Cott. 4. The focal length of a double C011VeX or double concave lens of glass, whose index of refraction is 1 500, is equal to the radius of the surface. Since by Prop. X111. Cor. BR : RV = Sin Incid.—Sin. Refract.: Sin. Incid. that is, BR : RV —1.5 — 1 : I —0.5 : I— I : 2, and RV=2 BR. But because BR=A r, we haA e A r : BR + A r : 1 :2 = RV : EG= 2 BR: EG. Con sequently, EG =BR.
COR. 5. In Plano-convex and plano-concave lenses of g/a9s, the focal length is equal to the diameter of the spherical surfare ; in this case the centre r of the plane surface is distant, and V r is parallel to ER. Consequently (112,1t V is a parallelogram, and EG = RV = 2 BR.
Con. 6. In Zircon, Sulphur, and other bodies, whose index of refraction is 2.000, it may bc shown by the same reasoning as in the two preceding, corollaries, that thc focal length of double convex and double concave lenses is equal to half the radius of either surface, and that in plano-convex lenses it is equal to the radius of the sphe rical surface.
When diverging or converging rays are incident upon a spherical lens, to find the focus of emergent rays.
Let a pencil of rays diverge from a radiant point Q, Figs. 7, 3, 9, 10, and fall upon a lens or sphere whose centre is E. Let F be the focus of parallel rays incident upon the lens in the opposite direction ; then make Ef= EF, and say, as QF : FE= Ef:fg, which gives FE x Ef f =--. Let f q thus found be set in the oppo QF site way from f that Q is from F, and the point g will be the focus of refracted rays.
Upon E as a centre, and with the radius EF or Ef describe the arches FG,f 3., cutting any ray QA g in G and g, and join EG and E g. Then, supposing rays such as GA to diverge from G, the convergent rays a g q will, by Prop. XIII. Cor. 2. he parallel to GE, and the triangles QGE, ggE will be similar. Hence QG : GE= Eg : g. But if we suppose QA to approach gradual ly to QE, g a will also approach to g E, and we shall ultimaiely have QC; = QF, GE= FE, EG = Ef, and g =f q. Consequently we have ultimately, or for rays neat- QE g, QF : FE= Ef:fy.
Coa. 1. When Q approaches to F, so as to coincide with it, the refracted or convergent rays become paral lel, or, which is the same thing, g recedes to an infinite distance ; and when Q passes to the othcr side of F, the focus g will also pass to the other side, or to the same side with F.
COR 2. On account of the similarity of the triangles QGE, QA 9, QG : QE = QA : Q g, or ultimately QF : QE= QC: Q g.
COR. 3. Since in the analogy QF : FE= Ef fq, the middle term, or the rectangle under FE and Ef is invariable, the distance f q varies reciprocally as FQ, and the points g , Q lie in opposite directions from/ and COR. 4. When the foci Q g lie on the same side of the lens, if the incident rays diverge from Q, the re fracted rays will also diverge from g ; and if the inci dent I ayS converge to Q, the refracted rays will also converge to y. When Q and g are on opposite sides of the lens, the contrary will happen. The points Q and I are called the Conjugate Foci.
Se noLium.
As our limits will not permit us to enter into any far ther details on this branch of the science, we shsll lay before our readers the formulx for the foci of lenses, as deduced analytically by the celebrated Dr. Halley.
Before we conclude this section on the focal lengths uf lenses, we shall describe the method employed by Dr. Smith for illustrating experimentally the motion of conjugate foci along the axis of a lens.
Having determined the focal distance EF of a con vex lens of glass E, Fig. 11. and fixed it on a thin piece of wood CE,' placed vertically on a long table or floor, draw a long line AB perpendicular to the board, and through the point C, and on this line, lay down the fo cal distance of the glass front C to F, and set the same distance from F to I, I to II, Il to III, Ecc. and also on the other side of C front C to f, from f to I, I to 2, 2 to 3. Let 4, 4, 1, &c. of the focal distance be next taken, and set front F towards I, and also from f to wards 1, and put the ntunbers 1, 4, Ste. to the points of division, as in Fig. 11. Things being thus prepared, darken the room, and place a candle at Q over the mat k I. The rays transmitted through the lens will converge at 9 upon a piece of paper held over the op posite mark 1. lf the candle is removed to II, and the paper to 1, the rays will there also be converged into a distinct image ; and in like manner it will be found, that when the candle is at III. the dihtinct image will be at at IV, and A, S:c. the effect remaining exactly the same if the paper and the candle be made to change places Hence it is obvious that f g varies reciprocally as FQ; that is, it increases in the same proportion as FQ increases, and vice vcrsa.