where the values of C, S are given by (14); whence

The maxima and minima values of I2 are determined by the

from which it follows, that the maxima and minima are determined by the roots of the equations V1 =0, J1 = 0. The roots of the former equation have been calculated by Lommel for the values у=π, 2π,...8π; and the results for y=π, y = 4π are shown in the following table.

=

From this table we see that when yπ, there is one minimum within the geometrical shadow; and that the exterior of the

+ '0636

0041 max.

0

⚫0017 min.

- 0709

.0056 max.

shadow is surrounded by a series of rings. We also see that as z increases, the absolute value of the intensity rapidly increases, whilst the differences between its maxima and minima values diminish. The general appearance on the screen accordingly consists of a bright spot at the centre, surrounded by an obscure circular ring of varying intensity which exists in the neighbourhood of the geometrical shadow; and at more distant points the intensity is sensibly uniform.

When y is equal to 47, the intensity within the geometrical shadow, and also the differences between the maxima and minima are very small. There is accordingly a well-defined shadow surrounding the central bright spot. As the boundary of the geometrical shadow is approached the intensity increases, and continues to increase rapidly after the boundary has been passed, until at some distance it becomes sensibly uniform.

It is a matter of common observation, that when the shadow of a well-defined object (such as the edge of a razor), is thrown upon a screen, the edge of the shadow is not sharply delineated, but an indistinct appearance is observed in its neighbourhood. This is accounted for by the foregoing theory, which shows that the intensity does not change abruptly to zero, but gradually diminishes, passing through a series of maxima and minima values.

On the Bessel's Function Jn+.

76. Before proceeding to discuss Lommel's method of dealing with two-dimensional problems of diffraction, it will be convenient to make a short digression for the purpose of considering the Bessel's function Jn+

When n is an integer, it is known that Jn (x) satisfies the equations

d2Jn
+

1 dJn
+1
x dx

dx2

..(26),

.(27),

..(28),

and the Bessel's function of order n +, will be defined to be a function which satisfies these three equations, when n+1 is written for n, where n is any positive or negative integer.

Writing n for n in (26), it may be written

Equation (29), being of the second order, has two independent solutions. One solution may be expressed in the form of the series

+

2.2n+ 3 2.4. (2n+3) (2n+5)

from which we see that x ̄1 sin x.

=

.(31),

We shall therefore define the Bessel's function Jn+, where n is zero or any positive integer, by the equation

From the manner in which Jn+ has been deduced, we see that it is a solution of (26); we have now to show that it satisfies (27) and (28). This can be at once done by substituting from (32) in the equations

n + 1⁄2 Jn + s − J n
Jn+} - Jn+}

J'n+} =

(n + 1) Jn+b

.(33),

which are what (27) and (28) become, when n+ is written for n..

77. We must now consider the function J-n-}, where n is zero or any positive integer. A series for containing negative powers of n, can easily be shown to be

+

2.4.6 (2n-1) (2n − 3) (2n − 5) - ... ... (34),

BESSEL'S FUNCTIONS.

-1

95

from which we see that 1 = x1 cos x. This series obviously represents a function different from (31), and therefore the series (31) and (34), each multiplied by an arbitrary constant, represent the complete solution of (30). If therefore we define the function Jn by the equation

it follows that J-n- is a solution of (26); and by substituting from (35) in the equations

-n-is

which are what (27) and (28) become, when -n- is written for n, it is at once seen that they are satisfied.

78. It also follows from (32) and (35), that

If in (30) we write y=x, it can be shown that

from which we see that J-n- is zero when x=∞, and infinite when x = 0.

79. We are now in a position to explain Lommel's method'. The light is supposed to diverge from a linear source O, and to be received on a screen. Let B be the projection of O on the

1 Abh. der II. Cl. der Kön. Bayer. Akad. der Wiss. vol. xv. p. 531.

screen, then if in the figure to § 63, we put = 0, BP = x, AQ= p, we may prove in precisely the same manner, that

and the origin of p is the intersection of the line OB with the wave-front at the point A. The integration extends over the effective portion of the wave.

80. The two principal problems, which we shall have to consider, are diffraction through a slit, and diffraction by a long narrow rectangular obstacle.

When the slit or obstacle, whose breadth is supposed to be equal to 2c, is parallel to the screen, and is symmetrically placed, so that its middle line is the intersection of the plane passing through the source and B, the integration will be from c toc in the case of a slit, and from ∞ to c, and ∞ to c in the case of an obstacle.

81. When the integration is from c toc, the odd parts of the integrals disappear, and we thus obtain

The integrals (39) cannot be evaluated in finite terms unless c; in this case, it may be shown by writing a = a(1 + i)/2} in the integral

that

where

2y

= C

π

π

[sin 4 p2 cos lpdp = c(") sin (2-1))

2y