Diverging rays are such as recede from each other, and whose directions meet if produced backwards. 2. The focus of a pencil of rays is that point towards which they converge, or from which they diverge. If the rays in a pencil, after reflection, or refraction, do not meet exactly in the same point, the pencil must be diminished; and the focus is tbe limit of the intersections of the extreme rays, when they approach nearer and nearer to each other, and at length coincide. In this case, the focus is usually called the geometrical focus. 'The focus is real, when the rays actually meet in that point; and imaginary, or virtual, when their directions must be produced to meet. 3. The axis of a pencil is that ray which is incident perpendicularly upon the reflecting or refracting surface. 4. The principal focus of a reflector, or refractor, is the geometrical focus of parallel rays incident nearly perpendicularly upon it. Fig. 4. Fig. 5. Prop. 6.- If a ray of light be reflected once by each of two plane surfaces, and in a plane which is perpendicular to their common intersection, the angle contained between the first and last directions of the ray, is equal to twice the angle at which the reflectors are inclined to each other. Let AB, CD, (fig. 5,) be two plane reflectors, inclined at the angle AGD; SB, BD, DH, the course of a ray reflected by them. Produce HD to 0, and SB till it meets DH in H. Then, because the _HBG= the < ABS= the < DBG, the whole angle DBH=2_DBG. In the same manuer, the DBO=24 BDC. And since the _ BGD= the BDC the < DBG, we have 2 _ BGD= 2 x BDC- 22 DBG the < BDO — the _ DBH = the < BHD.* PROP. 7.-Parallel rays, reflected at a plune surface, continue parallel. Case 1. When the angles of incidence are in the same plane, Let RS, fig. 6, be the reflecting surface; AB, CD the incident, BG, DH the reflected rays. Then the LABR=the < GBS, and the < CDR = the HDS; but, since AB and CD are parallel, the < ABR = the < CDR: therefore the < GBS = the <HIDS, and BG, DH are parallel (Euc 28. 1.) • Euc. 32. 1. C:sc 2. When the angles of incidence are in different planes. Let AB, C), lig 7, be the rays; BE, DP perpendiculars 10 the refircting surlace at tile poms of incidence; join BD, and let AB be rellected in the direction BG; also let DH be the intersection of the planes CDF, GBD. Fig. 7. F E А нс Then, since BE, Df, which are perpendicular to the same plane, are parallel (Euc. 6. 11), and AB, CD, are parallel, by the supposition, the a.yles of incidence ABE, CDF are equal (Euc. 10. 11); therefore the angles of reflection are equal. Again, since EB and FD are parallel, as also AB and CD, the planes ABG, CDH, are parallel (Euc. 15. 11), and th are intersected by the plane (BDH; conseguently DH paralle! to BG (Euc. 16. 11); therefore the angles EBG, FDH are equal (Euc, 10. 11); but the angle EBG is the angle of reflection of the ray AB; therefore the angle FDH is equal to the angle of reflection of the ray CD; and since DH is in the plane CDF, CD is reflected in the direction DH (Art. 18), which bas before been shown to be parallel to BG, PROP. 8.-If diverging or converging rays be reflected at a plane surface, the foci of incident and reflected rays are on eontrary sides of the reflector, and equally distant from it. Let QAB be a pencil of rays diverging from Q, and incident upon the plane reflector ACB; draw QC perpendicular to the surface; then will QC be reflected in the direction CQ (fig. 8). Let QA be any Fig. 8. other ray, and since a perpendicular to the surface at A is in plane with QC and QA (Euc. 6. and 7. 