We see here that the qualitative difference, in the case of heat reflected from a roughened surface of gold, disappears wholly when the rays fall upon the plate at a very small angle.

Similar experiments were made with silver, mercury, copper and brass the results are collected in the following Table :

heat diffusely reflected, from the unreflected heat, disappears when the incidence is very oblique. It is thus placed beyond doubt, that the properties of the reflected rays are dependent on the inclination at which they meet the reflecting surface.

The principal results of this investigation may be stated to be as follows:

1. Metals, as gold, silver and platinum, when in thin layers, are to be regarded as diathermanous bodies, which permit a portion of the calorific rays to pass through them; which portion naturally becomes less as the thickness of the layer increases.

In thus transmitting the calorific rays, certain metals, as gold and silver, exercise an elective absorption, similar to that of coloured transparent bodies upon light. Others, on the contrary, like platinum, act in the same manner upon all rays, and are therefore to be regarded as analogous to colourless bodies in the case of light.

2. In the case of diffuse reflexion, also, certain metals, such as gold, silver, mercury, copper and brass, similar to coloured and opaque bodies as regards light, exercise an elective absorption upon the calorific rays, in consequence of which the properties of the latter are altered. Others, on the contrary, for example, platinum, iron, tin, zinc, lead, alloy of lead and tin, German silver, reflect all kinds of calorific rays in the same proportion, exactly as colourless opaque bodies do with regard to light.

The properties which distinguish calorific rays reflected from metals, from unreflected heat, are so far dependent on the source of heat, that differences, for example, which exhibit themselves in a striking manner when solar heat is made use of, are diminished in the case of a Locatelli lamp, and completely disappear when the source of heat is a metallic cylinder not heated to redness.

The surface has the power either of causing the differences to appear in their maximum degree, or to disappear totally, according as the surface produces a diffuse or a regular reflexion.

The same is true of the change of the angle of incidence. In the case of a rough metallic surface, as the angle of the rays with the surface diminishes, the reflexion passes gradually from the diffuse to the regular, and at the same time the differences between the reflected and unreflected heat also gradually become less, until finally both have exactly the same character.

XLIII. On the Calculation of the Numerical Values of a certain class of Multiple and Definite Integrals. By Sir WILLIAM ROWAN HAMILTON, LL.D., M.R.I.A., F.R.A.S. &c., Andrews' Professor of Astronomy in the University of Dublin, and Royal Astronomer of Ireland*.

[1.] THE

SECTION I.

HE results, in part numerical, of which a sketch is here to be given, may serve to illustrate some points in the theory of functions of large numbers, and in that of definite and multiple integrals. In stating them, it will be convenient to employ a notation which I have formerly published, and have often found to be useful; namely the following,

with which I am now disposed to combine this other symbol,

(1)

(1)'

I shall also retain, for the present, the known notation of Vandermonde for factorials, which has been described and used by Lacroix, and in which, for any positive whole value of n,

[x]"=[x]"[x-m]”−m= [x]n+m : [x−n]", &c.; (4)'

which are extended by definition to the case of null and negative indices, and give, in particular,

It is easy, if it be desired, to translate these into other known notations of factorials, but they may suffice on the present

occasion.

[2.] With the notations above described, it is evident that

and more generally, that

tm+n

It"=

[m+n]"

Hence results the series,

(1 + I¿ + I¿2 + . .) 1 = (1 — I1)−1l=e';

Iet-et-1.

(9)'

The imaginary equation,

breaks up into the two real expressions,

cos t = (1+1,2) ̄11,

sin t=I(1+1,2)-'1.

The series of Taylor may be concisely denoted by the formula,

I¿D2f(x+t)=I¿f'(x+t) = f(x+t)—fx.

And other elementary applications of the symbol It may easily be assigned, whereof some have been elsewhere indicated. [3.] The following investigations relate chiefly to the function,

or

where

Fn, rt=1" (1+41,2)~*~*1 ;

Fn, rt=1" (1+41,2)='ft,

ft=Fo, ot=(1+41,2)~*1.

Developing by (5) and (6), and observing that

22" [−1}]" [0] ̃"[0]~2=(−1)" ( [0] ̄")3, . (12)′

we find the well-known series,

the function ft being thus equal to a celebrated definite integral, which is important in the mathematical theory of heat, and has been treated by Fourier and by Poisson.

[4.] It was pointed out* by the great analyst last named, that if there were written the equation,

then for large, real, and positive values of k, the function y✔k might be developed in a series of the form,

where a certain differential equation of the second order, which yk was obliged to satisfy, was proved to be sufficient for the successive deduction of as many of the other constant coefficients, A', A",. . and B', B", . . of the series, as might be desired, through an assigned system of equations of condition, after the two first constants, A and B, were determined; and certain processes of definite integration gave for them the following values,

so that when k is very large, we have nearly, as Poisson showed,

[5.] In my own paper on Fluctuating Functions†, I suggested a different process for arriving at this important formula of approximation, (15)", which, with some slight variation, may

*In his Second Memoir on the Distribution of Heat in Solid Bodies, Journal de l'Ecole Polytechnique, tome xii. cahier 19, Paris, 1823, pages 349, &c.

In the Transactions of the Royal Irish Academy, vol. xix. part 2, p. 313; Dublin, 1843. Several copies of the paper alluded to were distributed at Manchester in 1842, during the Meeting of the British Association for that year: one was accepted by the great Jacobi.