If LG is one of the rays incident at G, it. refracted ray GV will be determined by means of the index of re fraction of the medium, and a tangent drawn to the curve at G. NVe must therefore find in the line GV a point D, so that FD-I-az x DG shall be equal to a given line ; and in like manner for any other ray UM, refract ed into A10, we must find a point N, so that FN-Fm AIN shall be equal to the sante given line, nt represent ing thc index of refraction. For the demonstration of this proposition, which is not of much practical utility, we must refer the reader to Huygens's Traite de la Lu rniere, p. 114, 115.
The image of any object formed by a plane lens is equal and similar to the object, and in a similar situa tion on thc same side of the lens.
Let AabB be a plane lens, PQ the object, and PA, PC, two rays from P, incident upon the lens, and QB, QC, two rays from Q, incident upon the lens. The rays PA, QII, being supposed to fall perpendicularly, will pass on unretracted to a and b, but the rays PC, and QC, will be refracted at C in the direction C e, C d, and will again be refracted at thc second surface, so as to emerge at d and e, parallel to PC and QC, as if they had come from the points p and q, determined by draw ing e p and d g parallel to PC and QC. As the distances P p and Q y depend solely upon the thickness of the glass, and upon its refractive power, these will be in variable for the different points of the object, and con sequently p y will be equal and parallel to PQ.
An image formed by a prism is upright, and equal to the object, and lies on the same side of the prism, and at the same distance from it as the object, provided the refracting angle of the prism and its refractive power be small.
Let two rays PE, QE, from the extremities of the object PQ, pass through the point E, so near the sum mit of the prism ABE, that the distance between the points of incidence and emergence is very small. Now since the total refractions of the rays PEN, QEO are equal, they will cross each other, so that the angle PEQ=NEO=P E g formed by the emergent rays pro duced backwards, and because the distance Ep—EP, and E q=EQ. the image g will be upright, and of the same size as the object, and at an equal distance on the same side of the prism.
If an object placed before a lens has the form of a circular arch, whose centre is that of the lens, its image will be a similar concentric arch ; thc length of the ob ject will be to the length of the image as their dis tances from that centre, and the image will be erect or inverted, according as it lies on the same, or on the op posite side of the centres.
Let AB, Fig. '20. Plate CCCCXXIX. and Figs. 1, 2. Plate CCCCXXX. be the lens, and E its centre, and let the object PQR have the form of a circular arch, whdse centre is E. In PQR, take any point Q, and
having joined QE, let F be the principal focus of rays incident in the opposite direction. In QE, produced if necessary, take QF : FE =QE: E q, and setting q E front E to q, q will be the image of the point Q. Upon E as a centre with the radii EF, E g, describe the cir cular arcs p g r, and draw PE, RE meeting p q r in p and r ; then p q r will be the image of the object PQR. Sincc EP=EQ, and EG=EF, we hare EP--.J--EH=QE=t--EF ; that is, PH=QF ; but by the construction E p=E q, and QF : FE=QE : E ; con sequently PH : HF=PE : E p, or p is the image of the point P. In like manner it may be shown, the angle PER being- always supposed small, that every other point in PQR will have its image in a corresponding point of p q r. As the arches PQR, p y r subtend equal angles at E, they are similar. Since thc axes RE r, PE p of the incident pencils are reckoned straight lines, passing through E, the angle PER=p E r, and consequently PR : p r = PE :p E = QE : E g. As the points r and p are necessarily in the axes RE r, PE p, they will be on the same or on contrary sides of E, ac cording as their middle points Q q, are on the same or on contrary sides of F ; that is, the image will be erect or inverted, according as these points are on the same or on the contrary sides of E.
Cott. As the object PR approaches nearer to 3. straight line in proportion to the smallness of the angle which it subtends at E, a small rectilineal object placed at a considerable distance front the centre of the glass may be supposed to have very nearly a straight image.
The image of a rectilineal object, formed by a lens or sphere. is the arc of a conic section, being an ellipse, a parabola. or a hyperbola, according as the distance of the object is greater than, equal to, or lesa than, the prin cipal focal distance of the lens or sphere.
Let AB, Figs. 3, 4, 5. be a lens or sphere, whose centre is F, PQZ a rectilineal object placed before its focus F, in Fig. 3. between the focus and the lens, as in Fig. 4. or in thc focus, as in Fig. 5. Through E draw QE q, at right angles to PZ, and in PZ take any point P, and having drawn PE, produce it towards p. Let F be the principal focus of rays incident in the opposite direction, and upon E as a centre, with the radius EF, describe the circles FG/L, cutting PE in G. In the line PE p, take PG : PE=PE : P p, set ting P p to the left hand of P, if G is to the left hand of p, as in Fig. 4. then p is the image of the point P.
Draw p D parallel to Q g, and on account of the simi larity of the triangles PEQ, P p D.
PE : Pp = QE : D p Consequently, PG : PE =QE: D p.