If the intensity of light emanating from a luminous point, that is, the illumination of a unit of spherical surface having a unit radius, be represented by i, and a small plane, of which the area is a, be exposed to the same light at a distance r from the origin, and situated perpen dicular to the luminous ray, the quantity of light which it receives will be represented by ; but if the plane, instead of being perpendicular, be inclined to the direction of the ray at an angle a, the total illumi nation of the plane will then only be —I sin a, for a sin a is the area of the plane projected in a direction perpendicular to the ray, and this projection at the mime distance *mild evidently receive the whole of the light which fell on the inclined plane: we shall give a few examples of these formula.
lights, and 0, 8' the angles which r, r' make with the axis or lino joining the lights internally; then a representing an arc of the curve, the sines of the angles at which r, r' are inclined to an clement of the curve are r d °' ; and representing the Intensities as before, the its d a condition of equal illumination gives the d 0 d r'd r' r° • 77 : whence d i _ - i 37 . sin by trigonometry.
Integrating we find i cos O+i cos tr. const.. which (together with the common trigonometrical equations) gives the polar equation of the curve sought. We should obtain a negative sign, instead of a positive, if we supposed the curve equally illuminated on opposite sides.
Having now considered the laws of the emanation of light from points, we are next to consider its emanation from luminous surfaces, particularly when the direction of the light is oblique to that of the surface. To this end suppose A D, n c to be two planes of equal luminosity relative to a unit of either, and regarding only that portion of the light which emanates in the directions A D, B D, 0 D, perpendicular to A n produce A n to meet e 13 in the point a, and suppose the extent of u c to be taken such that B a= B A, then no will seem to the eye (receiving the rays in the directions A 0, n D, C D) to be of the same extent as its projection D a, or as that of D A ; but as its luminous surface is greater, it would appear brighter than B A in the ratio of n c to n A or no, if the intensity of the oblique emanation from e n were equal to that of the direct emanation from n A. Now we know by experience that it his only the same brightness as its projection, for if we take a bar of heated iron into a dark room, it appears no brighter when viewed obliquely than direct, the only observable dif ference being in apparent size, which is that of the projection of the bar on the line of vision : hence it follows that the emanation from a unit of the oblique surface is less than that of the direct, in the ratio of B a to n c, or, which is the same, as the sine of the angle of emana tion n c I) is to unity. After emanation it follows the same law as direct light, of diminishing in intensity inversely as the square of the distance. This law has been the subject of much contention, but we
may remark that something similar occurs in the action of electro dynamic currents, which, though they follow the Liw of the inverse square at different distances in a given direction, yet in different directions the intensity varies in a trigonometrical function of the directions of the currents acting and acted upon, and the line of junction. The Law above mentioned we should not be warranted in applying to luminous gases, as, for instance, the flame of a candle, since the light of the different parts freely then permeates the mass.
Suppose that A, A' represent two lights of the respective intensities and that r u, r s' are planes which bisect,the angles Al' A', e r A', respectively ; the angle a r s' is obviously then a right angle, and the plane r n car well as r n' will be equally illuminated at the point r by the two light., provided that provided '.12. be the con A 1" Al" A'h • then by Euclid, book vi., is equal to the same con A'D stunt, by which the point a may be found, and being still the n same, a' is similarly known; hence if on s n' as diameter it circle be deecribei, each point, such as r, will have the property that planes directed through it to either extremity of the diameter will be equally Blurninatal by the two lights ; but the different portions of the curve Weill do not pewees this property, which may be too readily supposed from the inaccurate statement of this question in optical treatises.
Let it now be proposed to find the nature of a curve, every element of which shall receive equal illuminations from two given lights. Let r, r' be the radii voctores to any point drawn from the two poles or Let A n C represent a small luminous plane, situated obliquely with respect to a point r and A n' its projection taken perpendicular to r A, and finally a b c, a similar plane to the latter taken at a distance r a= unity, the quantity of light emitted by A n c to the point r is the same as if it proceeded from A le c', and is therefore represented by . Area We' . Area a b c • 1. =t Area (a c), where i represents the r P as intensity of the given luminous plane ; hence if we have any luminous surface, we may, by dividing it into very small elements, transfer each element to another situated at a unit of distance from the illuminated point; in other words, we may substitute for this surface that portion of a spherical surface with radius unity which would be cut out by a conical surface having r for vertex and exactly enveloping the luminous surface. The calculation of the illumination of any small plane by a luminous surface of any figure is thus reduced to that arising from a portion of a spherical surface having that plane placed at its centre.