if the exterior angle at B is/n, one nth of a right angle;

and thence, by the previous integration in II. (27),

and so also along J A, j < u<a, q=Q; and measuring the arc s from A',

(39)

and if c is the breadth of the jet at J,

(40)

Qc = m, u = a'e",

a

u

a' . u

a' b'

И

will give the intrinsic equation of A'J or JA.

a"

a

a

b

a

a'

and if q denotes the velocity at I, where u∞, and d the distance between the two planes IA, I'A',

The figure can be duplicated about IB, and so give the motion when the channel is blocked by a wedge-shaped pier (Fig. 32).

=

When A' is carried along on I'A' up to J at an infinite distance, making a' j = 0, the wall I'A' is of infinite length, and the motion may be duplicated again about I'J, and so represent the flow of water through the piers of a bridge, wedgeshaped as at Westminster.

Duplicating the original figure once about I'J' will give the efflux from a channel with a conveying mouthpiece, as in Fig. 33.

A

B

---/

FIG. 33.

Make b = ∞, and the flow is obtained through the gap between two walls converging at an angle π/n; and now along the skin of the jet AJ, of breadth Qe at J,

the intrinsic equation of the jet AP.

In Helmholtz's original problem, where the liquid is drawn off between two parallel walls like a Borda mouthpiece (Fig. 30),

and if d is the outside distance between the walls,

so that the coefficient of contraction is 1.

In Helmholtz's next problem, where a slit of breadth d is made in a wall, through which the liquid escapes (Fig. 34),

As a general exercise on conformal representation, consider the application to the z diagram in Fig. 35, representing uniplane injector flow, with the associated w and 2 diagrams, where the z figure may be supposed duplicated in the median line IJ, and here

α

i (∞) >b> a > c> a' > b' > j > k >i(−∞). Next examine the change in the z diagram due to a rearrangement of the sequence of the constants

a, a', b, b', k.

DIGRESSION ON THE INTEGRAL CALCULUS.

Leaving the elliptic integral interest as leading too far, and passing over the analytical expression of w, as given merely

by the algebraical and logarithmic function, since

dw

du

can be resolved into a quotient and partial functions integrable immediately, we concentrate our attention on the determination of from the typical form

Returning through Paris in May, 1910, I took the opportunity of attending a lecture at the Sorbonne by M. Marchis, the new professor there of Aeronautics; and I found he devoted a whole lecture to the consideration of a single integral, which with Greek letters we can write

But as M. Marchis was allowed, not six, but some twenty or thirty lectures, he could go thoroughly into this detail, essential in the theoretical study of the aeroplane.

Marchis's integral is identified with the integration required in (27) II. for 2 by putting