Analytical expressions are obtained for the optimum
shapes which minimise the drag of a slender axisymmetric
body in free-molecular flow, provided the drag expression is
simplified using the slenderness assumption. The problem is
formulated as one of Mayer type in the calculus of variations
and solved by using the Buler-Lagrange equations together with
the transversality condition. The shapes derived are optimum
subject to constraints on thickness, length, wetted area and
volume. In the particular cases solved any two of these four
quantities are fixed while the remaining two are free. The
expression for the shape of the body when thickness is free
is obtained in closed form.