It will be found, however, that when the lens is used to its utmost limits in respect of covering power, whatever stop is employed, the angle included is approximately a constant for any extension of camera ; if the lens is used to cover a definite size of plate at different extensions of camera, it is obvious that the angle included on that plate is variable, diminishing as we increase the magnification.

On Equality of Illumination. - Speaking of lenses generally, one form may lend itself more readily than another to transmit the full pencil of light over a greater angle, but no lens can in reality give equal illumination over the entire plate. This inequality is an inherent defect of every form of lens producing an image that is to be received upon a plane at right angles to the axis of the lens. It is most apparent when the combinations of the lens-system are separated by a considerable distance, particularly if used without a diaphragm. The construction of the Telephotographic lens necessitates a comparatively large separation of the elements composing it, even when m, or-^' is small. Under these conditions the full pencil of rays from the front combination can only emerge from the back combination (without being cut off) over a very few degrees from the axis of the lens, or the centre of the plate. The tube of the lens then very rapidly commences to cut off the full pencil, and the worst conditions for equality of illumination are brought about. A lens is readily examined to test the field over which the full pencil is received, by focusing upon a bright object, such as a candle or small gas flame, and racking the screen in a little, until the image becomes a bright disc of light. If we now move the camera so that the image gradually passes from the centre towards the edge of the screen, and observe how far the disc continues circular, we can follow how far the best conditions for equality of illumination are fulfilled ; as soon as the disc commences to become cut off, the full oblique pencil is no longer received upon the plate. If this operation is conducted, in the first place, without a diaphragm, we shall find that equality of illumination can be attained over a greater angle as the size of the diaphragm decreases, but at the expense of intensity.

Even when the best conditions for equality of illumination are maintained, still the illumination of the plate rapidly decreases with the angle of obliquity. We state the case generally here as it is essential that we should choose our positive element, so that it, at any rate, shall transmit the full pencil of rays to the negative lens at the extreme obliquity.

Presuming that a full pencil passes through a lens at any obliquity of incidence, the quantity of light passing axially through the lens, as r r, is greater than that which passes obliquely, the latter varying as the cosine of obliquity. Again the oblique pencils r r are brought to a focus at f, more distant from the aperture of the lens l than the central pencil at f, the illumination varying according to the "law of inverse squares" when the plate is at right angles to rf, as at pp. Now l f and Lf vary with the secant of obliquity, and hence the illumination at f and f (on the plane p p) will vary as cos θ x 1/sec2θ = cos3 θ.

Fig.50.

The Use And Effects Of The Diaphragm And The Impro 84

But the illumination on pp as compared with that of pp decreases again in the ratio of cosθ (or as1/secθ). Hence the final illumination varies as cos4 0, or as the fourth power of the cosine of the angle of obliquity.

The following table shows the illumination of the plate given by an ideally perfect lens as regards transmitting the full incident pencil at various obliquities.

Angle of obliquity

Quantity of light passing through at the pupil

Illumination of image

- 8.

= cos θ.

=cos4 θ.

1.000

1.000

5

.996

.985

IO

.985

.941

15

.966

.870

20

.940

.780

25

.906

•675

30

.866

.562

40

.766

•344

45

.707

.250

5o

•643

.171

Now, although this table shows that the illumination rapidly decreases for the greater angles of obliquity (one half when 6o° and only one quarter when 900 are included upon the plate), the loss of light for small obliquities is seen to be very small up to 200 (that is 10° either side of the axis). The Telephotographic lens is seldom constructed to-include a larger angle than this, so that we can readily choose a positive element which shall transmit light most favourably for equality of illumination ; but the separation of the elements, their diameters, and the position of the diaphragm must determine the limits for this condition in the compound system.

On Distortion. - If we place a diaphragm at any distance either before or behind a single positive lens, distortion of the image produced by it takes place, due to the fact that no single lens can be made free from spherical aberration, curvature of field, etc. ; a reference to.

Fig. 51 will make the matter clear : a beam of rays parallel to the axis of the lens l forms a focus at a point f in the axis ; if the parallel beam R1RR2 falls upon the lens obliquely, its focus (or more correctly its approximate focus) will be found at a point f which is not situated in the same plane as f, but if these rays are produced they will meet the focal plane of the lens through f in the points r1 rr2 If we place a small diaphragm in contact with the lens, distortion will not take place, as both r1 r2 are symmetrically situated with respect to the ray Rr. If we remove the diaphragm from the position of contact, and place it in front of the lens in the position of s1 ,the rays R1 r only can traverse the upper portion of the lens l, and are received upon the plane through f in the points r r1: the effect being to displace the ray r1 r1 towards the centre of the image plane, the points which are most displaced being those furthest away from the centre; giving rise to what is known as "barrel shaped" distortion. If, on the other hand, we place the diaphragm in a position s2 behind the lens, it will be seen that the rays r r2 pass through the lower half of the lens l and are received upon the image plane in the points rr2 The effect is to displace every point outwards from the centre, giving rise to what is known as "pincushion" distortion. These effects of distortion are best illustrated in the