588 LIGHT the angle UPV. (This follows because OP, a radius of the sphere, is normal to the surface at P.) Hence it is Fig. 7. rigorously true that, if V be the intersection of PV with UOA, vp ou VP = UP The full consequences of this exact statement will be developed under OPTICS (GEOMETRICAL). For our pre sent purpose, an approximation will amply suffice. Let us suppose P to be so near to A that no sensible error is introduced by writing A for P in the above formula. This amounts to supposing the mirror s breadth to be very small in comparison with its radius of curvature. The formula now becomes VO_OU . AV"AU or, what is the same, AO-AV AU-AO AV AU and V is, to the degree of approximation above stated, independent of the position of the point P. If we call r the radius AO of the mirror, u = AU the distance of the source, and v = AV, the distance of the point V from the mirror, this becomes r-v u-r The formula, or the cut, shows at once that this relation between U and V is reciprocal; i.e., all rays from V, falling Conju- on the mirror, will be made to converge at U. These OC1 - points are therefore called conjugate foci. The simplicity of (a) is remarkable ; so, also, is that of its interpretation. For the rays passing from a source to a given object, like the mirror, are less and less divergent as the source is farther off. Hence (a) signifies that the (alge braic) sum of the divergences of the incident and reflected rays is equal to that divergence which the mirror can con vert into parallelism. In fact the rigorous geometrical relation may be written in the obvious form AVP + AUP = 2AOP, which, when all three angles are small, is simply (a). A similar state ment may easily be made in the case of refraction. Before we proceed to develop the consequences of this simple formula, we may point out that it is applicable to formula. a11 c . ases > to convergent rays falling on a concave mirror, to divergent rays falling on a convex mirror, &c., <fcc. The reader may easily verify this by trial for himself. But it follows at once from the necessary interpretation of the negative sign in geometry. Thus, if the mirror were convex, O would be to the left of A, as the figure is drawn; and AO, if formerly positive, would now be negative. Thus, for a convex mirror, the formula is General! zation jt If the incident rays be convergent, U is to the left of A, and therefore AU, or u, is negative ; and so on. We must now study the relative positions of U and V, in order to find tho sizo and position of the image for different positions of the object. Returning to the formula (a) above, we ses that the following pairs of values of u and v satisfy it : Infinite. Greater than r. r Less than r, greater than >: Greater than 0, less than |r. 0. Greater than ir, less than Greater than r. Infinite. Negative. 0. Thus, when the source is at a practically infinite distance Princ (as the sun or a star) the image is formed at a distance focus from the mirror equal to half its radius of curvature. It is then said to be in the principal focus. As the source comes nearer, the image comes out to meet it, and they coincide at the centre of curvature of the mirror. In fact, a ray leaving the centre of the mirror must meet the surface at right angles, and thus go back by the way it came. When the source comes still nearer, the image goes further off, until, when the source is at the principal focus, the image is at an infinite distance ; that is, the rays go off parallel to one another. This is the mode in which a concave reflector is employed for lighthouse purposes. When the source comes still nearer, the image is behind the mirror, i.e., the incident rays are so divergent that part of their divergence remains after reflexion. This remnant of divergence becomes greater and greater as the source is nearer to the mirror, i.e., the (then virtual) image comes closer to the mirror, which finally behaves, for a very near source, almost precisely like a plane mirror. All of these phenomena can be beautifully seen in a dark room by employing a beam of sunlight, rendered distinctly visible, in the fashion noted by Lucretius, by the motes in the air. For further explanation pictures are given (figs. 8, 9), Paths showing the course of the pencil of rays when (1) a real**** ^ i tii n ii an imay;e. and (2) a virtual image is formed by a concave mirror. It will bn seen at ones that, in the cases figured, the real image is inverted and less than the object, the virtual image erect and larger. In fact the size of a small object is obviously to that of its image in proportion to their distances from O, the centre of curvature of the mirror. Also the image is erect when it lies ou the same side
