This paper deals with the preliminaries essential for any theoretical investigation of three-dimensional sails—namely, with the two-dimensional flow of inviscid incompressible fluid past an infinitely-long flexible inelastic membrane. If the distance between the luff and the leach of the two-dimensional sail is
c
, and if the length of the material of the sail between luff and leach is (
c
+
l
), then the problem is to determine the flow when the angle of incidence
α
between the chord of the sail and the wind, and the ratio
l
/
c
are both prescribed; especially, we need to know the shape of, the loading on, and the tension in, the sail. The aerodynamic theory follows the lines of the conventional linearized theory of rigid aerofoils; but in the case of a sail, there is an additional equation to be satisfied which expresses the static equilibrium of each element of the sail. The resulting fundamental integral equation—the sail equation—is consequently quite different from those of aerofoil theory, and it is not susceptible to established methods of solution. The most striking result is the theoretical possibility of more than one shape of sail for given values of
α
and
l
/
c
; but there appears to be no difficulty in choosing the shape which occurs in reality. The simplest result for these realistic shapes is that the lift coefficient of a sail exceeds that of a rigid flat plate (for which
l
/
c
= 0) by an amount approximately equal to 0.636 (
l
/
c
)
½
. It seems very doubtful whether analytical solutions of the sail equation will be found, but a method is developed in this paper which comes to the next best thing; namely, an explicit expression, as a matrix quotient, which gives numerical values to a high degree of accuracy at so many chord-wise points. The method should have wide application to other types of linear equations.
