:

these lines and suppose the parallelogram whose breadth is dx to be divided into small parallelograms by lines parallel to the paper, the co-ordinates of two lines being y and y+dy, or the surface of the small parallelogram formed by the intersection of these with the others being dx. dy. Then the displacement which the little wave originating at this surface would cause at M will be represented by

Now PM2= (p- x)2 + (q − y)2 + (a + b − z)3,

which, since x2 + y2 + z2 = = a2,

is = (a+b)2 + a2 + p2 + q3 — 2px − 2qy − 2 (a + b) z.

But z=√(a2 — x3 — y3) = a

x2y nearly,

2a 2a

as x and y are supposed to be small even at their greatest

Call this Sx.dy.W: and let v be the sum of all the disturbances for the slice between the lines whose ordinates are X and xSx: w that for the whole surface. Since v increases by dx. W. dy upon increasing y by dy,

This integration is to be performed, and the limits of the values of y for the integral will be expressed in terms of x from a knowledge of the shape of the aperture. The effect of the narrow parallelogram being v or dxf, W, it is found in the same manner that the effect of the whole aperture is fa√, W. As sin (vt - B) and cos (vt-B) are not concerned in the

2π

λ

2π

λ

integration, the whole displacement may be expressed thus:

Let the integrals be E and F: then the whole displace

and hence (as in (14), (17), and (23)} the illumination is represented by E2 + F12.

proceeding therefore according to the precepts of (73), the first

thing to be done is to find

These integrals cannot generally be found in terms of y. Tables however have been formed expressing the numerical value of the integrals,

for different values of s, and there is no difficulty in applying them to the numerical expression of the first integral in our problem. But as the integral thus found is given in numbers, and not in an expression involving y and which can be

* Suppose U to be the function of unknown form which is the integral of S a given function of s. Then the values of U corresponding to sh and s+h

are

;

and therefore the value of the integral from s -h to s+h is

which can be easily calculated: and by taking h small enough, one or two terms will be sufficient. In this way the values of the integral can be computed for successive limits, and the sum will be the integral up to any given value of s. A table off, cos

this Tract.

and f, sin will be given at the end of

expressed in terms of x, it is seldom possible to perform the second integration.

75. In certain cases however this may be effected. For example, suppose the aperture to be a parallelogram whose sides are 2e and 2f in the direction of x and y, the line AO passing through its center. Let

and this is to be taken from y=-f to y=+f,

This will be the sum of the two numbers, in the table of

Let these be A,, A,: and after proceeding in a similar way for

πε

the sine, let the numbers in the table of s, sin correspond

2

ing to

s=

(2 (a + b))
λατ

f+

aq
a+b

Integrating with respect to x in exactly the same manner between x=-e and x=+e, and putting A,, A, for the

and s=

(2

ар

+

corresponding to

√ {2 (a + b)} (e-c76),

and B1, B, for those in the table of f, sin

the same,

λαό

F=

2 (a + b) { (4, +4) (4 ̧ + A,) − (B, + B,) (B ̧ +
+B)},

(B,+B) (4,+A)}.

2 (a + b) { (A,+4,) (B,+B) + (B, + B2) (42+

The intensity or E2 + F2

=

́ 4 (a + b) = { (4, + 4,)2 + (B, ̧ + B2)"}. {(4,+4)2+ (B,2+B)'}