Kirchhoff's result states that the motion of the medium, treated as homogeneous, is given by

(25)

N

ch Ω

=

cos a + √(W B w)

where a denotes the angle the stream at IJJ' makes with the plane AA', and N is a constant.

In the notation of this subject

(26)

z = x + yi, w =

φ + ψί,

denoting the velocity function at the points (x, y) such that аф the velocity q is given by the downward gradient- of the ds function, and is the conjugate stream function, constant along a stream line.

Also, if the velocity q at (x, y) makes an angle ◊ with Ox,

The function and 2 is now introduced, defined by

(33)

so that

(34)

♡>ф> ФВ, ФЕ

> ÞÈ› Þ1⁄2 − Þ is negative, √(4,4) imaginary,

B

But beyond B

(35)

ФВ>ф>- со, √(ÞB − ) and (ww) is real

(36)
or else

sin = 0, 0 = 0 along BA, = along BA';

as required along the stream line AJ and A'J'; and then

if the arc AP = s is measured from A, and

the intrinsic equation of the stream A'J' with 0<<a;

with a similar expression for A'J'.

For normal incidence (Fig. 5) a =90°, cos a = 0, and the curve AJ is the evolute of a catenary, given by the intrinsic equation

The theory requires the existence of the counter current BA' (nappe dorsale in French), passing over the attacking front edge A' of the plane AA'; this nappe dorsale is usually omitted in the diagram of popular explanation as insensible, but the existence is revealed in a photograph of a current, either of air or water.

and at a small angle a, in radians, we may replace

so that the counter current would still be hardly perceptible. But with normal incidence (Fig. 5),

Anticipating other results of Kirchhoff's theory, to be proved

of the ancient theory; also that the centre of pressure L is in front of F the centre of AA', between F and A', where

The look of these formulas (47) and (48) suggests a geometrical representation, as in Fig. 6, associated with the ellipse, whose focal polar equation is

Q2

Cg

Take FX as the unit to represent geometrically, the average

pressure in lb/ft2 on Newton's theory for normal incidence,

Draw the ellipse APL, with focus F, vertex A, semi-latus rectum FL, and eccentricity Draw the semicircle FQX on

if QP' is drawn parallel to FL; so that the curves L.PA L.P'X represent T and T" graphically.

giving L the centre of pressure when FL, is of the plane AA'; and L lies inside the middle

X

the breadth of AA'.

This can be verified experimentally with a plane plate AA' in a current of water; pivoted like a balanced rudder about an axis through L, and then measuring a.

If L coincides with F, the plane sets itself at right angles to the current with a = ; as L moves away from F, a diminishes to zero when FL = AA'; as L is placed still further away from F, the plane AA' still remains in the line of the stream.