to a Schwerd's lampblack grating. With the latter consistent results were obtained. But the crowding of the planes of polarization was towards the plane of diffraction; and when instead of measuring the azimuths of the planes of polarization of the incident and diffracted light, the incident light was polarized at an azimuth of 45° to the lines of the grating, and the diffracted light was divided by a double-image prism into two beams polarized in and perpendicularly to the plane of diffraction, it was the latter that was the brighter. From these experiments the conclusion seemed to follow that in polarized light the vibrations are in the plane of polarization. The amount of rotation did not very well agree with theory. The subject was afterwards more elaborately investigated by M. Lorenz*. He commences by an analytical investigation which he substitutes for that which I had given, which latter he regards as incomplete, apparently from not having seized the spirit of my method. He then gives the results of his experiments, which were made with gratings of various kinds, especially smoke gratings. His results with these do not confirm those of Holtzmann, and he points out an easily overlooked source of error, which he himself had not for some time perceived, which he thinks may probably have affected Holtzmann's observations. Lorenz's results like mine were decisively in favour of the supposition that in polarized light the vibrations are perpendicular to the plane of polarization. He found as I had done that the results of observation as to the azimuth of the plane of polarization of the diffracted light agreed very approximately with the theoretical result, provided we imagine the diffraction to take place before the light reaches the ruled lines.]

* Poggendorff's Annalen, Vol. 111 (1860) p. 315, or Philosophical Magazine, Vol. 21, p. 321.

[From the Transactions of the Cambridge Philosophical Society. Vol. IX. Part 1.]

ON THE NUMERICAL CALCULATION OF A CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES.

[Read March 11, 1850.]

IN a paper "On the Intensity of Light in the neighbourhood of a Caustic," Mr Airy the Astronomer Royal has shewn that the undulatory theory leads to an expression for the illumination involving the square of the definite integral

П

COS (w3 — mw) dw,

2

where m is proportional to the perpendicular distance of the point considered from the caustic, and is reckoned positive towards the illuminated side. Mr Airy has also given a table of the numerical values of the above integral extending from m = -4 to m = +4, at intervals of 0.2, which was calculated by the method of quadratures. In a Supplement to the same paper+ the table has been re-calculated by means of a series according to ascending powers of m, and extended to m=56. The series is convergent for all values of m, however great, but when m is at all large the calculation becomes exceedingly laborious. Thus, for the latter part of the table Mr Airy was obliged to employ 10-figure logarithms, and even these were not sufficient for carrying the table further. Yet this table gives only the first two roots of the equation W= 0, W denoting the definite integral, which answer to the theoretical places of the first two dark bands in a system of spurious rainbows, whereas Professor Miller was able to observe 30 of these bands. To attempt the computation of 30 roots of the equation W= 0 by + Vol. vii. p. 595.

*Camb. Phil. Trans. Vol, vi. p. 379.

means of the ascending series would be quite out of the question, on account of the enormous length to which the numerical calculation would run.

After many trials I at last succeeded in putting Mr Airy's integral under a form from which its numerical value can be calculated with extreme facility when m is large, whether positive or negative, or even moderately large. Moreover the form of the expression points out, without any numerical calculation, the law of the progress of the function when m is large. It is very easy to deduce from this expression a formula which gives the ith root of the equation W=0 with hardly any numerical calculation, except what arises from merely passing from (m/3), the quantity given immediately, to m itself.

The ascending series in which W may be developed belongs to a class of series which are of constant occurrence in physical questions. These series, like the expansions of e, sin x, cos x, are convergent for all values of the variable x, however great, and are easily calculated numerically when x is small, but are extremely inconvenient for calculation when x is large, give no indication of the law of progress of the function, and do not even make known what the function becomes when x = ∞. These series present themselves, sometimes as developments of definite integrals to which we are led in the first instance in the solution of physical problems, sometimes as the integrals of linear differential equations which do not admit of integration in finite terms. Now the method which I have employed in the case of the integral W appears to be of very general application to series of this class. I shall attempt here to give some sort of idea of it, but it does not well admit of being described in general terms, and it will be best understood from examples.

Suppose then that we have got a series of this class, and let the series be denoted by y or f (x), the variable according to ascending powers of which it proceeds being denoted by x. It will generally be easy to eliminate the transcendental function f (x) between the equation y = f (x) and its derivatives, and so form a linear differential equation in y, the coefficients in which involve powers of x. This step is of course unnecessary if the differential equation is what presented itself in the first instance, the series

being only an integral of it. Now by taking the terms of this differential equation in pairs, much as in Lagrange's method of expanding implicit functions which is given by Lacroix*, we shall easily find what terms are of most importance when x is large: but this step will be best understood from examples. In this way we shall be led to assume for the integral a circular or exponential function multiplied by a series according to descending powers of x, in which the coefficients and indices are both arbitrary. The differential equation will determine the indices, and likewise the coefficients in terms of the first, which remains arbitrary. We shall thus have the complete integral of the differential equation, expressed in a form which admits of ready computation when x is large, but containing a certain number of arbitrary constants, according to the order of the equation, which have yet to be determined.

For this purpose it appears to be generally requisite to put the infinite series under the form of a definite integral, if the series be not itself the developement of such an integral which presented itself in the first instance. We must now endeavour to determine by means of this integral the leading term in f(x) for indefinitely large values of x, a process which will be rendered more easy by our previous knowledge of the form of the term in question, which is given by the integral of the differential equation. The arbitrary constants will then be determined by comparing the integral just mentioned with the leading term in f(x).

There are two steps of the process in which the mode of proceeding must depend on the particular example to which the method is applied. These are, first, the expression of the ascending series by means of a definite integral, and secondly, the determination thereby of the leading term in f(x) for indefinitely large values of x. Should either of these steps be found impracticable, the method does not on that account fall to the ground. The arbitrary constants may still be determined, though with more trouble and far less elegance, by calculating the numerical value of f (x) for one or more values of x, according to the number of arbitrary constants to be determined, from the ascending and descending series separately, and equating the results.

In this paper I have given three examples of the method just described. The first relates to the integral W, the second to an infinite series which occurs in a great many physical investigations, the third to the integral which occurs in the case of diffraction with a circular aperture in front of a lens. The first example is a good deal the most difficult. Should the reader wish to see an application of the method without involving himself in the difficulties of the first example, he is requested to turn to the second and third examples.

for different values of m, especially for large values, whether positive or negative, and in particular to calculate the roots of the equation W= 0.

2. Consider the integral

where

is supposed to lie between - π/6 and +π/6, in order that the integral may be convergent.

Putting

1

x= (cos - √1 sin 0) z,

we get dx = (cos 0 −√— 1 sin 0) dz, and the limits of z are 0 and ∞o; whence, writing for shortness

* The legitimacy of this transformation rests on the theorem that if ƒ (x) be a continuous function of x, which does not become infinite for any real or imaginary, but finite, value of x, we shall obtain the same result for the integral of f(x) dx between two given real or imaginary limits through whatever series of real or imaginary values we make x pass from the inferior to the superior limit. It is unnecessary here to enunciate the theorem which applies to the case in which f(x) becomes infinite for one or more real or imaginary values of x. In the present case