SHADOWS. If rays pass in straight lines, the main phenomena of shadows are at (MCC ex plained. A small source of light, a 'point Will cast sharp shadows of any opaque object which stops the rays. If the source of light is large, like a window, there will he cer tain points in the shadow of an opaque object. close to the object. which are not reached by any rays, there will also lie points which are reached by rays from part of the source only, and then there will be points which receive rays from the whole source. 'Those points which do not receive any rays lie in the 'umbra' or shadow Proper. while those which receive rays from only a portion of the source lie in the `penumbra.' Thus. an eclipse of the sun by the moon is toted at a point of the earth's surface which passts through the cast out into space )my the moon obscuring the ray:; it Is called pa' fiin at a point on the earth if that point passes thromdi the penumbra only; it is called 'atiaular' if at any time during the eclipse there can be seen a ring of the sun's surface extending past the moon's disk. 'Pin-hole photography' is an other phenomenon which follows at once from the passage of rays in straight lines.
If rays from a point-source fall upon a reflecting surface, each ray individually will obey the laws of reflection. If the incident rays forth a small solid cone symmetrical about the line drawn through the point-source and perpendicular to the surface, they are said to form a liomucentrie' pencil; if, however, the axis of the cone of rays is oblique to the surface, it is called an 'astigmatic pencil.' I'lle rays thrilling a homocentric pencil after reflection form another such pencil. i.e. they either eon rerge to a point on the perpendicular to the sur face ir dicerge in such a manner that if the rays be produced back of the surface they will inect in a point on the perpendicular to the surface. This vertex of the cone of reflected rays is called the 'focus' or 'image' of the point-source: it is a 'rear focus or 'image' in the former of the above cases; 'virtual,' in the latter. The rays forming an astigmatic pencil after reflection do not in general have a point focus or image. but have as foci two short straight lines at right angles to each other and a short distance apart: they are called 'focal lines.' and may be either real or virtual. A few special cases will be dis •ussed briefly.
Moir Surfaee.-1.,et P Al be the section made by the paper of a plane surface perpendicular to the paper: let 0 be the point surface; 0 Al 0', the perpendicular dropped from 0 upon the surface; 0 P, any incident ray; P Q, the reflected ray; P, a perpendicular to the surface at P; 1' Cr. the con tinuation of Q P backward. By the laws of reflection the angles 0 P S and Q P S are equal. Therefore by the laws of geometry the point 0' is at a distance 0' M back of the surface equal to that of the point 0 in front of it and further the position of 0' is entirely independent of the di rection of the ray 0 P, and is therefore the same for all rays. Consequently. all rays
diverging from 0, both homo •entric and astigmatic pencils.
proceed after reflection at the surface as if they had come originally from the direction of 0'. 0' is there fore the virtual image of 0. Similarly if an ex tended object is emitting light toward a plane mirror each point of the object will have a vir tual image in the surface at the same distance be hind it as it itself is in front of the surface. There will thus be a virtual image of the object of the same size.
phcrical Surface.—Let P.M N be the section made by the paper of the concave spherical sur face; 0, the point-source of light; C, the centre of the sphere of which the surface is part; 0 CM, the perpendicular to the mirror (i.e. the 'axis' for the point 0) ; 0 1', any ray of a homo centric pencil; C P, the radius of the sphere drawn to 1', and therefore perpendicular to the surface at 1'; P h 0', the reflected ray; C F A, a line drawn through C parallel to the ray O P. By the laws of reflection, the angles O I' C and 0' P C are equal; then by ordinary laws of geometry the point b' where the reflected ray 0' P intersects the radius (_' A divides this radius in two equal parts, and the position of the point 0' on the line 0 C AI is independent of the in cident ray provided that P is near M. Therefore all the rays of a homocentrie pencil from 0 form another homocentric pencil on reflection with its vertex at. 0': this point is therefore a real focus or image of O. If the distances OM. 0' C are called u, r, r respectively, they are con nected by the formula 1 1 2 +— = U r This formula can be shown to apply to any position of the point 0 and to either a concave or a convex mirror. provided only that the pencil of rays is homocentric.
Let 0 S and 0'1' he two rays of an astigmatic pencil, and let S P, and T U be their reflected rays. They intersect in a point R not on the axis. If a small pencil of rays is considered as falling on an elementary area of the mirror near 'I' and S, the reflected rays will combine to produce a line perpendicular to the paper at R and another line lying in the paper across the axis; these are the two 'focal lines,' real in this case.
If all the rays falling on the whole concave mirror are considered. they will form by reflec tion a bright point at the focus 0'. is the apex of a bright curved surface made up of the focal lines such as was produced at R owing to an elementary area of the mirror. This pointed surface is called the 'caustic.' surface. A section of such a caustic formed by reflection at a cylin drical mirror is often seen on the surface of a glass of milk or a cup of coffee. The fact that astigmatic pencils do not bring the rays to the same focus as that of the homocentric pencil is said to be due to 'spherical aberration.' If any small object 0 Q is placed in front of a concave mirror. as show's'', its image will he 0' Q', as is evident from the figure. The image is therefore real, inverted, and diminished in size. The linear size of the image divided by that of the object equals 11— r