To determine whether all the rays proceeding from O are refracted through O′, we investigate the triangle OPO′. We have p/p′ = sin u′/sin u. Substituting for sin u and sin u′ the values found above, we obtain p′/p = c′ sin i/c sin i′ = n′c′/nc. Also c = OC = CS + SO = −SC + SO = s − r, and similarly c′ = s′ − r. Substituting these values we obtain
| p′ | = | n′(s′ − r) | , or | n(s − r) | = | n′(s′ − r) | . |
| p | n(s − r) | p | p′ |
(6)
To obtain p and p′ we use the triangles OPC and O′PC; we have p2 = (s − r)2 + r2 + 2r(s − r) cos φ, p′2 = (s′ − r)2 + r2 + 2r(s′ − r) cos φ. Hence if s, r, n and n′ be constant, s′ must vary as φ varies. The refracted rays therefore do not reunite in a point, and the deflection is termed the spherical aberration (see [Aberration]).
Developing cos φ in powers of φ, we obtain
| p2 = (s − r)2 + r2 + 2r(s − r) { 1 − | φ2 | + | φ4 | − | φ6 | + ... }, |
| 2! | 4! | 6! |
and therefore for such values of φ for which the second and higher powers may be neglected, we have p2 = (s − r)2 + r2 + 2r(s − r), i.e. p = s, and similarly p′ = s′. Equation (6) then becomes n(s − r)/s = n′(s′ − r)/s′ or
| n′ | = | n | + | n′ − n | . |
| s′ | s | r |
(7)
This relation shows that in a very small central aperture in which the equation p = s holds, all rays proceeding from an object-point are exactly united in an image-point, and therefore the equations previously deduced are valid for this aperture. K. F. Gauss derived the equations for thin pencils in his Dioptrische Untersuchungen (1840) by very elegant methods. More recently the laws relating to systems with finite aperture have been approximately realized, as for example, in well-corrected photographic objectives.