LEMMA.
There is within every double convex or double concave lens, a certain point called its centre, through which every ray that passes will have its incident and emer gent parts parallel ; but in plano-convex or plano-con cave lenses, this point is removed to the vertex of the convex or concave smface; and in meniscuses and con cavo-convex lenses, it is removed a little way out of them, and tics nearest the surface which has the greatest curvature.
Let II r, Fig. 23-26. be the centres of the surfaces A a of these lenses, and RE r their axes. Draw any two of their radii RA,r a, parallel to each other, and joining A, a the point where this line intersects, the axis will be the point E above described. For since the triangles REA, r F. a, ale similar, RA :r a-12E : r E, and there lore by GEon. Prop. IV. or V. R A =–, ra:ra E (or R : r E ; and as the three lirst terms ol this tmalogy are invariable, the last r E must also be invariable. 'fence it follows, that to whatever points in the surfitces of the lens the parallel radii RA, r a, are drawn, the line A a produced, if necessary, will always cut the axis R r in the same point E.
If we now suppose the ray A a to pass hoth ways out of the lens, it will be refracted equally, and in contrary directions; because P,A, r a being perpendiculars to the surfaces at A and a, the angles of incidence of the ray A a, or a A, will be equal. Consequently AQ will he pa rallel to a q. In plano-convex or plano•coneave leleteR, one of the radii RA or r a becomes infinite, and conse quently parallel to the axis RE, the other radius will therefore also coincide with the axis, and the points A, E, or a, c, will coincide in the vertex of the convtx or conclave siirface.
Con. \Viten the thickness of a lens is inconsiderable, and when a ray falls nearly perpendicularly upon it, the path of the ray through E, viz, QAE a y may be taken as a straight line passing through the centre E of the lens ; for the perpendicular distance of AQ and a g di minishes both with the thickness of the lens, and with the obliquity of the ray to the axis.
To find the principal focus. or the focus of parallel rays falling perpendicularly upon a lens whose thickness is inconsidt.rable.
In the six different kinds of lenses, shown in Figs. 1-6. let E be their centre, R r their axis. passing
through the centres P., r of their surfaces. Let g EG be a line parallel to the rays incident upon the stirlace B. Draw the radius BR parallel tog E, and in BR pro duced let V be the focus of the rays after their first re fraction at the surface B, and having joined V r, the point G where it cuts g E will be the focus of rays re tracted by the lens. Since V is the focus of the rays incident upon the second surface A, the refracted rays must have their focus in some point of the line V r, drawn through r perpendicular to that surface, conse quemny the focus of all the ralfs must be at thc intersec tion G of the ray g EG with the ray V r.
Cox. 1. If the inclination of the incident rays to the axis of the lens is gradually diminished till they become parallel to it, the focal distance EF will be equal to EG. In passing from GE to FE, the first and second foci V and G will describe circular arcs \"1, GF, having their centres in R and E. For as RV is to RB, as the lesser of the sines of incidence and refraction is to their differ ence, the line EG will be invariable, being to the given line RV in the given ratio ol E to r R, on account of the struilar triangles EG r, RV r.
COR. 2. The following rule, for determining the focal distance of a thin lens, is deducible from the similarity of the triangles EG r, RV r. Aa the interval between thc centres of the surfaces, or R r, is to the radius of the second surface r E, so is the continuation of the first radius to the first focus, viz. RV or RT, to the focal dis tance of the lens EG or EF, which must be on the same side as the emergent rays, if the lens is thicker in the middle than at its edges, and on the opposite side, if it is thinner in the middle ; that is, COR. 3. The distance of the principal focus of a lens from its centre F, is the same on each side of the lens, that is, EF and E f are equal, when rays fall on both sides of the lens. Continue E r to t, the first focus of rays falling- parallel upon the surface ; and since r : r E — RT = EF, we shall for the same reason have r R : RE = r t : Ef. Consequently, since the rectan gle under r E. RT, and also RE, r t, are equal, we shall have Ef = EF ; for r E : r t, and RE : RT, in the same given ratio.