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the first and third become equal when w = }; that is, when A happens as often as B in n trials: a result which we might have looked for à priori. It also appears that when w is less than }, it is more likely that p should exceed a than fall short of it; which is in accordance with another result of the theory, namely, that the chance of drawing
Let w = 1 — k where k is small (not being less than }).
Let the limits be a = 1 – Ak and b = 1; where X is greater than unity, or u is negative, and v is positive and infinite. We have also
so that if A happen n (1 – K) times out of n, the presumption that its probability lies between 1 – Ak, and 1 is
Next, let w = } – k, where k is small (not being less than }), and
let a = 0, b = }; that is, required the presumption that the less frequent (slightly) of two events is the less probable. Then u is infinite and negative, and v is positive and derived from
AUGUSTUS DE MORGAN. UNive Rs1ty College, LoNDoN, December 30, 1837.
THE formula (1) at the beginning of this paper is unnecessarily complex, seeing that if 1 + 3 were written instead of l in the limits of the integral, the increment of the latter would differ from the additional term only by a quantity of such an order as was rejected in the approximation.
If then v and w be the components of n (or v +w), which are as the probabilities of A and B in a single trial, the probability that
A will happen a number of times between v + 1 and v – l in n trials is
which is of the same order of exactness as the formula given by
Laplace, and is somewhat more symmetrical and less difficult to calculate.
Perhaps it may not be here out of place to notice that the usual approximation to the product 1.2. 3....... a may be made very much
more exact without being rendered materially more difficult to calculate. As follows: instead of
iá; ass: " ") since the third term of the series is half the square of the second. The approximation is so close that even if we take a = 1, the error is very little more than the five hundredth part of the whole.
XXI. On the Diffraction of an Object-glass with a triangular Aperture. By S. EARNsh Aw, M. A. Of St John's College, Cambridge.
[Read December 12, 1836.]
The general adoption of Fresnel's theory seems to indicate that the scientific world is convinced that the Newtonian theory is inadequate to the explanation of the phaenomena of Diffraction; and that the theory which ascribes them to reflexion at the edges of the obstacle is equally unsatisfactory; and hence it is that the phaenomena of this class have been declared by Sir J. Herschel, to form the strongest points of the undulatory theory of light. Professor Airy also at the end of his paper on “The Diffraction of an Object-glass with a circular Aperture,” (Phil. Trans. Vol. v.) has thus stated his opinion of the importance of these phaenomena in the present state of science: “The investigation of cases of diffraction similar to that discussed here, appears to me a matter of great interest to those who are occupied with the examination of theories of light.” This sentiment was expressed in 1834, and since that time I am not aware that any thing has been done in the comparison of theory with experiment in this class of phaenomena. It is true, theory may have been applied to certain cases of diffraction, but it does not appear that the persons who have so applied it have ever contemplated more than merely to shew that theory gave a result something like the observed phaenomenon. Such an inference being wholly useless in the present state of the theory of light, there is still need for the minute discussion of particular cases of diffraction, and for the impartial comparison of the results of theory and experiment. I have selected for this purpose, the diffraction of light at the object-glass of a telescope with a triangular aperture, for two reasons,—because the phaenomenon is very singular and very beautiful; —and because Sir J. Herschel has declared that “to represent analytically the intensity of the light in one of the discontinuous rays, will call for the use of functions of a very singular nature and delicate management.” His description of the phaenomenon is as follows. (Encyclop. Metrop. Light. Art. 772.)
“When the dbject-glass of the telescope was limited by a diaphragm so that the aperture was in form of an equilateral triangle, the phaenomenon seen by viewing a star through the telescope was extremely beautiful: it consisted of a perfectly regular, brilliant, sixrayed star, surrounding a well-defined circular disc of great brightness. The rays do not unite to the disc, but are separated from it by a black ring. They are very narrow and perfectly straight; and appear particularly distinct in consequence of the ToTAL destruction of all the diffused light, which fills the field when no diaphragm is used : a remarkable effect, and much more so than the mere proportion of the light stopped.”
Let us suppose the aperture of the telescope an isosceles triangle, one of whose equal sides = a ; the perpendicular from the vertical angle upon the base = 3c, and the inclination of either side to the base = a. Let the image of the star be received upon a screen passing through the focus of the object-glass; and take the projection of the centre of gravity of the triangular aperture upon the screen for the origin of coordinates; the axes of a and y upon the screen being respectively perpendicular and parallel to the projection of the base of the triangular opening, and the axis of z coinciding with the axis of the telescope, and passing through the centre of gravity of the aperture. Suppose b = focal length of the object-glass, and let a, y, z be co-ordinates of any point P in the wave surface, which emerges from the object-glass, and tends to the origin of co-ordinates as its focus;
Let p, q be co-ordinates of any point M in the screen: then the disturbance caused at M by the element 3a. Öy of the wave surface may be represented by
But as o refers only to the intensity of the light, which emanates from P, when it reaches M, it will not vary sensibly with the variation of PM in the problem under consideration, since both the triangle and its image are small. No appreciable error, therefore, will be committed if for the purpose of simplifying our calculations we suppose PM constant in this term, and assume the disturbance at M due to the element 8a. 8 y to be
or. , sin . (et – PM).
Hence the disturbance at M due to the element 3a. 3/ is represented by
in 27 pa, a 43/ 3a. 3/ sin X (ot- B + b " %). The whole disturbance at M will be found by integrating this expression, first with regard to y, between the limits y = – (2C — a cota, and y = (2e – a cot a ; and then with regard to a between the limits a = – c and a = 20.