Q

Beyond a', from a' to j, log

= 0, and 0 diminishes from

to a, so that j' rejoins j, and the circuit is complete.

Then in accordance with the fundamental theorem (1) of conformal representation,

ΦΩ

du

(23)

is composed of the factors (u — b) ̄1', (u —a) ̄*, (u — a') ̄* ;

In the neighbourhood of ub, when the chief variation is due. to u-b, we may replace u-a, u-a' by b-a, b-a', and put

and since increases by as u diminishes through b, equation (14) shows that

by theorems, of the Integral Calculus, which ought to be familiar to the student of this subject.

Over the plane AA', a >u>a', 0=0, dz = dx,

2

and integrating with respect to u, from a' to a,

= cos2 a (a — a')2 + sin2 1⁄2 a (a — a')2 + sin a ‡ π (a — a')2,

since the last integral represents graphically sin a times the area of a circle on the diameter a-a'; and

To express any length such as A'P = x, as a function of u, it is convenient to introduce a variable angle 4 (not to be confused with the velocity function ), as shown in the diagram (Fig. 11);

BA'

and - is obtained as in (41), Lecture I., by putting = a. AA'

As the velocity diminishes from Q on the skin of the jet at atmospheric pressure to q something less in the interior, the

increases by the same amount; so that the gauge pressure in the interior, the excess over atmospheric pressure, becomes

b) du,

dx is given by

Q2

2

(u

2 C (1-8) dx = C (1) 2 (wb) du

Cg

2 Cg

and integrating for the thrust T (PA') over PA',

π

(1 + ‡ 7 sin a) = √" (1 − ‡ cos a sin2 ☀ + § ☀ sin a

COS a + sin a

LA'

= = (π
= +

(52)

AA'

FL

(53)

=

ΑΑ'

1-24

1 cos a

6