The general form of the Smith-Helmholtz equation

The Smith-Helmholtz equation μ sin θy = μ' sin θ'y' is only applicable to infinitesimal objects and images. When both field and aperture are large, in addition to the ordinary equation involving lengths measured along a ray, there are three exact equations of the type xx' a + μ'L'x'b - μLxc - μLμ'L'd = 0, bc - ad = I, all expressing the same correspondence. This correspondence is primarily geometrical rather than optical. The coefficient a in all these equations is the secondary power of the system for the ray considered. When the instrument is axially symmetrical the coefficients in two of the equations have the values b = c = I, d = 0. The third equation can be made to take the same form by moving the origins to suitable points on the axis of the instrument. The equations are considered in their application to asymmetrical instruments. In telescopic systems axial symmetry is, in general, necessary for the Smith-Helmholtz equality to be applicable. From the general equations a great variety of formulae for computing conjugate points can be obtained. Equations of the same one-one class can be found for other projective relations, but, in general, they are less useful than the corrected Smith-Helmholtz equations, because when the linear and angular variables are associated as desired the positions of the origins must be computed instead of being recognisable by inspection.