be described
as A + a and A + b, so that a and b denote the specific differences
which discriminate the families from one another. Now following in
imagination the family of the type A + a, let us begin with the case
where the specific difference a is well marked. As we cast our eyes
along the series forming the family, we find the difference a becoming
less conspicuous. It gradually dwindles until it disappears; beyond this
point it either becomes reversed, or else the type has ceased to be
a possible one. In our shorthand we have started with A + a, and have
watched the characteristic a dwindling to zero. When it vanishes we have
reached a type which may be specified as A; beyond this point the type
would be A - a or would be impossible.

Following the A + b type in the same way, b is at first well marked, it
dwindles to zero, and finally may become negative. Hence in shorthand
this second family may be described as A + b,... A,... A - b.

In each family there is one single member which is indistinguishable
from a member of the other family; it is called by Poincare a form of
bifurcation. It is this conception of a form of bifurcation which forms
the important consideration in problems dealing with the forms of liquid
or gaseous bodies in rotation.

But to return to the general question,--thus far the stability of these
families has not been considered, and it is the stability which renders
this way of looking at the matter so valuable. It may be proved that if
before the point of bifurcation the type A + a was stable, then A + b
must have been unstable. Further as a and b each diminish A + a becomes
less pronouncedly stable, and A + b less unstable. On reaching the point
of bifurcation A + a has just ceased to be stable, or what amounts to
the same thing is just becoming unstable, and the converse is true
of the A + b family. After passing the point of bifurcation A + a has
become definitely unstable and A + b has become stable. Hence the point
of bifurcation is also a point of "exchange of stabilities between the
two types." (In order not to complicate unnecessarily this explanation
of a general principle I have not stated fully all the cases that may
occur. Thus: firstly, after bifurcation A + a may be an impossible type
and A + a will then stop at this point; or secondly, A + b may have been
an impossible type before bifurcation, and will only begin to be a real
one after it; or thirdly, both A + a and A + b ma