Lastly, suppose a point p, to be given on plane b (Fig. 29) Let and choose any element of surface ds, on plane a. us suppose that rays from different points of this element pass through p,, and that they are produced to the plane c. Then the magnitude of the element of surface dse, in which all these rays meet plane c, is found, using the same symbols as before, to be as follows: From this we see that the two corresponding elements in this case bear exactly the same relation to each other, as the two elements which are obtained when we have an element ds, given in plane b, and then, having assumed as the origin of the rays a point first in plane a, and secondly in plane c, determine in each case the element of surface in the third plane corresponding to the element ds. § 7. Fractions formed out of six quantities to express the Relations between Corresponding Elements. In the last section we have only employed the first of the three pairs of equations in § 5. We can however employ the two other pairs (2) and (3) in the same manner. Each pair leads us to three quantities of the same kind as those already denoted by A, C, and E. These quantities serve to express the relations between the elements of surface. Of the nine quantities thus obtained, however, there are four which are equal to each other, whereby the actual number is reduced to six. The expressions for these six are here placed together for the sake of convenience, although three of them have been already given. X ас dy.dy. dx dy dy dx Tab dTab dTab a ac (d2 (Tac-Tas) x d2 (Tac-Tas) _ [d2 (Tae— Tab) (dxa)* (dy) dxadya By help of these six quantities every relation between two elements of surface can be expressed by three different fractions, as may be shewn in tabular form as follows: It is easily seen that the three horizontal rows relate to the three cases, in which the given point through which the rays must pass is taken either in plane a, plane c, or plane b. Of the three vertical rows of fractions, which express the relations between the elements of surface, the first is deduced from equation (1) of § 5, the second from (2) and the third from (3). Since the three fractions, which express a given relation between two elements of surface, must be equal to each other, we have the following equations between the six quantities: Our further investigations will be performed by means of these six quantities; and since every relation between two elements of surface is expressed by three different fractions, we can always choose amongst these the fraction most suitable for each special case. III. DETERMINATION OF THE MUTUAL RADIATION, WHEN THERE IS NO CONCENTRATION OF RAYS. § 8. Magnitude of the Element of Surface corresponding to ds on a plane in a particular position. We will first consider the case to which Kirchhoff's expression refers, and seek to determine how much heat two elements send out to each other, on the assumption that every point of one element receives from every point of the other one ray and only one; or at most a limited number of particular rays, which may be considered separately. dsc Given two elements ds, and ds, in planes a and c (Fig. 30), we will first determine the heat which ds, sends to ds. For this purpose let us suppose the intermediate plane b to lie parallel to plane a at a distance p, which is so small, that the part which lies between these two planes of any ray passing from dsa to ds may be considered as a straight line, and the medium through which it passes as homogeneous. Let us now take any point in element ds, and consider the pencil of rays which passes from this point to the element ds. This pencil will cut plane b in an element ds, whose magnitude is given by one of the three fractions in the uppermost horizontal row of equations (II.). Choosing the last of these we have the equation ds ̧= 1⁄2ds. dsa dsb Fig. 30. The quantity C may in this case be brought into a specially simple form, on account of the special position of plane b. For this purpose let us follow Kirchhoff in choosing the system of co-ordinates in b so as to correspond exactly with that in the parallel plane a; i.e. let the origins of both lie in a common perpendicular to the two planes and let each axis of one system be parallel to the corresponding axis of the other. Let r be the distance between two points lying on the two planes, and having co-ordinates xa, ya and x, y, respectively. Then r = √ p2 + (ï¿− xa)2 +(Yo−Ya)2. ..(15). Let us now suppose a single ray to pass from one of these points to the other; then, since its motion between the two planes is supposed to be rectilinear, the length of its path will be simply represented by r; and if we denote by v its velocity in the neighbourhood of plane a, which by the assumption will remain nearly constant between a and b, then the time which the ray expends in the passage will be given by the equation C = v.n. 2 d2r d2r dr X dy.dx.). v ̧2\dxdx ̧ˆdy.dy, dx.dy, Substituting for r its value as given by (15), we obtain If we denote by the angle which the indefinitely small pencil of rays, which starts from a point on ds, makes with the normal at that point, then cos 0=2; and the last equation also takes the form § 9. Expressions for the quantities of Heat which ds and ds, radiate to each other. When the magnitude of the element of surface ds, is determined, the quantity of heat which ds. sends to ds, can be easily expressed. From every point of ds, an indefinitely small pencil of rays goes to ds; and the solid angle of the cone made by the pencil from each of these points may be taken as the same. The magnitude of this angle is determined by the magnitude and position of that element ds, in which the cone cuts plane b. To express this angle geometrically, let us suppose that a sphere of radius p is drawn round the point from which the rays start, and that within this sphere we may consider the path of the rays as being rectilinear. If do is |