The stop is placed, as shown in the figure, for the purpose of cutting off the outside rays of the pencils, and thereby reducing the amount of spheric,a1 aberration in each pencil. It will easily be seen that the position of the stop for any particular pencil depends upon its obliquity ; and that the greater the obliquity the nearer the stop must be placed to the lens. In practice, the stop should be placed so as best to suit the most oblique pencils ; but the inaccuracy thus introduced with respect to the less oblique pencils is hardly appre ciable.
It will be seen by the foregoing figure that the amount of light in the image depends not only on the size of the stop, but also on the obliquity of the pencil, being greatest in the case of the direct pencil, and diminishing as the obliquity increases. For this reason there is not in the common view lens absolute equality of illumination, the centre of the field having the most light and the edges the least.
From the case of the single plano-convex lens, the transition is easy to that of the same lens achromatized, and thence to the achro matic meniscus. In these two latter cases it may be considered as very approximately true that the field for distant objects is a sphere concentric with the posterior convex surface of the lens.
If the stop be placed immediately in contact with the lens the radius of the field is shortened and the centre of the posterior surface itself, and not of the sphere of which it is a portion, becomes the centre of the field. This is explained in the article on the " Optical centre ;" q. v. There would, however, be no distortion in this case, while there is considerable distortion when the stop is put at a distance from 'the lens, in order to get a flatter field. See " Distortion." The general form and principle of the common view lens having now been described, it remains to give the exact formula for its con struction.
The data are :-1st, the focal length of the lens 4 2nd, the indices of refraction for the crown and flint glasses ; 3rd, the dispersive powers of the glasses ; and 4th, the radius of the front surface. Let these quantities be expressed by the following symbols : F =focal length of compound lens.
pi=refractive index of flint glass.
/12= crown glass.
D = ratio of dispersive power of front glass to back glass.
R=radius of anterior surface of lens.
The unknown quantities are : 8=radius of inner surface of lens.
t= posterior surface of back lens.
fi= focal length of front lens.
= 97 back lens.
Then, the equations which connect these quantities are 1 1 1 - = . . (1) = D . . . (2) A /2 1 1 1 1 1 1 - (PI 1) = (P2"-- 1) ) (4) R 8 t By which four equations, the four unknown quantities may be de termined.
It is immaterial in the above formula whether the flint or crown lens be placed in front, so as to receive the incident rays. If the flint lens be placed in front, the formula gives the common view lens which has been in use for a number of years. If the crown lens be placed in front, the formula gives the lens shown by the second of the figures at page 255.
We have said that the radius of the front lens is arbitrary, but it depends upon the size of the lens and the work it is intended to do. As the lens increases in size the foefd length remaining the same, the stop is placed further in front of it, the field becomes flatter, and there is more distortion ; at the same time the front surface must become flatter. When these conditions are reversed, i. e. as the lens becomes smaller, the stop is placed nearer to it, the field becomes more concave, \ere is less distortion, and the front surface should then become more concave. Some little latitude is therefore allowable in the radius of the front surface, and it may be considered as arbitrary, within certain limits ; the mean being about 3 feet radius for a three-inch lens.
The second equation must be fulfilled in order that the lens may be achromatic. No attempt is made to cure spherical aberration except by the stop. The central pencil from Q would have a finer focus, q, if the small central portion of the lens through which it passes could be turned with its convex side to the incident rays. See what has been said before respecting the aberration of a plano-convex lens. The remark is equally applicable to the compound lens which is nearly plano-convex. The large lens of an opera glass, which is compound and convexo-plane, is placed with the convex side to the objects ; and for the same reason the front lens of the portrait com bination, which is nothing but a view lens reversed, has its convex side to the objects.