sidered as variable when differentiating according to 1, whilst the second co-ordinate y, of the same point, and the co-ordinates x, y, of the other, are assumed to be constant,and where similarly in differentiating according to x, this is taken as the only variable-can thus represent no real quantity of finite value. Therefore in this case we must find an expression of somewhat different form from Kirchhoff's; and for this purpose we must first consider some questions similar to those considered by Kirchhoff in arriving at his expres sion. II. DETERMINATION OF CORRESPONDING POINTS AND CORRESPONDING ELEMENTS OF SURFACE IN THREE PLANES CUT BY THE RAYS. § 5. Equations between the co-ordinates of the points in which a ray cuts three given planes. Let there be three given planes a, b, c (Fig. 25), and in each of them let there be a system of rectangular co-ordinates, which we may call respectively xaya, Xoyb and xy. Let us take a point Pa in plane a, and a point p, in plane b, and consider a ray as passing from one of these to the other; then to determine its path we have the condition that the time, which the ray expends in traversing it, must be less than it would expend in traversing any other neighbouring path. Call this minimum time Ts. It is a function of the co-ordinates of pa and p, i.e. of the four quantities xaya, xy. Similarly let T be the time of the ray's passage between two points Pa and p., in planes a and c; and let T be the time of its passage between two points p, and p., in planes b and c. T. is a function of the co-ordinates xaya, xeye; and T. is a function of the co-ordinates xy, ye ac ab' Pa Fig. 25. ac Now as a ray, which passes between two of these planes, will in general cut the third plane also, we have for each ray three points of section, which are so related to each other that any one of them is in general determined by the other two. The equations which serve for this determination may be easily deduced from the above-mentioned condition. Let us first suppose that points p, and p, are given beforehand, and that the point is still unknown in which the ray cuts the intermediate plane b. This point, to distinguish it from other points in the plane, we will call p'. Take any point whatever p, in this plane, and consider two rays, which we may call auxiliary rays, passing the one from pa to po, and the other from p, to p.. In Fig. 25 these rays are shewn by dotted lines, and the primary ray, which goes direct from p. to p., by a full line*. If, as before, we call the times expended by these two rays T and T., the value of the sum of these times Tab+ The will depend on the position of and Po therefore, since the points p, and p, are assumed to be given, it may be considered as a function of the co-ordinates xy of point p.. Among all the values, which this sum may assume if we give to point p, various positions in the neighbourhood of the minimum value must be that which is obtained by making p, coincide with p',, in which case the auxiliary rays merely become parts of the direct ray. We therefore have the following two equations to determine the co-ordinates of this point p': bc As Tab and T., in addition to the co-ordinates of the unknown point p, contain also the co-ordinates of the known points p. and P., we may consider the two equations thus Pa established as being simply two equations between the six co-ordinates of the three points in question. These equations, therefore, can be applied, not merely to determine the coordinates of the point in the intermediate plane from those * These lines are shewn curved in the figure, to indicate that the path taken by a ray between two given points need not be simply the straight line drawn between the two points, but a different line determined by the refractions or reflections which the ray may undergo: it may thus be either a broken line, made up of several straight lines, or (if the medium through which it passes changes its character continuously instead of suddenly) a continuous curve. of the two other points, but generally to determine any two of the six co-ordinates from the other four. C Next let us assume that the points p, and p. (Fig. 26), in which the ray cuts planes a and b, are given, and that the point is yet unknown in which it cuts plane c. This point let us call p'., to distinguish it from other points in the same plane. Take any point p, in plane c, and consider two auxiliary rays, one of which goes from pa to p., and the other from Pa Po to p.. In Fig. 26 these are again shewn dotted, while the primary ray is shewn full. Let T. and T be the times of passage of these auxiliary rays. Then the value of the difference Tae- The depends on the position of p. in plane c. Among the various ac ac P& Fig. 26. values obtained by giving to p, various positions in the neighbourhood of p', the maximum must be that obtained by making p. coincide with p'. For in that case the ray passing from p. to p, cuts the plane b in the given point p,, and is therefore made up of the ray which passes from pa to p, and of that which passes from p, to p.. Accordingly we may put Pa Pa Hence the required difference is given in this special case by Tac-Tre = Tab If on the other hand p. does not coincide with p', then the ray which passes from Pa to Pc does not coincide with the two which pass from p. to p, and from p, to pe; and since the direct ray between Pa and Po travels in the shortest time, we must have Pa Ро Tae < Tab + Toe and therefore we have in general for the required difference the inequality Tac-Toe<Tab ас This difference is thus generally smaller than in the special case where p. lies in the continuation of the ray which passes from pa to p, and this special value of the difference thus forms a maximum*. Hence we have the two following conditions: If we lastly assume that the points p, and P. in the planes b and c are given beforehand, while the point in which the ray cuts the plane a is still unknown, we obtain by an exactly similar procedure the two following conditions: In this way we arrive at three pairs of equations, each of which serves to express the corresponding relation between the three points in which a ray cuts the three planes a, b, c; so that if two of the points are given the third can be found, or, more generally, if of the six co-ordinates of the three points four are given the other two may be determined. § 6. Relation between Corresponding Elements of Surface. We will now take the following case. Given on one of the planes, say a, a point pa, and on another, say b, an element of surface ds; then if rays pass from pa to the different points of ds,, and if we suppose these rays produced till they cut the third plane c, they will all cut that plane in general within another indefinitely small element of surface, which we will call ds. (Fig. 27). Let us now determine the relation between ds, and dsc. * In Kirchhoff's paper (p. 285) it is stated of the quantity there considered, which is essentially the same as the difference here treated of, except that it refers to four planes instead of three, that it must be a minimum. This may possibly be a printer's error, and in any case an interchange of maximum and minimum in this place would have no further influence, because the principle used in the calculations which follow, viz., that the differential coefficient=0, holds equally for a maximum or a minimum. In this case, of the six co-ordinates which relate to each ray (viz. those of the three points . in which the ray cuts the three As the form of the element ds, may be any whatever, let it be a rectangle dx, dy, and let us find the point in plane c corresponding to every point in the outline of this rectangle. We shall then have on plane c an indefinitely small parallelogram which forms the corresponding element of surface. The magnitude of this parallelogram is determined as follows. Let A be the length of the side which corresponds to the side da, of the rectangle in plane b, and let (x) and (y) be the angles which this side makes with the axes of co-ordinates. Then Again, let μ be the other side of the parallelogram, and let (x) and (uy) be the angles it makes with the axes. Then we have = dye dyo dy dy Let (u) be the angle between the sides λ and μ. Then we have cos (u) = cos(x) cos (ux) + cos (y) cos (uy) |