different instants of the passage of S'. If these do not suffice 
to elongate $ to an eccentricity greater than 0'8826, the tides raised in S 
merely die down after S' has receded. If however the impulses received 
from S' once elongate S to an eccentricity greater than 0‘8826, unstable 
motion follows, quite independently of the subsequent motion of S', and the 
sequence of events in S is that described and sketched in fig. 60. 
If aS^' is supposed to describe an approximately rectilinear path past $ 
with a constant relative velocity v, I have found that the closed distance of 
safe approach R is given by 
where M' is the mass of S' and p is the density of S, supposed uniform. 
If we assign to v the value v 2 = 7 (M + M')/R, which is appropriate to a 
circular orbit at a distance R, the limiting distance R takes the form 
This value of R is somewhat less than the value given by equation 
In passing we may notice that the limit of safety fixed by equation