Bauhaus-Universität Weimar

Helmholtz's treatise on physiological optics. Volume 1. Edited by James P. C. Southall. Translated from the 3rd German edition
Helmholtz, Hermann von
55, 56.] 
§9. Laws of Optical Imagery 
Generalization of the preceding results. Now let us inquire how the 
focal points, principal points and nodal points, as above defined, are 
applicable in the case of a single spherical refracting surface. 
The focal points are the points where rays intersect in one medium 
that proceed parallel to the axis in the other medium. The positions 
of the focal points can be found from equations (3a) and (3b), because 
the magnitudes denoted by F>, F>> and G>, G>> are the distances of the 
focal points from the vertex and centre of the refracting surface, 
The focal planes are the transversal planes perpendicular to the axis 
at the focal points. Since the conjugate point to either focal point is 
infinitely far away, the same statement must be true with respect to 
any point in one of the focal planes, provided it is not too far from the 
axis. In other words, rays which meet in a point in either focal plane 
will be parallel to each other in the other medium. 
The characteristic property of the principal points is that object and 
image lying in the principal planes, which are perpendicular to the axis 
at these points, have precisely the same dimensions in all respects. 
For these planes, therefore, ß> = ß>>. According to equations (6b), this 
means that here f> =f>> =0, which conditions must both be satisfied, 
as shown by equations (3d). Consequently, the two principal points 
of a spherical refracting surface coincide with each other at the vertex, 
so that object and image are here coincident. 
The peculiar property of the nodal points is that a ray which is 
directed towards the first nodal point before refraction will proceed 
through the second nodal point after refraction in the same direction as 
before. The two nodal points of a spherical refracting surface also 
coincide, both being at the centre of the surface; because any ray 
whose path in the first medium is directed towards the centre of cur¬ 
vature will proceed into the second medium not only in the same 
direction but along the same straight line. 
The constructions given above which were based on the properties 
of these special points and planes are, therefore, applicable also to the 
case of a single re¬ 
fracting surface. 
The process is P 
simpler here be¬ 
cause every point 
in the first prin¬ 
cipal plane is its own image, and also because a ray directed to 
the first nodal point proceeds straight on along the same line. 
The two equations (3c), similar in form, enable us to determine 
the abscissa of the axial image-point, with reference, however, to two


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