that it is not necessary for the truth of this proportion that the rays should finally converge, and meet at one point, but they may also be divergent, in which case their directions meet in one point when produced backwards, and form what is called in optics a virtual image.

If we take the special case in which the medium at the point of starting and of re-union is the same (e.g. where the rays issue from an object which is in air, and, after certain refractions or reflections, form an image which really or virtually is also in air), then vv, and n = n; whence we have

=

:: ds. ds.

If we add the further condition, that the pencil of rays makes equal angles with the two elements of surface (e.g. that both are right angles), then we have

In this case the angles of the cones formed by the pencil of rays at the object and at the image stand simply in the inverse ratio of the areas of the corresponding elements of object and image.

In the valuable demonstration which Helmholtz has given in his "Physiological Optics "* of the laws of refraction in the case of spherical surfaces, he seeks to connect with these the case of the refractions which take place in the eye; and he finds in page 50, and extends further in page 54, an equation which expresses the relation between the size of the image and the convergence of the rays, for the case in which the change of direction of the rays takes place by refraction or by reflection at the surface of co-axial spheres, and in which the rays are approximately perpendicular to the planes which contain the object and the image. The author however believes that the relation has not before been given in its general form, as in proportions (29) and (30).

Karsten's Universal Encyclopedia of Physics.

VI. GENERAL DETERMINATION OF THE MUTUAL RADIATION BETWEEN TWO SURFACES, IN THE CASE WHERE ANY CONCENTRATION WHATEVER MAY TAKE PLACE.

§ 15. General View of the Concentration of Rays. We must now extend our investigation so as to embrace not only the extreme case, in which all the rays which issue from a point on plane a within a certain finite cone unite again in one point forming a conjugate focus on plane c, but also every conceivable case of the concentration of rays.

Pa

To obtain a closer view of the phenomena of concentration, we may use the following definition. If rays issue from any point p. and fall on plane c, and if these rays when close to that plane have such directions that on one part of the plane the density of the impinging rays is indefinitely great compared to the mean density, then at this part there is concentration of the rays which issue from P. With this definition we may easily treat mathematically the case of concentration. Between point pa and plane c take any intermediate plane b, which is so placed that there is no concentration in it of the rays issuing from p.; and also that its relation to plane c is such, so far as we are concerned, that the pencils of rays issuing from points on one of those planes suffer no concentration on the other. Now consider an indefinitely small pencil, which starts from p, and cuts the planes b and c. Let us compare the areas of the elements ds, and ds, in which these planes are cut. If ds, vanishes in comparison to ds, so that we may put

ds.
= 0......

ds

.....(31),

this is a sign that there is a concentration of rays, in the sense defined above, at plane c.

Let us now return to equations (II.) of § 7. The equations in the first horizontal row are those that refer to the present case and of the three fractions, which there represent the ratio of the elements of surface, the first applies to our case, because under the assumption made as to the position of the intermediate plane we may determine A and E in the ordi

nary manner. We have thus the equation

ds.
ds,

с
=

E
A

This fraction can only equal zero if the numerator E is zero, since under the assumption made as to the position of the intermediate planes the denominator A cannot be indefinitely large. We have then as a mathematical criterion, whether the rays issuing from p, suffer concentration at plane c or not, the condition

E=0........

which must be fulfilled where there is concentration.

.(32),

Now assume conversely that on plane c a point p, is given, and that we have to decide whether the rays issuing from this point suffer concentration at any part of plane a. Then

we have in the same way the condition

dsa

ds

=

0; and since by

we arrive at the same final condition

E=0.

It is in fact easy to see that when the rays issuing from a point on plane a suffer concentration at plane c, then conversely the rays issuing from the latter point must suffer concentration on plane a.

Since equations (13) express the relations which hold between the six quantities A, B, C, D, E, F, we may apply. those equations to ascertain what becomes of B, D and F, in the case where E = 0, whilst A and C have finite values. By those equations

Hence it follows that all three quantities must in the present case be indefinitely great.

§ 16. Mutual Radiation of an Element of Surface and of a Finite Surface, through an Element of an Intermediate Plane.

We will now attempt so to determine the ratio between the quantities of heat which two surfaces radiate to each

other, that the result must hold in all cases, independently of the question whether there is any concentration of heat

or no.

a

For greater generality we will substitute for the planes a and c, as hitherto considered, two surfaces of any kind, which we may call Sa and Sc. Between them let us take any third surfaces, which need only fulfil the condition that the rays which pass from Sa to se, or vice versâ, suffer no concentration in St. Now choose in s any element ds, and in s, an element ds, so situated that the rays which pass through it from ds will when produced strike the surface s.. Then we will first determine how much heat ds, sends through ds, to the surfaces, and how much heat it receives back from 8, through the same element of the intermediate plane. To ascertain the first mentioned quantity of heat, we have only to determine how much heat ds, sends to ds,, since, by our assumption as to the position of ds,, all this heat after passing ds must strike the surfaces. This quantity of heat may be expressed at once by means of our previous formulae. Suppose a tangent plane to be drawn to s at a point of the element ds, and similarly a tangent plane s, at a point of ds; and consider the given elements of surface as elements of these planes. If in these tangent planes we take two systems of co-ordinates xa, ya, and x, y, and if we form the quantity C by means of the third of equations (I.), then the required quantity of heat, which ds, sends through ds, to the surface s., is given by the expressions

a

Next with regard to the quantity of heat which ds receives through ds, from the surface s, the relations of the points in 8, from which these rays issue, are not in general so simple as that which holds in the special case, where ds. has an optical image ds, lying on s, and therefore is also itself the optical image of ds. If we choose a known point P. on the intermediate element ds, and consider the rays which pass through this point from all points of ds, we have an indefinitely small pencil of rays, cutting s, in a certain element of surface. It is this element which sends rays to ds, through the selected point p. But if we now choose

another point of ds, as the vertex of the pencil, we arrive at a somewhat different element on the surface s Thus the rays, which ds, receives from s, through different points on ds, do not all issue from one and the same element

of se

Since however the area of ds, may be any whatever, nothing hinders us from supposing it so small, that it is an indefinitely small quantity of a higher order than ds. In this case, if the vertex of the pencil changes its position within ds,, then the element of s, which corresponds to ds. will change its position through a distance so small that in comparison with the dimensions of the element it is indefinitely small and may be neglected. Hence in this case the element ds, which we obtain when we choose any point whatever p, on element ds,, and make it the vertex of the pencil of rays issuing from ds, may be considered as the part of ds, which exchanges rays with ds, through ds. The area of this element ds. is easily found from what precedes. Let us suppose as before that a tangent plane to the surface s is drawn at p, and that tangent planes to the surfaces s, and s are drawn at points on the elements ds, and ds, respectively; and let us consider the two latter elements as elements of the tangent planes. Take systems of co-ordinates on these three tangent planes, and form the quantities A and C, as given by the first and third of equations (I.). Then by equation (II.) we have

The quantity of heat which ds, sends to ds,, and which, as mentioned above, may be considered as the quantity which ds, receives from the surface s through ds,, is expressed by

a

A
ev2 = ds. ds;

or, substituting for ds, its value given above,