Page:Harold Dennis Taylor - A System of Applied Optics.djvu/35

 virtual; and all intrinsically negative distances are drawn in dotted lines, with their virtual extensions drawn lighter.

Theorem of Central Projection

Having now the formulæ relating to axial pencils of rays, we may next consider the case like that shown in Figs. 7 a and b.

Besides the conjugate axial pencils Q1..q1, let another point of origin Q2, in the case of the collective lens, or another apex of convergence Q2, in the case of the dispersive lens, be taken at some small but appreciable distance away from the axis, such that Q1 and Q2 are on a plane perpendicular to the axis. It is evident that a ray drawn from Q2 through the centre of the lens will pass straight on, as it is crossing two elements of surfaces which are parallel and practically touching. If a straight line from Q2 is therefore drawn through the centre of the lens and produced until it cuts the other so-called conjugate focal plane q1..q2 (which is perpendicular to the axis

and passes through ql, the conjugate focus to Q1), then the point of intersection q2 is where the conjugate image of the point Q2 is formed. That is, the centre of the lens is always in a straight line between any point Q2 or Q3 of a plane object and its conjugate image q2 or q3. This theorem is capable of a further extension, as shown in Figs.8a and b, Plate III.

Here are two cases in which the pencil of rays from Q2 (here drawn in solid lines) is eccentric; that is, none of the rays of the eccentric pencil actually pass through the centre of the lens owing to the stop s being interposed. But it is assumed that the rays constituting such an eccentric pencil are but a part of a larger pencil of rays filling the whole lens; and since the lens is assumed so small that all the rays refracted through it from any one point are caused to converge to or diverge from one and the same image point, therefore these eccentric rays may be regarded as coming under the same law, and the conjugate points Q2 and q2 may be considered to be strictly on a straight line of projection drawn through the centre of the lens. Thus

the pencils of rays are assumed to be homocentric—that is, all the rays constituting each pencil are assumed to diverge from or converge to one point. From this it follows that the distance $$\scriptstyle{q_1..q_2=(Q_1..Q_2)\frac{v}{u}}$$ and the scale of any conjugate image formed of the plane

Q1..Q2 is $$\scriptstyle{\frac{v}{u}}$$ times the scale of the original. The scales of image and object are in direct ratio to their axial distances from the lens centre.