Limitations.Although this theorem, which is a part of the larger Gauss theory, is in its nature only true for minute angles of obliquity and for exceedingly narrow pencils, which never have more than a very small degree of eccentricity, yet it is of the highest importance when we proceed to ascertain that very important function of a more or less complex combination of lenses, known as the equivalent focal length.
Corrected lens system. Theorem untrue for the parts, but true for the whole.While the theorem is of little practical worth when applied to simple uncorrected lenses of substantial aperture, yet, for a combination of lenses yielding a flat and rectilinear image, it becomes absolutely true in the sum for the series, since the departures from its truth in any one lens are in that case neutralised by contrary departures from its truth in the other lenses.
Thick Lenses
Gauss and Listing.
We may now proceed to deal with the case of lenses of considerable thickness as measured along the axis. This subject was long ago worked out by Gauss (about 1838) and Listing (about 1868), and it will suffice to recapitulate here the most important results, although perhaps arriving at them by methods differing from theirs, but more convenient for our purpose. Let Figs. 9a, b, c, d, e, f, and g represent various forms of lenses, of central thicknesses A1..A2, and radius c1..r1 for first spherical surface, and c2..r2 for second surface. It is obvious that if any two radii c1..r1 and c2..r2 are drawn parallel to one another and joined by the straight line r1..r2, then the latter will cut the axis at the point C, so that we have two similar triangles c1Cr1 and c2Cr2, and two similar mixtilinear triangles CAlrl and CA2r2, and the distance C..Al:C..A2::c1..r1:c2..r2, and moreover the straight line r1..r2 cuts the first surface or its tangent at r2, at exactly the same angle as it cuts the second surface or its tangent at r1. If, therefore, r1..r2 represents a ray of light, it will obviously, if refracted out of the surface at r1 be deviated from the direction r2..r1 by exactly the same angle as it would be deviated from the direction r1..r2 if refracted outwards at the point r2, only the deviation will be in opposite directions. Hence the ray after refraction at r1 will pursue a course r1..t1, and after refraction at r2 will pursue a course r2..t2, and these refracted rays are parallel to one another. If, then, r1..t1 and r2..t2 are produced backwards (if necessary) to cut the axis at two points p1 and p2, we then get again two similar mixtilinear triangles Principal points or nodal points. r1A1p1 and r2A2p2, and again have A1..p1:A2..p2::c1..r1:c2..r2. These two points p1 and p2 are the two principal points of the lens or nodal points (sometimes
