To estimate 
,
we will use
Rayleigh's cutting method
(see Rayleigh
[14],
Maxwell
[8]).
This method belongs to a class of related methods known collectively
as
Rayleigh's short-cut method.
For a general discussion of the short-cut method
see Doyle and Snell
[6].
(In the references given here,
Rayleigh's method is applied to conductance problems;
the generalization from conductance problems to bass-note problems
is straight-forward.)
The cutting method is based on Rayleigh's cutting law, one form of which is the following:
Proposition.
If M is obtained by gluing together parts of the boundary
of another manifold 
,
then
Proof.
This follows from the definition of
that we have adopted.
qed