emergence be parallel to one another. They will therefore all be parallel to a'F'.

Hence if TB meet the first Principal Plane at B, and if BB' drawn parallel to the axis meet the other Principal Plane at 3', then B'N' drawn parallel to a'F" will be the emergent ray produced by the incident ray TB.

Let TB produced and the corresponding emergent ray meet the axis of the system at the points N, N' respectively. Then we see from the figure that the triangles TFN and a'H'F' are equal in all respects; therefore

FN=H'F' = constant.

Hence the position of N is independent of the position of T; therefore N is a fixed point, and its distance from F is equal to the second focal distance.

In a similar way it may be shown that

therefore N' also is a fixed point independent of T, and is at a distance from F' equal to the first focal distance. The points N and N' are clearly the Nodal Points referred to in Art. 136.

138. If the extreme media be the same, the two focal distances are equal; hence, as we have already noticed, the points N and N' coincide with H and H'.

139. From the figure we have also

NN' = ßß' = HH'.

Hence the distance between the Nodal Points is equal to the distance between the Principal Points.

140. We have also

HN=H'N' = H'F' - HF.

141. When the Nodal Points have been determined, we may with their help very readily determine the direction of the emergent ray produced by a given incident ray, and also the position of a point conjugate to a given one (fig. 24).

With the usual notation we will suppose XTa to be an incident ray meeting the Focal Plane through F at T. Join TN. Let aa' parallel to the axis meet the second Principal Plane at a'. Through a' draw a'T'X' parallel to TN. Then a'T'X' is the direction of the emergent ray produced by the incident ray XTα, and X, X' are a pair of conjugate points.

Otherwise: Draw N'T" parallel to XTa and meeting the Focal plane through F' at the point T; then 'T' is the direction of the emergent ray.

If an object lens be situated on the Nodal Plane through N its image will be situated on the Nodal Plane through N', and it may easily be proved that the linear dimensions of object and image are to one another inversely as the indices of refraction of the first and last media.

CHAPTER VI.

THE DIFFERENT FORMS OF LENSES.

142. WE will now apply some of the results of Chapter II to determine the positions of the Principal Points and Foci for the five most important forms of the simple lens.

We will consider, (1) a double convex lens, (2) a planoconvex lens, (3) a double concave lens, (4) a plano-concave lens, (5) a meniscus; and finally we will consider (6) the case of two mirrors placed upon the same axis, and facing one another, in such manner as we find in Gregory's and Cassegrain's telescopes.

143. In Arts. 55 and 64, it was shown that the distances of the Foci and of the Principal Points from the vertices of the lens are given by the formulæ

We will now suppose the refractive index of the outside medium to be unity, and that of the substance of the lens

to be μ. The values of
The values of p and become therefore

μ - 1

r

Also for the reduced thickness -t we will snbstitute its

t

value, -, in terms of the absolute thickness.

μ

These formulæ belong to such a lens as is described in fig. 14, which we have chosen for our standard. To obtain the corresponding formulæ for any one of the particular forms, we have merely to make the proper changes in the signs of the radiir and s.

We shall assume, and the and the assumption is practically correct, that r and s are both greater than the thickness of the lens.

We may now proceed to consider the six cases in turn.

144. Case I. A double convex lens.

In this case s is negative, and we must therefore change the sign of s in the formulæ given above.

We see then, -t being a positive quantity, that AH is positive and A'H' negative, and that both are numerically less than the thickness of the lens. Consequently both the Principal Points are situated within the material of the lens.

Also, since the sum of the distances AH and A'H' considered numerically is found to be greater than the thickness,

it follows that the points A, A', H, H', are disposed in the order

A, H', H, A'.

The focal length of the lens is negative, therefore the focal length of the equivalent lens is negative. Consequently a double-convex lens is a convergent one; that is to say, the deviation of any ray passing through it will be in the direction of the axis.

145. Case II. A Plano-convex lens.

In this case one of the radii, r suppose, is infinite; and the others is negative. The formulæ therefore become

The Principal Points are therefore situated, one at the vertex of the curved surface, and the other in the interior of the lens.

and this is also the focal length of the equivalent lens. Hence the lens is convergent.

146. Case III. A double concave lens.

In this case is negative. Hence we see that AH is positive and less than the thickness; so that the point H lies within the substance of the lens. Similarly H' also

must lie within the lens.

Again AH+A'H' is numerically less than t, therefore the points A, A, H, H' are disposed in the order

A, H, H', A'.