  The joint NASA-ESA mission LISA relies crucially on the stability of the
three spacecraft constellation. All three spacecraft are on heliocentric and
weakly eccentric orbits forming a stable triangle. It has been shown that for
certain spacecraft orbits, the arms keep constant distances to the first order
in the eccentricities. However, exact orbitography shows the so-called
`breathing modes' of the arms where the arms slowly change their lengths over
the time-scale of a year. In this paper we analyse the breathing modes (the
flexing of the arms) with the help of the geodesic deviation equations to
octupole order which are shown to be equivalent to higher order
Clohessy-Wiltshire equations. We show that the flexing of the arms of LISA as
given by the `exact' solution of Keplerian orbits, which gives constant
armlengths to the first order in eccentricity and whose maximum flexing
amplitude is $\sim 115,000$ km, can be improved, by tilting the plane of the
LISA triangle slightly from the proposed orientation of $60^\circ$ with the
ecliptic to obtain a maximum flexing amplitude of $\sim 48,000$ km, reducing it
by a factor of $\sim 2.4$. The reduction factor is even larger if we consider
the corresponding Doppler shifts, for which the reduction factor reaches almost
a factor of 6. We solve the second order equations and obtain the general
solution. We then use the general solution to establish the optimality of the
solutions that we have found.
