The shortest way to obtain the result is that of writing [Equation] with tanα = *u*_{1}, tanβ = *u*_{2}.

The actual slope of the line *P*′_{1}*P*′_{2} is [Equation] The last term, however, only introduces a second-order correction in the small term δυ_{2}.

The condition is necessary, but not sufficient: the knowledge of the abscissa alone, in fact, does not give the position of the line, but only the direction.

For instance,E. P. Adams, "Smithsonian Mathematical Formulae" (Washington, 1947), p. 30–33.

In the meridian case, we spoke indifferently of "equinormal" or "equitangent" pairs. In the present case, two concurrent pairs of rays are equinormal but not equitangent, as the tangents can lie in different planes.

It must be remembered that the mere condition of coplanarity of the two rays imposes that (*p*_{2} - *p*_{1})/(*u*_{2} - *u*_{1}) = (*q*_{2} - *q*_{1}) / (υ_{2}- υ_{1}).

The convention ∂*u*/∂β=*u* implies that *all* the partial derivatives of *u* with respect to β be *u*; so that the finite variation Δ*u* corresponding to a finite Δξ is given by MacLaurin's expansion as Δ*u*=*u*{Δβ+(Δβ)^{2}/2! + (Δβ)^{3}/3!+ … = *u* {*e*^{Δβ}-1}, as should be.

Reference is made to Fig. 12 for symbols of points: in the present case, a figure cannot be drawn, as the points are complex.

This means, physically, that the absolute angle between the entrance ray and the normal differs from its meridian projection by a second-order angle.

The reader will remark that the formula differs from formula (20) established in Sec. 13. This is due to the fact that the entrance ray of Fig. 17 is directed to the left while that of Fig. 7 goes from left to right. The lines of the two rays being the same, the *normal* of Fig. 7 is the tangent of 17, and vice versa.

In other words, the line is the locus of the points (*u*,υ) that map physical lines passing through *x*_{0}, *y*_{0}.

The formula is valid for the mirror by making ν_{1} = - ν_{2}.

A shorter way of approach is indicated in Sec. 29.

This is a general property of the present step by step procedure.

It is found that *d*_{3}ξ = aλ + *d*_{2}ξ, and *d*_{2}ξ = Bλ + *d*_{2}ξ, where a and B are rational expressions in terms of the constants *u*_{10}, *u*_{20}, etc. ...

The *minus* sign is taken to conform with Fig. 10 where the displaced point *B*_{2} is assumed to be *on the right* of *B*_{0} when facing the first surface.

The "first differentials" are zero as the displaced ray meets the first surface in the *design point*.

One surface was obliged, so that a single design condition could be imposed: in the actual case, Abbe's condition was accepted.