LIGHT. 240 LIGHT. form a iiomocentric' pencil; if, however, the a.is of the cone of rays is oblique to Ihc surface, it is called an "asligiualic pencil.' The rays forming a Iiomocentric pencil after retlection form another such pencil, i.e. they cither con- verge lo a point on the perpendicular to the sur- face or tliierye in such a manner that if the rays be produced back of the surface they will meet 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 "real' focus or "image' in the former of the above cases; 'virtual,' in the latter. The rays forming an astigmatic ])cncil after retlection do not in general have a j)oint 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- cussed brielly. Plane Surface. — I.,et P JI be the section made by the paper of a plane surface perpendicular to the jKiper: let O be the point surface; () M O', the perjjendicular dropped from <) upon the surface; OP. any incident ray; I' (.), the reflected ray; S P, a perpendicular to the surface at P ; P O'. the con- tinuation of Q P backward. Bv the laws of reflection the angles OPS and Q P S are eqinil. Therefore by the laws of geometry the point O' is at a distance O' il back of the surface equal to that of the point ( ) in front of it ; and further the jiosition of O' is entirely independent of the di- rection of the ray O P, and is therefore the same for all rays. Consequently, all rays diverging from O. i)(]tli homo- centric and astigm:itic pencils, jjroceed after rcllection at the surface as if they had come from the direction of O'. O' is thcre- irtual image of O. Similarlv if an ex- 'axis' for the point O) ; P, any ray of a homo- centric pencil; (J P, the radius of the sphere drawn to P, and therefore |)crpendicular to the surface at P; P K O', the reflected ray; C 1 A, a line drawn through C parallel to the ray OP. By the laws of reflection, the angles OPC and O' P C are equal; then by ordinary laws of geometry the point !•' where the reflected ray O' P intersects the radius C A divides this radius in two equal parts, and the i)osition of the point O' on the line C JI is independent of the in- cident ray provided that P is near Jl. Therefore all the raj's of a Iiomocentric pencil from O form another Iiomocentric ]iencil on reflection with its vertex at ()': this point is therefore a real focus or image of 0. If the distances O .. O' M, CM are called », i', r respectively, they arc con- nected by the formula 1.1 2 This formula can be shown to apply to any position of the point O and to either a concave or a convex niirroi . provided only that the pencil of rays is hoinocentric. Let O S and T be two rays of an astigmatic jiencil, and let 8 R and T K be their rellected 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 T and S, the reflected rays will combine to j)roduce a line perpendicular to the ))aper at R and another line lying in the i);ipcr 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 O'. which is the apex of a bright curved .surface made up of the I-|i;. I. originally fore the vi 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. Spherical Surface. — Let P JI 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; O C M, the perpendicular to the mirror (i.e. the Fig. 3. 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 Iiomocentric pencil is said to be due to 'spherical aberration.' If any small object O Q is placed in front of a concave mirror, as shown, its image will be C 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 • . u — r Refraction. One must distinguish between homocentrie and astigmatic pencils in refraction as well as in reflection ; for a homocentrie pen- cil of incident rays produces a homocentrie pencil of refracted rays, thus giving a point-focus, either real or virtual ; and an astigmatic pencil I
