Page:Harold Dennis Taylor - A System of Applied Optics.djvu/39

 also called Gauss points), and, as we have seen, have this important property, that any ray which, while outside the lens, passes through the first principal or nodal point, will, after passage through the lens, emerge on the other side in a direction parallel to its first direction, and radiating from the second principal point ; moreover, the same ray, while traversing the interior substance of the lens, passes ex hypothesi through the geometric centre of the lens or the point C.

As a corollary from the above principle, it follows that if we wish to know the relative sizes or scales of conjugate images formed by thick lenses, we must then measure the focal distances of such images from the principal points of the lens. The focal distance of the first image or object, virtual or otherwise, formed by the entering rays must be measured from the first principal point $$p_l$$ and the distance of the second image formed by the emergent rays must be measured from the second principal point $$p_2$$, when the sizes of the images will be in direct ratio to those focal distances. Our theorem of central projection still holds good, with this modification, viz. that the centre of the lens presents two aspects, or two different positions, according to whether the lens is viewed from one side or the other. Regarded from the left hand the centre of the lens is practically the first principal point $$p_l$$ but regarded from the right hand the centre of the lens is practically the second principal point $$p_2$$, and these two points are but the refracted images of the geometric centre C of the lens. That is, $$p_l$$ is the conjugate image of C by refraction at the first surface, and $$p_2$$ is the conjugate image of C by refraction at the second surface. Therefore the distances $$A_1 p_l$$ and $$A_2 p_2$$ may be derived from the Formula II.,

$\textstyle{ \frac{\mu}{u'} = \frac{\mu -1}{r}-\frac{1}{u} }$,

in its more special application to Figs. 4f and 4g. At the first surface we have $$u=A_1..p_l$$ (Fig. 9a), which by convention is a minus quantity, while $$A_1..C=u'$$, and is a plus quantity, and $$A_1 c_1=r$$. Let r and s = first and second radii of curvature respectively, and let the thickness be denoted by t, therefore

$\textstyle{ \frac{\mu}{A_1..C} = \frac{\mu -1}{r}-\frac{1}{A_1..p_1} }$,

and

$\textstyle{ \frac{1}{A_1..p_1} = \frac{\mu}{A_1..C}\frac{\mu -1}{r} }$;

but

$\textstyle{ A_1..C = t\frac{r}{r+s} }$,