11), QA will be reflecte plane. Produce CA to D, and make the angle DAO eqoal to to angle QAC, then will_AO be the reflected ray. Produce OA, QC, till they meet in q. Then, since the <qAC = the LOAD = the LQAC, and also the <qCA = the < QCA, and the side CA is common to the two triangles QCA, CAq, the side QC is equal to Cq, In the same manner it may he shown, that every other reflected ray in the pencil, will, if produced backwards, meet the axis in g; that is the rays, after reficction, diverge from the focus q. IF QABE be a pencil of rays coverging to 9, they will, after reflectiou at the sarface ACB, converge to Q; therefore, in this case also, the foc! of incident and reflected rays are on contrary sides of the reflector, and equally distant from it. Cor. 1. The divergency, or convergency of rays, is not altered by reflection at a plane surface. Cor. 2. In the triangles QAC, CA9, Aq is equal to QA; if therefore any reflected ray AO be produced backwards to q, making Aq=AQ, 4 is the focus of reflected rays. Cor. 3. If the incident rays QA, Qa, be 'parallel, or the distance of Q from the reflector be increased without limit with respect to Aa, the distance of q is increased without limit, or the reflected rays are parallel. PROP. 9.-1f parallel rays be incident nearly perpendicularly upon a spherical reflector, the geometrical focus of reflected rays is the middle point in the axis between the surface and centre. Let ACB be a spherical reflector, whose centre is E; DA, EC, two rays of a parallel pencil incident upon it, of which EC passes through the centre, and is therefore reflected in the disaction CE; join EA, arred in the plane DACE, make the angle EAq equal to the angle DAE, and DA will be reflected in the direction Aq (fig. 9); draw GAT' in the same plane, touching the reflector in A, and let it meet EC produced in T. Then, since the < EAq=the DAE= the < A EQ (Euc. 29. 1), Eq=Aq; also, the <IAT=the < DAG (Art. 19) = the ATQ (Euc. 29. 1); therefore Aq=qT; consequently Eq=qT; that is, 9 bisects ET the sec nt of the arc AC. Now let DA approach to EC, and the arc AC will ecrease, and its secant, at length, become cqual to the radius; consequently the limit of the intersections of Aq and CE is F. the middle point between E and C. If the rays be incident upon the convex side of the reflector, the reflected rays must be produced backwards to meet the axis; and, in this case, F, the midule point between E and C, may be shown to be the limit of the intersections of CE and Aq, as before. Cor, 1. As the arc AC decreases, L9, or Fq, decreases. Thus, when AC is 60°, Eq= EC; and when AC is 45°, Aq is perpendicular to EC, and Eq : EC ::1: v2. Cor. 2. If different pencils of parallel rays be respectively incident, nearly perpendicularly, upon the reflector, the foci of reflected rays will lie in the spherical surface SFV, whose centre is E and radius EF.Fig. 10. Fig. 9. Fig. 10. Cor. 3. If the axes EA, EC, EB, of these pencils, lie in the same plane, the foci will lie in the circular arc SFV. Cor 4. If any point S, in the arc SFV, whose radius EF is on half of EC, he the focus of a peucil of rays incident nearly perpendicularly upon the reflector, these raus will be reflected parallel to cach other, and to EA the axis of that pencil. Prop. !0.- When diverging or converging rays are incident neurly perpendicularly upon a spherical reflector, the distance of the focus of incident rays from the principal focus, measured along the axis of the pencil, is to the distance of the principal focus from the centre, as this distance is to the distance of the principal focus from the geometrical focus of reflected rays. Let ACB, fig. 11, be the spherical reflector, whose centre is E; Q ibe focus of the rays; QA, QC, two rays of the pencil, of which QC passes thronghi the centre E, and is therefore reflected in the direction CQ; joia EA, and, in the plane QACE, make the angle EA equal to the angle EAQ; then the ray QA will be reflected in the direction Aq.' Draw DA parallel to QC, and make the angle EAe equal to the angle EAD; bisect EC in F. Then, since the DAE= the < EAe, and the < QAE = =the < EA7, the < DAQ, or its equal AQe, is equal to the LeAq; also, the < geA is common to the two triangles A Qe, Aqe; Therefore they are similar, and Qe : A :: A : eq; or, since eA=E QE : E:: eE : eq. Now let ihe arc AC be diministed without limit, or the ray QA be incident nearly perpendicularly, then e coiveides with F; and the limit of the intersections of CQ and Aq, is determined by the proportion QF : FE :: FE : F9. The diagram, fig. 11, is constructed for the casc in which diverging rays are incicient upon a concave spherical surlace, and the same demonstration is applicable when the incident rays converge, as is represented in fig. 12. If the lines DA, QA, EA, eA, 9A, be produced, the figures serve for those cases in wbich the rays are incident upon the convex surface. Cor. 1. If y be the focus of incident, Q will be the focus of reflected rays; and Q and q are called conjugate foci. Cor. 2. If the distance QF be very great when compared with FE, FQ is very small when compared with it. Thus, if the rays diverge from a point in the sun's disc, and fall upon a reflector wbose radius does not exceed a lew feet, F and y may, for all practical purposes, be considered as coincident. Cor. 3. Wben Q coincides with E, all the rays are incident perpendicu. larly upon the reflector, and therefore they are reflected perpendicularly, or q coincides with E. Cor. 4. The point e bisects the secant of the arc AC. Cor. 5. Since Qe: eE :: E: eg, hy composition or division, Qe: QE :: eE : Eq; alternately, Qe : eE :: QE : Eq; and, when QA is idcident nearly perpendicularly, QF : FE :: QE Eq. Cor. 6. Since EA bisects the angle QAq (or PAT), QA : Aq :: QE : E9 (Euc 3 6 ); and, when QA is incident nearly perpendicularly. QC : Cq :: QE : E9. That is, the distances of the conjugate furi ficm the centre, are proportional to their distances from the surface. Cor. 7. Since QE : Eq :: QF : FE, and QE : Eq :: QC : Cq, e have, ultimately, QF : FE :: QC : Cq. Cor. 8. As the arc AC decreases, Eq, the distance of the intersection of the reflected ray, and the axis from the centre decreases, unless Q coin, cide with E, or lie between E and e. For, Qe : eE :: QE : En, and as AC decreases Ee decreases; therefore, when Q is an e E produced, the terms of the ratio of greater inequality, Qe : Ee, are equally diminished, and that ratio, or its equal QE : Eq, increases; and, since QE is invariable, Eq decreases. When Q is in Ee produced, as AC decreases Qe increases, and Ee decreases; therefore the ratio of Qe : Ee, or of QE : Eq increases ; and consequently, as before, Eq decreases. But when Q lies between E and e, as AC decreases the terms of a ratio of less inequality Qe : £e are equally diminished ; therefore that ratio, or its equal QE: Eq, decreases; and since QE is invariable, Eq increases. When Q coincities with E, q also coincides with it, whatever be the magnitude of the arc AC. PROP. 11.-The conjugate foci, Q and q, lie on the same side of the principal focus ; they move in opposite directions, and meet at the centre and surface of the reflector. Since QF : FE :: FE : Fq, we have QF X Fq=FE? ; thai is, Q and q are so situated that the rectangle under QF and Fq is invariable. Also, when Q coincides with E, q coincides with it (fig. 13); in this case tben, QF and Fq are measured in the same direction from F; and, since their rectangle is invariable, they must always be measured in the same direction. Fig. 14, P That Q and q move in opposite directions may thus be proved: the rectangle QF X Fq is invariable; and therefore as one of ihese quantities increases, the other decreases ; also, Q and q lie the same way from the fixed point F, they must therelore move in opposite directions. Having given the place of Q, and FE the focal length of the reflector, to determine the place of the conjugate focus y, we must take QF : FE :: FE : F9, and measure FQ and Fq in the same direction from F. Thus, when Q, the focus of incident rays, is further from the reflector than E, and on the same side of it, FQ is greater than FE, therefore FE is greater than Fq; or ļ, the focus of reflected rays, lies between F and E. When Q is between E and F, 9 lies the other way from E; and whilst Q moves from E to F, 7 moves in the opposite direction from E an infinite distance. When Q is between F and C, QF is less than FE or FC; therefore FC is less than Fq; anıl, since F'Q and Fq are measured in the saine direction from F, 9 is on the convex side of the reflector.--Fig 14, |