See, for example, the relevant sections in E. Goursat, A Course in Mathematical Analysis (Dover, New York, 1959), Vols. I and II.

If need be, the restrictions placed on the surfaces can be modified without significantly altering the results of Sections 3 and 4. For example, the degree of q(y†, z†) can take any other value without altering the results at all. Until there is a practical requirement for such modifications, however, there is no point in being more general at this stage.

That ∇F is never zero can be seen as follows: Since ∂F/∂x is equal to RT∂G/∂x†and R is invertible, ∂F/∂x vanishes only when ∂G/∂x†does. If G is given by Eq. (2.4), the first component of ∂G/∂x† is unity, and the result follows immediately. On the other hand, if G is given by Eq. (2.3), the first component of ∂G/∂x† vanishes only if x†= P(y†, z†), and, under this condition the remaining two components are simply A2∂q/∂y†and A2∂q/∂z†. If these two components also vanish, q(y†, z†) itself must be equal to zero since, on account of homogeneity, q= ½ (y†∂q/∂y†+ z†∂q/∂z†). Any point for which x†= P(y†, z†) and q(y†, z†) = 0 cannot lie on the surface since, by inspection of Eq. (2.3), at such a point G(x†) is equal to −A2and A is necessarily nonzero.

In practice, this means that, in passing through, say, a symmetric system of spherical surfaces, a ray may intercept one of the spheres at the furthest point from the axis and even cross over onto what might be referred to as the “other face of the sphere,” yet the aberration series taken about the axis can still be convergent. In fact, the contrived system discussed in Ref. 6 (see the comment at the end of Sec. 7.1) exhibits this rather surprising property. As a related aside, I remark that a ray in such a system may, in fact, become normal to the axis at some stage (and even reverse the sense of the axial component of its direction) with the aberration series converging all the while. This also follows from the discussion in Sections 2 and 3 in which no spatial direction is given a special status.

Notice that, if (a·∇)2F′also happens to vanish at Ps, Eq. (3.7) should be modified so that the term that is cubic in δλ appears and the singularity takes on a different form (caused by grazing the surface at a point of inflection). As is made clear in Section 1, singularities of this type and countless like them are of no interest here.

If this divergence is associated with surface i, u is, in fact, parallel to αi−1(Ps).

If this divergence is associated with surface i, v is, in fact, parallel to ∇Fs.