Our purpose in cutting M into tubes is to reduce
our n-dimensional eigenvalue problem to a bunch of 1-dimensional
inhomogeneous string problems (Sturm-Liouville problems).
An inhomogeneous string is described by two functions 
and 
,
telling its cross-section and its density as a function of length.
When we come to consider the strings that arise from our tubes,
we will want to set 
,
to indicate that the mass per unit length is proportional to the
cross-section.
For the moment we will distinguish from ,
not so much for the sake of generality as to make clearer
their differing roles.
Definition.
If
is measurable, and
whenever 
,
we will say that is
neither too big nor too small.
Thus (iii) above states that for almost all 
,
is neither too big nor too small.
Definition. Let
be measurable,
with and neither too big nor too small.
Then we define
as the infimum over 
of the Rayleigh quotient
